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this paper is the second in a series on cobordism ( = concordance ) groups of a natural class of high - dimensional links . chapter @xmath5 of the first work  @xcite discusses background to the problem at greater length but we summarize here some of the key ideas . a knot is an embedding of spheres or topological manifolds according to taste , with the understanding that @xmath6 is permitted exotic structures if one selects the smooth category . ] @xmath7 . the following are generalizations :    * a @xmath2-component _ link _ is an embedding of @xmath2 disjoint spheres @xmath8 * a _ boundary link _ is a link whose components bound disjoint @xmath9-manifolds . the union of these @xmath9-manifolds is called a _ seifert surface_. * an _ @xmath10-link _ is a pair @xmath11 where @xmath12 is a link and @xmath13 is a homomorphism from the fundamental group @xmath14 of the link complement @xmath15 onto the free group @xmath10 on @xmath2 ( distinguished ) generators such that some meridian of the @xmath16th link component is sent to the @xmath16th generator . not every link is a boundary link ; a link @xmath12 can be refined to an @xmath10-link @xmath11 if and only if @xmath12 is a boundary link . let us call a homomorphism @xmath17 permissible if it sends some meridian of the @xmath16th link component to the @xmath16th generator . there may be many permissible homomorphisms for a given boundary link but if @xmath13 and @xmath18 are permissible then @xmath19 where @xmath20 is some `` generator conjugating '' automorphism of @xmath10 ( cappell and shaneson  @xcite , ko  ( * ? ? ? * p660 - 663 ) ) . homomorphisms @xmath21 correspond to homotopy classes of maps from the link complement @xmath22 to a wedge of @xmath2 circles and the permissible homotopy classes correspond , by the pontrjagin - thom construction , to cobordism classes of seifert surfaces ( rel @xmath12 ) . every knot is a ( @xmath5-component ) boundary link and admits precisely one permissible homomorphism , namely the abelianization @xmath23 among the three generalizations above it is the theory of @xmath10-links which seems to bear the closest resemblance to knot theory . although one does not hope for a complete classification of knots or @xmath10-links in higher dimensions much is known about their classification up to the equivalence relation known as cobordism ( or concordance ) . two links @xmath24 and @xmath25 are called _ cobordant _ if there is an embedding @xmath26 \subset s^{n+2}\times[0,1]\ ] ] which joins @xmath27 to @xmath28 . one requires is also required to meet @xmath29 and @xmath30 transversely . ] that @xmath31 for @xmath32 and @xmath33 but no such requirement is made when @xmath34 . boundary links are said to be _ boundary cobordant _ if there is a cobordism @xmath35 whose components bound disjoint @xmath36-manifolds in @xmath37 $ ] . two @xmath10-links @xmath38 and @xmath39 are called cobordant if there is a pair @xmath40 \backslash li ) \to f_\mu)\ ] ] such that the restrictions of @xmath41 to @xmath42 and @xmath43 coincide with @xmath44 and @xmath45 ( up to inner automorphism ) . the cobordism classes of knots form an abelian group @xmath46 under ( ambient ) connected sum but this operation does not extend to links in any obvious way . if one attempts to add links @xmath24 and @xmath25 there are many inequivalent choices of connecting arc from the @xmath16th component of @xmath24 to the @xmath16th component of @xmath25 . however when @xmath47 connected sum @xmath48+[l_2,\theta_2]$ ] of _ cobordism classes _ of @xmath10-links is well - defined ; one can remove the ambiguity in the choice of paths by assuming , perhaps after some surgery , that @xmath49 and @xmath50 are isomorphisms . the set @xmath51 of cobordism classes of @xmath10-links is therefore an abelian group . when @xmath52 is even , @xmath51 is in fact the trivial group  @xcite ; we sketch a proof in  ( * ? ? ? * ch14.1 ) . on the other hand j.levine obtained a complete system of invariants for odd - dimensional knot cobordism groups @xmath53 for @xmath3  @xcite and showed that each is isomorphic to a countable direct sum @xmath54 for all @xmath3 and all @xmath55 . both  ( [ iso_class_of_knot_cobordism_gp ] ) and  ( [ iso_class_of_f_link_cobordism_gp ] ) were deduced from a purely algebraic reformulation of @xmath10-link cobordism associated to seifert surfaces : it was proved by levine  @xcite in the knot theory case @xmath56 and by ko  @xcite and mio  @xcite independently in the general case that @xmath0 is isomorphic to the `` witt group of seifert forms '' . in the notation of the present paper , which we explain more carefully in sections  [ section : define_hermitian_forms_and_witt_gp ] ,  [ section : seifert_modules ] and  [ section : cobordism_invariants_seifert ] , @xmath57 the symbol @xmath58 denotes was denoted @xmath59 in  @xcite . ] a category of `` seifert modules '' designed to contain the homology modules of seifert surfaces among the objects ( see notation  [ sei_notation ] ) . in the case @xmath56 an object in @xmath58 is a finitely generated free @xmath60-module @xmath61 together with an endomorphism @xmath62 which carries information about how a seifert surface is embedded . if @xmath63 then the definition of seifert module also includes a direct sum decomposition @xmath64 which reflects the connected components of a seifert surface . the intersection form in a seifert surface is an isomorphism @xmath65 in @xmath58 which satisfies @xmath66 . such @xmath67-hermitian forms are the generators of the witt group @xmath68 . the relations say that certain `` metabolic forms '' are identified with zero ; see definitions  [ define_metabolic ] and  [ define_witt_group ] below . although seifert surface methods are convenient in explicit computations , it is preferable to define @xmath10-link invariants without making a choice of seifert surface . in the present paper we focus instead on the covering space @xmath69 of a link complement determined by the homomorphism @xmath70 . this approach sits more naturally in homology surgery theory and is more amenable to generalization from boundary links to arbitrary links or other manifold embeddings . we take as starting point the identification @xmath71 where @xmath72 is a category designed to contain homology modules of the cover @xmath73 ( see definition  [ define_f_link_module ] and notation  [ flk_notation ] ) . the objects in @xmath72 are certain modules over the group ring @xmath74 $ ] of the free group ; they are called @xmath10-link modules in the present paper although they are more commonly known as link modules . the @xmath10-equivariant poincar duality in @xmath75 leads to a @xmath76-hermitian form @xmath77 in the category @xmath72 . this is the blanchfield - duval form of the title , originally introduced by blanchfield  @xcite in the knot theory case @xmath56 . the identity  ( [ identify_f_link_cobordism_with_w(flink ) ] ) was proved by kearton for @xmath56  @xcite and by duval  @xcite for @xmath55 . cappell and shaneson earlier identified the cobordism group @xmath51 with a @xmath78-group , an obstruction group in their homology surgery theory  @xcite . the identification of this @xmath78-group with the witt group @xmath79 was due to pardon  @xcite , ranicki  @xcite and smith  @xcite for @xmath56 and to duval  @xcite for @xmath55 . more general results of vogel  @xcite on homology surgery and universal localization are stated elsewhere in this volume  @xcite . an outline of their application to @xmath51 is given in  ( * ? ? ? * ch1,4.4,5.3 ) . universal localization plays two roles in this paper . firstly the `` augmentation localization '' of the group ring @xmath74 $ ] of the free group appears in the definition of the blanchfield - duval form , our main object of study . secondly , we prove that the category @xmath72 of @xmath10-link modules is ( equivalent to ) a universal localization of the category @xmath58 of seifert modules . our first aim is to use  ( [ identify_f_link_cobordism_with_w(flink ) ] ) to distinguish the elements of  @xmath0 . we define complete invariants ( and secondary invariants if certain primary invariants vanish ) by analyzing the witt groups @xmath80 . we proceed in three steps , explained in more detail in section  [ section : blanchfield_form_invariants ] , which run parallel to steps 2 , 3 and 4 in chapter 2 of  @xcite :    1 . obtain a direct sum decomposition of @xmath80 by `` devissage '' . one must prove that @xmath81 is an abelian category in which each module has a finite composition series . 2 .   use hermitian morita equivalence to show that each summand of the group @xmath80 is isomorphic to some group @xmath82 where @xmath83 is a division ring of finite dimension over @xmath84 . 3 .   recall from the literature invariants of each @xmath82 .    in the knot theory case @xmath56 there is one summand of @xmath80 for each maximal ideal @xmath85 $ ] which is invariant under the involution @xmath86 . the generator @xmath87 is often called an alexander polynomial . the division ring @xmath83 coincides with the quotient field @xmath88/(p)$ ] and @xmath82 is the witt group of hermitian forms over @xmath83 ( compare milnor  @xcite ) .    the following theorem and corollary are restated and proved in section  [ section : blanchfield_form_invariants ] ; see theorem  [ bd_form_invariants_for_w_suffice : precise ] and corollary  [ bd_form_invariants_for_flinks_suffice : precise ] . [ bd_form_invariants_for_w_suffice : vague ] the invariants ( and secondary invariants ) defined in section  [ section : blanchfield_form_invariants ] are sufficient to distinguish the elements of the witt groups @xmath89 of blanchfield - duval forms with coefficients in @xmath84 . [ bd_form_invariants_for_flinks_suffice : vague ] let @xmath90 and suppose @xmath91 and @xmath92 are the blanchfield - duval forms for the @xmath1-dimensional @xmath10-links @xmath38 and @xmath39 respectively . these two @xmath10-links are cobordant if and only if all the invariants ( and possible secondary invariants ) of @xmath93 \in w^{(-1)^{q+1}}(\bl(\q))\ ] ] defined in section  [ section : blanchfield_form_invariants ] are trivial . corollary  [ bd_form_invariants_for_flinks_suffice : vague ] follows from  ( [ identify_f_link_cobordism_with_w(flink ) ] ) and the fact that the canonical map @xmath94 is an injection , which we deduce from theorem  [ main_covering_construction_theorem ] at the end of section  [ section : overview_of_bd_form_invariants ] . corollary  [ bd_form_invariants_for_flinks_suffice : vague ] is also a consequence of theorem  [ theorem_relating_invariants : vague ] and theorem  b of  @xcite . our second aim is to understand the algebraic relationship between the seifert forms and the blanchfield - duval form of an @xmath10-link and prove that the cobordism invariants defined in  @xcite using seifert forms are equivalent to those defined in section  [ section : blanchfield_form_invariants ] via the blanchfield - duval form . example  [ worked_example_seifert ] gives a sample calculation of the seifert form invariants in  @xcite . in the knot theory case @xmath56 , the relationship between seifert and blanchfield forms has been investigated extensively by kearton  @xcite , levine  @xcite , farber  @xcite and ranicki  ( ( * ? ? ? * ch32),@xcite ) . for @xmath95 k.h.ko  @xcite used geometric arguments to obtain a formula for cappell and shaneson s homology surgery obstruction in terms of the seifert form . a formula for the blanchfield - duval form in terms of the seifert form , again based on geometric arguments , can also be found in cochran and orr  ( * ? ? ? * thm4.2 ) in the slightly more general context of `` homology boundary links '' . m.farber related seifert and blanchfield - duval forms of @xmath10-links in a purely algebraic way  @xcite . although the present paper is logically independent of his work , we take up a number of his ideas in sections  [ section : seifert ] and  [ section : covering_construction ] , providing a systematic treatment in the language of hermitian categories . whereas farber takes coefficients in a field or in @xmath60 , in these sections we allow the coefficients to lie in an arbitrary associative ring @xmath96 . the first step is to show that an @xmath10-link module admits a canonical seifert module structure ( cf  ( * ? ? ? * p193 ) ) . an @xmath10-link module @xmath97 is not in general finitely generated ( or projective ) as an @xmath96-module so we introduce a larger category @xmath98 which contains @xmath99 as a full subcategory ( see notation  [ sei_notation ] ) . we obtain a `` forgetful '' functor @xmath100 for example , in the case @xmath56 of knot theory , an object in @xmath101 is a module @xmath102 over the ring @xmath103 $ ] of laurent polynomials with a presentation @xmath104)^m\xrightarrow{\sigma } ( a[z , z^{-1}])^m \to m\to 0\ ] ] such that @xmath105 is an isomorphism . the seifert module @xmath106 is the @xmath96-module @xmath102 together with the endomorphism @xmath107 . if @xmath108 is a field , farber defined , for each @xmath109 , the `` minimal lattice ''  ( * ? ? ? * p194 - 199 ) of @xmath102 , a seifert submodule of @xmath106 which is of finite @xmath110-dimension . we prefer to work directly with @xmath106 which is defined regardless of the coefficients and avoids technicalities of farber s definition . his minimal lattice becomes isomorphic to @xmath106 after one performs a universal localization of categories which we describe a few paragraphs below .    given a seifert surface for an @xmath10-link one can construct the free cover by cutting the link complement along the seifert surface and gluing together infinitely many copies of the resulting manifold in the pattern of the cayley graph of @xmath10 . figure  1 illustrates the geometric construction in the case of a @xmath111-component link .    the algebraic analogue of this geometric construction is a functor @xmath112 from seifert modules to @xmath10-link modules ( see definition  [ define_covering_construction ] ) . since @xmath113 takes values in the larger category @xmath98 we expand the domain of @xmath4 to @xmath98 , by necessity replacing @xmath101 by a larger category @xmath114 . this process of enlargement stops here for there are functors @xmath115    we show in section  [ section : adjunction ] that @xmath4 is left adjoint to @xmath113 . roughly speaking , this means that @xmath116 is the `` free '' @xmath10-link module generated by the seifert module @xmath61 ( with respect to the functor @xmath113 ) . in other words , @xmath4 is universal ( up to equivalence ) among functors from seifert modules to @xmath10-link modules . returning our attention to the subcategories @xmath99 and @xmath101 whose definitions involve a `` finitely generated  projective '' condition we show that @xmath4 is compatible with the notions of duality in @xmath99 and @xmath101 , extending @xmath4 to a `` duality - preserving functor '' between `` hermitian categories '' @xmath117 ( see definitions  [ define_hermitian_category ] and  [ define_dpres_funct ] and proposition  [ extend_b_to_duality_pres_functor ] ) . the following theorem concerns the induced homomorphism of witt groups : @xmath118 recall that by wedderburn s theorem , a ring @xmath96 is semi - simple and artinian if and only if it is a product of matrix rings over division rings . = 2 @xmath119    [ main_covering_construction_theorem ] if @xmath96 is a semi - simple artinian ring then ( [ witt_map_induced_by_b ] ) is an isomorphism . the map ( [ witt_map_induced_by_b ] ) will be considered for more general rings @xmath96 in subsequent work ( joint with a.ranicki ) . it follows from the isomorphisms  ( [ identify_f_link_cobordism_with_w(seifert ) ] ) and  ( [ identify_f_link_cobordism_with_w(flink ) ] ) above that ( [ witt_map_induced_by_b ] ) is an isomorphism when @xmath120 . [ theorem_relating_invariants : vague ] the duality - preserving functor @xmath121 identifies the seifert form invariants of  @xcite with the blanchfield - duval form invariants of section  [ section : blanchfield_form_invariants ] . theorem  [ theorem_relating_invariants : precise ] below is a more precise statement of theorem  [ theorem_relating_invariants : vague ] . the invariants of  @xcite are outlined in section  [ section : cobordism_invariants_seifert ] . theorem  [ main_covering_construction_theorem ] is proved in two stages . the first stage is to establish that , for any ring @xmath96 , there is an equivalence between @xmath121 and a certain universal localization of hermitian categories . taking up farber s terminology we call a seifert module @xmath122 primitive if @xmath123 . we denote by @xmath124 the category of primitive modules . one may write @xmath125 the category quotient @xmath126 is universal among functors which make invertible morphisms whose kernel and cokernel are primitive . in particular , primitive modules in @xmath98 are made isomorphic to @xmath127 in @xmath128 . since @xmath129 is left adjoint to @xmath113 it follows that @xmath4 exhibits the same universal property as @xmath130 although only `` up to natural isomorphism '' ( proposition  [ b_is_universal_localization_up_to_equivalence ] ) . we conclude that @xmath4 is equivalent to @xmath130 and , with a little extra work , establish that @xmath131 is equivalent to a universal localization of hermitian categories ( theorem  [ bbar_is_equiv_on_fgproj_cats ] and proposition  [ bbar_is_eq_of_hermitian_cats ] ) . in the knot theory case @xmath56 the category @xmath98 coincides with the category of ( left ) modules over the polynomial ring @xmath132 $ ] in a central indeterminate @xmath133 . setting @xmath134 , the functor ( [ introduce_universal_cat_loc ] ) is the central localization @xmath135\otimes_{a[s]}\functor$ ] from the category of @xmath132$]-modules to the category of @xmath135$]-modules ; see farber  ( * ? ? ? * thm2.6 ) and ranicki  @xcite . pere ara recently gave an independent proof  ( * ? ? ? * thm 6.2 ) using farber s minimal lattice that if @xmath108 is a field then @xmath136 is equivalent to a localization of @xmath137 by a category of primitive modules ( for all @xmath95 ) on a set @xmath138 takes the place of the group ring @xmath139 $ ] in the present paper ; the category denoted @xmath140 in  @xcite plays the role of @xmath136 . consequently , there is only one kind of `` trivially primitive '' module ( denoted @xmath141 in  @xcite ) as compared with the two kinds in  @xcite and section  [ section : structure_of_primitives ] below . ara also related the modules in @xmath140 to modules over the leavitt algebra @xmath12 . by definition , @xmath12 is the universal localization of @xmath142 which makes invertible the map @xmath143 ( compare the sato condition , lemma  [ sato_condition ] below ) . the category @xmath140 turns out to be equivalent to the category of finitely presented @xmath12-modules of finite length . ] . + the second stage in the proof of theorem  [ main_covering_construction_theorem ] involves the analysis of primitive modules . a seifert module @xmath61 is called `` trivially primitive '' if the endomorphism with which it is endowed is either zero or the identity . in proposition  [ identify_prim_infty ] we show that every primitive module @xmath144 is composed of ( possibly infinitely many ) trivially primitive modules . restricting attention to @xmath99 we show that if @xmath96 is semi - simple and artinian then every primitive module is composed of finitely many trivially primitive modules in @xmath99 ( cf farber  ( * ? ? ? * ,  7.10 ) ) . the proof of theorem  [ main_covering_construction_theorem ] is completed in section  [ section : pf_of_witt_isomorphism_thm ] by establishing that the witt group of the subcategory @xmath145 of primitive modules is trivial . see proposition  [ witt_isomorphism_for_hermitian_quotient ] and part 2 .  of lemma  [ witt_group_of_primitives_is_zero ] . the definitions of blanchfield - duval form invariants in section  [ section : blanchfield_form_invariants ] parallel the author s seifert form invariants in  ( * ? ? ? the three steps outlined above to analyze @xmath89 were applied to @xmath146 in  @xcite . theorem  [ theorem_relating_invariants : vague ] is proved in section  [ section : pf_of_eq_of_invariants ] by checking that the duality - preserving functor @xmath121 respects each of these three steps . let us summarize the contents of this paper . section  [ section : blanchfield_form ] discusses @xmath10-link modules over an arbitrary ring @xmath96 and uses universal localization ( cf vogel  @xcite and duval  @xcite ) to describe hermitian structure in the category @xmath101 . we define the witt groups @xmath147 of blanchfield - duval forms . in section  [ section : blanchfield_form_invariants ] we set @xmath148 and define invariants of @xmath149 , obtaining intrinsic cobordism invariants for @xmath10-links . we discuss each of the steps 1 - 3 . listed above , reformulating and proving theorem  [ bd_form_invariants_for_w_suffice : vague ] and corollary  [ bd_form_invariants_for_flinks_suffice : vague ] .    in section  [ section : seifert ] we define seifert modules and seifert forms with coefficients in an arbitrary ring @xmath96 . we treat a worked example of the invariants defined in  @xcite and we define a forgetful functor @xmath150 . section  [ section : covering_construction ] begins to study the functor @xmath151 from seifert modules to @xmath10-link modules . we prove that @xmath4 is left adjoint to @xmath152 and show that @xmath4 factors through a category equivalence @xmath153 . we describe the structure of the primitive modules  those which are sent to zero by @xmath4  and outline a construction of the localization @xmath154 . we construct a duality - preserving functor @xmath131 which is natural in @xmath96 and factors through an equivalence @xmath155 of hermitian categories . if @xmath96 is a semi - simple artinian ring we give a simplified description of the primitive modules and the universal localization of hermitian categories . section  [ section : proof_that_b_induces_witt_isomorphism ] contains a proof of theorem  [ main_covering_construction_theorem ] and a reformulation and proof of theorem  [ theorem_relating_invariants : vague ] . [ [ acknowledgments ] ] acknowledgments : + + + + + + + + + + + + + + + +    the invariants in section  [ section : blanchfield_form_invariants ] came into being during my thesis work at the university of edinburgh under the guidance of andrew ranicki . i am also indebted to andrew for several more recent conversations and e - mails and for encouragement to complete this paper . i am grateful to john baez and james dolan for helpful discussions in category theory , and to pere ara and to the referee for their comments and corrections . i thank the london mathematical society for financial support to attend the icms workshop on `` noncommutative localization in algebra and topology '' in april 2002 and the edinburgh mathematical society who financed my visit to edinburgh in august 2002 . let @xmath96 be an associative ring with @xmath5 . modules will be left modules except where otherwise stated . let @xmath156 $ ] denote the group ring of the free group @xmath10 ; an element of @xmath156 $ ] is a formal sum of elements of @xmath10 , with coefficients in @xmath96 . note that elements of the group @xmath10 commute with elements of @xmath96 and @xmath156\cong a\otimes_\z \z[f_\mu]$ ] . the symbols @xmath157 and @xmath158 will be used for three slightly distinct purposes but the meaning will be clear from the context . firstly , @xmath158 denotes the inclusion of @xmath96 in @xmath156 $ ] . secondly @xmath158 denotes the functor @xmath159\otimes_a v$ ] from the category of @xmath96-modules to the category of @xmath156$]-modules . for brevity we write @xmath160 $ ] in place of @xmath156\otimes_a v$ ] . thirdly , we use @xmath158 to denote the inclusion of a module @xmath61 in @xmath160 $ ] given by @xmath161\otimes_a v = v[f_\mu].\ ] ]    in the opposite direction @xmath162\to a$ ] denotes the ring morphism which sends every element of @xmath10 to @xmath163 and is the identity on @xmath96 . we also write @xmath157 for the functor @xmath164}\functor$ ] from the category of @xmath156$]-modules to the category of @xmath96-modules . thirdly , @xmath165\to v$ ] denotes the morphism @xmath166=a[f_\mu]\otimes_a v \xrightarrow{\epsilon\otimes1 } a\otimes_a v \cong v.\ ] ] note that the composite @xmath167 of ring morphisms is the identity @xmath168 and the composite @xmath167 of module morphisms is the identity on @xmath61 . the composite @xmath167 of functors is naturally isomorphic to the identity functor on @xmath96-modules and we sometimes suppress the natural isomorphism identifying @xmath164}(a[f_\mu]\otimes_a v)$ ] with @xmath61 . [ define_f_link_module ] an _ @xmath10-link module _ is an @xmath156$]-module @xmath102 which lies in an exact sequence : @xmath169\xrightarrow{\sigma } v[f_\mu]\to m\to 0\ ] ] such that @xmath61 is an @xmath96-module and @xmath170 is an isomorphism . as we remarked in the introduction , the examples of @xmath10-link modules in the literature are more often called `` link modules '' . note that if @xmath61 is a finitely generated @xmath96-module then the @xmath10-link module @xmath102 is finitely generated as an @xmath156$]-module but usually not as an @xmath96-module ( see lemma  [ sato_condition ] below ) .    it will be helpful to make the following observation about the definition of @xmath10-link modules : the condition that @xmath171 is an isomorphism implies that @xmath172 is an injection ( see lemma  [ sigma_is_injective ] below ) . [ flk_notation ] let @xmath114 denote the category of @xmath10-link modules and @xmath156$]-module homomorphisms . thus @xmath114 is a full subcategory of the category of @xmath156$]-modules . let @xmath173 denote the category of modules with a presentation  ( [ presentation_of_f - link_module ] ) such that @xmath61 is a finitely generated projective @xmath96-module and @xmath171 is an isomorphism . the morphisms in @xmath101 are , as usual , the @xmath156$]-module morphisms so @xmath101 is a full subcategory of @xmath114 . we show in lemma  [ bl(functor)_is_functor ] below that @xmath174 and @xmath175 are functorial in  @xmath96 . the following lemma gives an alternative characterization of @xmath10-link modules . let @xmath176 denote generators for @xmath10 . ( sato  @xcite)[sato_condition ] suppose @xmath102 is an @xmath156$]-module which has a presentation @xmath177\xrightarrow{\sigma } v'[f_\mu ] \to m\to0.\ ] ] for some @xmath96-modules @xmath61 and @xmath178 . the augmentation @xmath179 is an isomorphism if and only if the @xmath96-module homomorphism @xmath180 is an isomorphism . there is an exact sequence @xmath181)^{\oplus\mu}\xrightarrow{\gamma } a[f_\mu ] \xrightarrow{\epsilon } a\to0\ ] ] where @xmath182 for all @xmath183 $ ] . now ( [ tree_presentation ] ) is split ( by @xmath158 ) when regarded as a sequence of right @xmath96-modules so the functors @xmath184 and @xmath185 lead to a commutative diagram @xmath186         & 0       \ar[d ] & 0\ar[d ] & \\ 0\ar[r ] & ( v[f_\mu])^{\oplus\mu }    \ar[r]^{\sigma^{\oplus\mu}}\ar[d]_{1\otimes\gamma } &    ( { v'}[f_\mu])^{\oplus\mu }    \ar[r]\ar[d]_{1\otimes\gamma } & m^{\oplus\mu}\ar[d]^\gamma \ar[r ] & 0 \\   0\ar[r ] & v[f_\mu ] \ar[r]^\sigma\ar[d]_\epsilon & v'[f_\mu ] \ar[r ] \ar[d]_\epsilon & m\ar[d]\ar[r ] & 0 \\ 0\ar[r ] & v \ar[r]^{\epsilon(\sigma)}\ar[d ] & v ' \ar[r]\ar[d ] & 0 \\   & 0 & 0 & } \ ] ] in which the first two rows and the first two columns are exact . a standard diagram chase ( e.g.  ( * ? ? ? * p49 ) ) shows that the third row is exact if and only if the third column is exact and the lemma follows . we discuss next completions of @xmath156$]-modules , which we shall need both to refine definition  [ define_f_link_module ] ( in lemma  [ sigma_is_injective ] ) and later to study the universal augmentation localization of @xmath156 $ ] ( see lemma  [ i_sigma_injective_for_af ] ) . let @xmath187\to a)$ ] . if @xmath188 is an @xmath156$]-module one defines @xmath189 an @xmath156$]-module morphism @xmath190 maps @xmath191 to @xmath192 for each @xmath52 and therefore induces a homomorphism @xmath193 . the natural isomorphisms @xmath194/i^n)\otimes_{a[f_\mu ] } n \to n/(i^nn)$ ] induce a map @xmath195}\otimes n   \to \widehat{n}$ ] but the latter is not in general an isomorphism .    in the examples with which we are most concerned , @xmath196=a[f_\mu]\otimes_a v$ ] for some @xmath96-module @xmath61 . one can describe @xmath197}$ ] as a module of power series as follows . let @xmath138 and let @xmath198 denote the free ring generated by @xmath22 ( in other words the ring of `` polynomials '' in non - commuting indeterminates @xmath199 ) . let @xmath200 so that the elements of @xmath96 are formal sums of words in the alphabet @xmath22 with coefficients in @xmath96 . let @xmath201 be the @xmath202-module @xmath203 and denote by @xmath204 the submodule whose elements are formal sums of words of length at least @xmath52 with coefficients in @xmath61 . we may now define the @xmath22-adic completion @xmath205 in which an element is a formal power series in non - commuting indeterminates @xmath199 with coefficients in @xmath61 .    [ identify_free_completions ] there is a natural isomorphism @xmath206}$ ] . the required isomorphism is induced by the isomorphisms @xmath207}{i^nv[f_\mu ] } & \cong \frac{v\langle x\rangle}{x^nv\langle x\rangle } \\ z_i & \mapsto 1+x_i \\ z_i^{-1 } & \mapsto 1-x_i + x_i^2 \cdots + ( -1)^{n-1}x_i^{n-1}\qedhere\end{aligned}\ ] ]    [ completion_is_injection ] if @xmath61 is an @xmath96-module , the canonical map @xmath160\to \widehat{v[f_\mu]}$ ] is an injection . the argument of fox  ( * ? ? ? * corollary 4.4 ) implies that @xmath208}=0 $ ] and the lemma follows . [ one_plus_tau_invertible ] if @xmath61 is an @xmath96-module and @xmath209\to v[f_\mu]$ ] is an @xmath156$]-module morphism such that @xmath210 then @xmath211)\subset iv[f_\mu]$ ] and the map @xmath212}\to \widehat{v[f_\mu]}$ ] is invertible . the commutative diagram @xmath213 \ar[d]_{\epsilon } \ar[r]^{\tau } & v[f_\mu ] \ar[d]^{\epsilon } \\ v \ar[r]_{\epsilon(\tau ) } & v } \ ] ] implies that if @xmath214 then @xmath215)\subseteq \operatorname{ker}(\epsilon : v[f_\mu]\to v ) = \operatorname{ker}(\epsilon\otimes 1:a[f_\mu]\otimes_a v \to a\otimes_a v)\ ] ] since the surjection @xmath162\to a$ ] is split by @xmath158 we obtain @xmath215)\subseteq i\otimes_a v = i(a[f_\mu]\otimes_a v)=iv[f_\mu].\ ] ] by induction , @xmath216)\subset i^nv[f_\mu]$ ] for all @xmath52 so @xmath212}\to \widehat{v[f_\mu]}$ ] has inverse @xmath217    [ sigma_is_injective ] if @xmath61 and @xmath178 are @xmath96-modules and @xmath218\to v'[f_\mu]$ ] is an @xmath156$]-module homomorphism such that @xmath179 is an isomorphism then @xmath172 is an injection and the induced map @xmath219}\to\widehat{v'[f_\mu]}$ ] is an isomorphism . let @xmath220\to v[f_\mu]$ ] . now @xmath221}$ ] so we may write @xmath222 where @xmath214 . now @xmath223 } \to \widehat{v[f_\mu]}$ ] is an isomorphism by lemma  [ one_plus_tau_invertible ] so @xmath219}\to\widehat{v'[f_\mu]}$ ] is an isomorphism . the commutative diagram @xmath224 \ar@{>->}[d ] \ar[r]^{\sigma ' } & v[f_\mu]\ar@{>->}[d ] \\ { \widehat{v[f_\mu ] } } \ar[r]_{\sigma ' } & { \widehat{v[f_\mu]}}}\ ] ] implies that @xmath225\to v[f_\mu]$ ] and @xmath218\to v'[f_\mu]$ ] are injections .    [ bl(functor)_is_functor ] a homomorphism @xmath226 of rings induces functors @xmath227\otimes_{a[f_\mu]}\functor   & : \bl_\infty(a)\to\bl_\infty(a ' ) \quad \mbox{and}\\ a'[f_\mu]\otimes_{a[f_\mu]}\functor & : \bl(a)\to\bl(a')\end{aligned}\ ] ]    if @xmath228 then @xmath102 has a presentation  ( [ presentation_of_f - link_module ] ) such that @xmath171 is invertible . applying @xmath229\otimes_{a[f_\mu]}\functor$ ] one obtains an exact sequence @xmath230\otimes_{a[f_\mu]}v[f_\mu]\xrightarrow{1\otimes\sigma } a'[f_\mu]\otimes_{a[f_\mu]}v[f_\mu ] \to a'[f_\mu]\otimes_{a[f_\mu ] } m \to 0.\ ] ] the naturality of the identifications @xmath231 } v[f_\mu ] ) \cong a'\otimes_a v \cong a'\otimes_{a'[f_\mu]}(a'[f_\mu]\otimes_{a[f_\mu ] } v[f_\mu])$ ] implies that @xmath232 . but @xmath233 is an isomorphism so @xmath234 is injective by lemma  [ sigma_is_injective ] . thus @xmath229\otimes_{a[f_\mu ] } m\in\bl_\infty(a')$ ] . the argument for @xmath175 is similar , for if @xmath61 is a finitely generated projective @xmath96-module then @xmath235 is a finitely generated projective @xmath236-module . recall that an involution on a ring @xmath96 is a map @xmath237 such that @xmath238 , @xmath239 and @xmath240 for all @xmath241 . if @xmath96 is a ring with involution then the category @xmath101 can be endowed with a notion of duality . but let us begin with simpler examples :    [ duality_functor_on_a_proj ] suppose @xmath96 is a ring with an involution . let @xmath242 denote the category of finitely generated projective ( left ) @xmath96-modules . there is a duality functor defined on modules by @xmath243 for @xmath244 and on morphisms by @xmath245 for @xmath246 . in short , @xmath247 note that @xmath248 is a left @xmath96-module with @xmath249 for all @xmath250 , @xmath251 and @xmath252 . [ finite_abelian_group_duality ] the category of finite abelian groups admits the duality functor @xmath253    a finite abelian group @xmath102 bears similarity to an @xmath130-link module in that there exists a presentation @xmath254 and @xmath255 is an isomorphism . a more explicit description of the duality functor in example  [ finite_abelian_group_duality ] is the following :    [ make_finabgp_duality_explicit ] there is a natural isomorphism @xmath256    suppose @xmath102 is a finite abelian group . the short exact sequence @xmath257 gives rise to a long exact sequence @xmath258 which is natural in @xmath102 . the presentation  ( [ present_fin_ab_gp ] ) implies that @xmath259 so @xmath260 is an isomorphism . the following general definition subsumes examples  [ duality_functor_on_a_proj ] and  [ finite_abelian_group_duality ] and the category @xmath101 which we wish to study :    [ define_hermitian_category ] a hermitian category is a triple @xmath261 where    * @xmath262 is an additive category , * @xmath263 is an ( additive ) contravariant functor and * @xmath264 is a natural isomorphism such that @xmath265 for all @xmath266 . the functor @xmath267 is called a duality functor . we usually abbreviate @xmath261 to @xmath262 and identify @xmath61 with  @xmath268 via @xmath269 . it follows from definition  [ define_hermitian_category ] that if @xmath262 is an abelian hermitian category then @xmath267 is an equivalence of categories and hence respects exact sequences . if @xmath96 is a ring with involution then there is a unique involution on @xmath156 $ ] such that @xmath270 for each @xmath271 and such that the inclusion of @xmath96 in @xmath156 $ ] respects the involutions . the category of finitely generated projective @xmath156$]-modules therefore admits a duality functor as in example  [ duality_functor_on_a_proj ] . returning to @xmath101 , duality is defined in a manner analogous to example  [ finite_abelian_group_duality ] . [ define_duality_on_flink_modules ] define @xmath272}(\functor , a[f_\mu ] ) : \bl(a)\to \bl(a)$ ] . note that if @xmath102 has presentation  ( [ presentation_of_f - link_module ] ) then @xmath273}(m , a[f_\mu])$ ] has presentation @xmath274)^*\xrightarrow{\sigma^ * } ( v[f_\mu])^*\to m^\wedge\to 0\ ] ] where @xmath275)^*=\operatorname{hom}_{a[f_\mu]}(v[f_\mu],a[f_\mu])\cong \operatorname{hom}_a(v , a)[f_\mu]= v^*[f_\mu]$ ] ; see lemma  [ lemma : duality ] and remark  [ explicit_formula_for_mwedge_pres ] below . the natural isomorphism @xmath166\to ( v[f_\mu])^{**}\ ] ] induces a natural isomorphism @xmath276 with @xmath277 . there is also a more explicit description of @xmath278 which is analogous to lemma  [ make_finabgp_duality_explicit ] ; see lemma  [ lemma : duality ] below . unlike @xmath60 , the ring @xmath156 $ ] is in general highly non - commutative so universal localization will be required . let @xmath279 be a ring ( associative with unit ) and let @xmath280 be a set of ( isomorphism classes of ) triples @xmath281 where @xmath282 and @xmath283 are finitely generated projective @xmath279-modules . in our application @xmath279 will be @xmath156 $ ] and @xmath280 will contain the endomorphisms @xmath218\to v[f_\mu]$ ] such that @xmath61 is finitely generated and projective as an @xmath96-module and @xmath171 is an automorphism of @xmath61 . a homomorphism @xmath284 is said to be _ @xmath280-inverting _ if @xmath285 is invertible for each morphism @xmath286 . there exists a universal @xmath280-inverting homomorphism was denoted @xmath287 in  @xcite ] which , for consistency with the other papers in the volume , will be denoted @xmath288 . the universal property is that every @xmath280-inverting homomorphism @xmath289 may be written uniquely as a composite @xmath290 if @xmath279 is commutative and each @xmath291 is an endomorphism then the localization is the ring of fractions @xmath292 whose denominators lie in the multiplicative set @xmath293 generated by the determinants of the morphisms in @xmath280 : @xmath294 more general constructions of @xmath295 may be found in  ( * ? ? ? * ch4 ) , ( * ? ? ? * p255 ) or  @xcite . if @xmath296 $ ] and @xmath280 is defined as above , the inclusion of @xmath156 $ ] in @xmath195}$ ] is @xmath280-inverting ; see lemmas  [ completion_is_injection ] and  [ sigma_is_injective ] . by the universal property of @xmath156_\sigma$ ] there is therefore a commutative diagram @xmath297\ar@/_1.5pc/[rr ] \ar[r]^{i_\sigma } & a[f_\mu]_\sigma \ar[r]^(0.35)\gamma & \widehat{a[f_\mu]}\cong a\langle\langle x\rangle\rangle } .\ ] ] where the natural isomorphism @xmath195}\cong a\langle\langle x\rangle\rangle$ ] is defined as in lemma  [ identify_free_completions ] . the image of @xmath298 is the ring @xmath299 of rational power series ( see  @xcite ) . if @xmath96 is a field or a principal ideal domain then @xmath298 is known to be injective so @xmath156_\sigma$ ] can be identified with @xmath299 ( cohn and dicks  ( * ? ? ? * p416 ) , dicks and sontag  ( * ? ? ? * thm 24 ) , farber and vogel  @xcite ) . in the knot theory case @xmath56 , if @xmath96 is commutative then the localization is a ring of fractions @xmath300_{\sigma } ~\cong~\{p / q\ \mid\ p , q\in a[z , z^{-1 } ] , q(1)\ \mbox{is invertible}\}~\cong~a_\rat[[x]].\ ] ] and @xmath298 is injective . however , there exist non - commutative rings @xmath96 such that @xmath298 is not injective  ( * ? ? ? * prop 1.2 ) . diagram  ( [ completion_and_localization ] ) and lemma  [ completion_is_injection ] imply :    [ i_sigma_injective_for_af ] the localization @xmath301\to a[f_\mu]_\sigma$ ] is injective . the following is a generalization of lemma  [ sigma_is_injective ] :    [ sigma_injective ] if @xmath302 is injective then each @xmath286 is injective . suppose @xmath303 is in @xmath280 . there is a commutative diagram @xmath304^{i_\sigma\otimes 1 } \ar[d]_{1\otimes\sigma } & r_\sigma\otimes_r p_1 \ar[d]^{1\otimes\sigma}\\ r\otimes_r p_0 \ar[r]_{i_\sigma\otimes 1 } & r_\sigma\otimes_r p_0}\ ] ] and @xmath305 is an isomorphism . if @xmath295 is injective then @xmath306 is also injective so @xmath172 is injective . [ lemma : duality ] suppose @xmath307 is an injection and @xmath308 with @xmath286 . the ( right ) @xmath279-module @xmath309 is isomorphic to @xmath310 . 2 .   there is a natural isomorphism @xmath311 . if @xmath279 has involution one can regard these right modules as left modules . ( compare example  [ finite_abelian_group_duality ] ) . + 1 . by lemma  [ sigma_injective ] , the map @xmath172 is injective so @xmath102 has presentation @xmath312 there is therefore an exact sequence @xmath313 + 2 .  the short exact sequence of @xmath314-bimodules @xmath315 induces a long exact sequence of right @xmath279-modules @xmath316 which is natural in @xmath102 . it remains to prove that @xmath317 . the presentation  ( [ present_m ] ) gives rise to the long exact sequence @xmath318 there is a natural isomorphism @xmath319 induced by @xmath302 and , in particular , a commutative diagram @xmath320^{(\operatorname{id}\otimes\sigma)^*}\ar[d]_\cong & { \operatorname{hom}_{r_\sigma}(r_\sigma\otimes_r p_1,r_\sigma ) } \ar[d]^\cong \\ { \operatorname{hom}_r(p_0,r_\sigma)}\ar[r]_{\sigma^ * } & { \operatorname{hom}_r(p_1,r_\sigma ) } } \end{gathered}\ ] ] the upper horizontal arrow is an isomorphism since @xmath321 is an isomorphism so the lower horizontal arrow is also an isomorphism . thus @xmath322 as required . [ explicit_formula_for_mwedge_pres ] if @xmath323 is injective and @xmath286 then the sequence @xmath324 is exact by lemma  [ sigma_injective ] . now lemma  [ lemma : duality ] gives an exact sequence @xmath325 let us give an explicit formula for @xmath326 . there is a short exact sequence of right @xmath279-module chain complexes @xmath327 & \operatorname{hom}_r(p_0,r ) \ar[r]\ar[d]_{\sigma^ * } & \operatorname{hom}_r(p_0,r_\sigma)\ar[r]\ar[d]_{\sigma^*}^\cong &   \operatorname{hom}_r(p_0,r_\sigma / r ) \ar[r]\ar[d]_{\sigma^ * } & 0 \\   0\ar[r ] & \operatorname{hom}_r(p_1,r ) \ar[r ] & \operatorname{hom}_r(p_1,r_\sigma)\ar[r ] & \operatorname{hom}_r(p_1,r_\sigma / r)\ar[r ] & 0 } \ ] ] and the natural isomorphism @xmath328 is inverse to the boundary map in the induced long exact homology sequence from the kernel of the right - most @xmath329 to the cokernel of the left - most @xmath329 . written out at length , the map @xmath330 is the composite @xmath331{\cong }   \operatorname{hom}_{r_\sigma}(r_\sigma\otimes p_0,r_\sigma ) \\ \xrightarrow{\cong } \operatorname{hom}_r(p_0,r_\sigma)\to\operatorname{hom}_r(p_0,r_\sigma / r),\end{gathered}\ ] ] the image of which lies in the submodule @xmath332 . suppose @xmath333 and @xmath334 . choose @xmath335 such that @xmath336 and write @xmath337 . now @xmath338    combining lemmas  [ lemma : duality ] and  [ i_sigma_injective_for_af ] we have    [ identify_mwedge_hom_to_loc ] there is a natural isomorphism of contravariant functors @xmath339}(\functor , a[f_\mu])\cong\operatorname{hom}_{a[f_\mu]}(\functor , a[f_\mu]_\sigma / a[f_\mu]):\bl(a)\to\bl(a ) . \qedhere\ ] ]      as we noted in the introduction ( equations  ( [ identify_f_link_cobordism_with_w(seifert ) ] ) and  ( [ identify_f_link_cobordism_with_w(flink ) ] ) ) , the cobordism group @xmath51 can be identified with a witt group of seifert or blanchfield - duval forms . let us recall the definition of a hermitian form in a hermitian category and the appropriate definition of witt group . let @xmath340 or @xmath341 . a _ @xmath342-hermitian form _ in a hermitian category @xmath343 is a pair @xmath344 where @xmath65 and @xmath345 . if @xmath77 is an isomorphism then @xmath77 is called _ non - singular_.    for example , in the category @xmath101 a @xmath342-hermitian form is an @xmath156$]-module isomorphism @xmath346_\sigma / a[f_\mu])$ ] such that @xmath347 ( we suppress the natural isomorphism @xmath348 ) . an object @xmath61 in a hermitian category @xmath343 is called _ self - dual _ if @xmath61 is isomorphic to @xmath349 . if there exists a non - singular @xmath342-hermitian form @xmath344 then @xmath61 is called _ @xmath342-self - dual_.    when one has a suitable notion of exact sequences in a hermitian category one can define the witt group of the category . for simplicity , suppose @xmath262 is a full subcategory of an abelian category @xmath350 , so that every morphism in @xmath262 has kernel , image and cokernel in @xmath350 . suppose further that @xmath262 is admissible in @xmath350 in the following sense : if there is an exact sequence @xmath351 and the modules @xmath61 and @xmath352 lie in @xmath262 then @xmath178 lies in @xmath262 . in section  [ section : seifert ] below we consider a serre subcategory of an abelian category which is defined by @xmath353 if and only if @xmath266 and @xmath354 . for the present we maintain greater generality ; in particular an admissible subcategory @xmath262 is not required to be an abelian category . [ define_metabolic ] let @xmath340 or @xmath341 . a non - singular @xmath342-hermitian form @xmath344 is called metabolic if there is a submodule @xmath355 such that i ) @xmath12 and @xmath356 are in @xmath262 and ii ) @xmath357 . by definition @xmath358 where @xmath359 is the inclusion . [ define_witt_group ] the witt group @xmath360 is the abelian group with one generator @xmath361 $ ] for each isomorphism class of non - singular @xmath342-hermitian forms @xmath362 subject to relations @xmath363=[v,\phi]+[v'',\phi '' ] } ,   & \mbox{if $ ( v',\phi')\cong ( v,\phi)\oplus ( v'',\phi'')$ } \\ { [ v,\phi]=0 } , & \mbox{if $ ( v,\phi)$ is metabolic}. \end{cases}\ ] ] two forms represent the same witt class @xmath361=[v',\phi']$ ] if and only if there exist metabolic forms @xmath364 and @xmath365 such that @xmath366    for example , a non - singular @xmath342-hermitian form @xmath367 in the category @xmath60-proj ( see example  [ duality_functor_on_a_proj ] ) is metabolic if there exists a summand @xmath368 with @xmath369 .    in the category @xmath101 a metabolizer @xmath12 for a form @xmath370 need not be a summand but @xmath12 and @xmath371 must lie in @xmath101 and one must have @xmath369 and @xmath372 if @xmath373 . now that the notation is defined , we repeat equation  ( [ identify_f_link_cobordism_with_w(flink ) ] ) : @xmath374      a functor between hermitian categories which respects their structure is called duality - preserving . our first examples will be functors induced by a morphism of rings with involution . duality - preserving functors will also play an essential role in later sections ( see theorem  [ main_covering_construction_theorem ] above and theorem  [ hermitian_morita_equivalence ] below ) . [ define_dpres_funct ] a duality - preserving functor from @xmath261 to @xmath375 is a triple @xmath376 where    * @xmath377 is a functor , * @xmath378 is a natural isomorphism , * @xmath379 or @xmath341    such that @xmath380 for all @xmath381 . we sometimes abbreviate @xmath376 to @xmath382 . the composite of duality - preserving functors is defined by @xmath383    [ simplest_ring_change_example ] a homomorphism @xmath384 of rings with involution induces a duality - preserving functor @xmath385 from the category @xmath242 of finitely generated projective @xmath96-modules to the category @xmath386 of finitely generated projective @xmath236-modules . explicitly @xmath387 for all @xmath388 , @xmath389 and @xmath252 . we are particularly concerned with the category @xmath101 :    [ change_of_rings_bl ] a homomorphism @xmath384 of rings with involution induces a canonical duality - preserving functor @xmath390\otimes_{a[f_\mu]}\functor,\upsilon,1):\bl(a)\to\bl(a').\ ] ]    the natural isomorphism @xmath391\otimes_{a[f_\mu ] } \functor^\wedge\to ( a'[f_\mu]\otimes_{a[f_\mu ] } \functor)^\wedge$ ] will be defined in the course of the proof . we saw in lemma  [ bl(functor)_is_functor ] that there is a functor @xmath229\otimes_{a[f_\mu]}\functor$ ] from @xmath101 to @xmath392 . the dual @xmath278 of a module @xmath97 has presentation @xmath274)^*\xrightarrow{\sigma^*}(v[f_\mu])^ * \to m^\wedge \to 0\ ] ] ( see remark  [ explicit_formula_for_mwedge_pres ] ) . applying example  [ simplest_ring_change_example ] to @xmath393\to a'[f_\mu]$ ] one obtains a natural isomorphism @xmath394}:a'[f_\mu]\otimes_{a[f_\mu]}(v[f_\mu])^ * \cong   ( a'[f_\mu]\otimes_{a[f_\mu]}v[f_\mu])^*\ ] ] and hence a commutative diagram @xmath395 & a'[f_\mu]\otimes_{a[f_\mu ] } ( v[f_\mu])^ * \ar[r]\ar[d]^{\cong } & a'[f_\mu]\otimes_{a[f_\mu ] } ( v[f_\mu])^ * \ar[d]^{\cong } \ar[r ] & a'[f_\mu]\otimes_{a[f_\mu ] } m^\wedge \ar[r]\ar@{.>}[d]^{\upsilon_m } & 0 \\ 0\ar[r ] & ( a'[f_\mu]\otimes_{a[f_\mu ] } v[f_\mu])^ * \ar[r ] & ( a'[f_\mu]\otimes_{a[f_\mu ] } v[f_\mu])^ * \ar[r ] & ( a'[f_\mu]\otimes_{a[f_\mu ] } m)^\wedge \ar[r ] & 0.}\ ] ]    we must check that the induced isomorphism @xmath396\otimes_{a[f_\mu ] } m^\wedge\to ( a'[f_\mu]\otimes_{a[f_\mu ] } m)^\wedge\ ] ] is independent of the choice of presentation @xmath172 and that @xmath397 is natural with respect to @xmath102 . if @xmath398 denotes the chain complex @xmath160\xrightarrow{\sigma}v[f_\mu]$ ] and @xmath399 denotes an alternative choice of resolution for @xmath102 , say @xmath400\xrightarrow{\sigma'}v'[f_\mu])$ ] then the identity map @xmath401 lifts to a chain equivalence @xmath402 . the naturality of @xmath403 in example  [ simplest_ring_change_example ] implies that the diagram @xmath404\otimes c^ * \ar[r]\ar[d ] & a'[f_\mu]\otimes(c')^*\ar[d ] \\ ( a'[f_\mu]\otimes c)^ * \ar[r ] & ( a'[f_\mu]\otimes c')^ * } \ ] ] commutes . the horizontal arrows induce the identity map on @xmath229\otimes m^\wedge$ ] and @xmath405\otimes m)^\wedge$ ] respectively , so the vertical arrows induce the same map @xmath406 . the naturality of @xmath397 follows similarly from the naturality of the transformation @xmath403 in example  [ simplest_ring_change_example ] . identifying @xmath278 with @xmath407_\sigma / a[f_\mu])$ ] by proposition  [ identify_mwedge_hom_to_loc ] , an explicit formula for @xmath406 is @xmath408\otimes_{a[f_\mu]}\operatorname{hom}\left(m,\frac{a[f_\mu]_\sigma}{a[f_\mu]}\right ) & \to \operatorname{hom}\left(a'[f_\mu]\otimes_{a[f_\mu ] } m\ , \frac{a'[f_\mu]_\sigma}{a'[f_\mu]}\right ) \\ a'_1\otimes \theta&\mapsto ( a'_2\otimes m\mapsto a'_2\nu(\theta(m))\overline{a'_1})\end{aligned}\ ] ]    we conclude this section by noting the effect of duality - preserving functor on witt groups :    [ duality_preserving_functor_induces ] a duality - preserving functor @xmath409 which respects exact sequences induces a homomorphism of witt groups @xmath410&\mapsto[g(v),\psi_vg(\phi)]\end{aligned}\ ] ]    see for example  ( * ? ? ? * p41 - 42 ) . in  @xcite the author defined invariants of the cobordism group @xmath0 of @xmath10-links using the identification ( [ identify_f_link_cobordism_with_w(seifert ) ] ) due to ko  @xcite and mio  @xcite of @xmath0 with a witt group of @xmath2-component seifert forms , denoted @xmath68 below ( @xmath411 ) . to distinguish @xmath10-links one first chooses a seifert surface for each and then computes invariants of the associated seifert forms .    in the present section we define @xmath10-link cobordism invariants via duval s identification @xmath412 . the definitions will parallel those in  @xcite and we shall prove in section  [ section : proof_that_b_induces_witt_isomorphism ] that the invariants obtained are equivalent . whereas seifert forms are convenient for computing the invariants in explicit examples , the blanchfield - duval form has the advantage that it is defined without making a choice of seifert surface . let @xmath340 or @xmath341 . the inclusion @xmath413 induces a duality - preserving functor @xmath414 ( see lemma  [ change_of_rings_bl ] above ) and hence a homomorphism of witt groups @xmath415 it follows from theorem  [ main_covering_construction_theorem ] that  ( [ zintoq_w(bdform)_morphism ] ) is an injection ; see the proof of corollary  [ bd_form_invariants_for_flinks_suffice : precise ] below . we proceed to compute @xmath416 in three steps , which were outlined in the introduction . we list them again here in more detail :    1 .   _ devissage_. we prove that @xmath81 is an abelian category with ascending and descending chain conditions . recall that a module @xmath417 is called simple ( or irreducible ) if @xmath418 and there are no submodules of @xmath102 in @xmath81 other than @xmath127 and @xmath102 . if @xmath102 is a simple module then @xmath419 denotes the full subcategory in which the objects are direct sums of copies of @xmath102 . recall that @xmath102 is called @xmath342-self - dual if there is an isomorphism @xmath420 such that @xmath421 . we obtain , by `` hermitian devissage '' , the decomposition @xmath422 with one summand for each isomorphism class of @xmath342-self - dual simple @xmath10-link modules @xmath102 . let @xmath423 denote the projection of @xmath416 onto @xmath424 . morita equivalence_. for each @xmath342-self - dual simple module @xmath102 we choose a non - singular @xmath342-hermitian form @xmath425 . we obtain by hermitian morita equivalence an isomorphism @xmath426 where @xmath427}m$ ] is the endomorphism ring of @xmath102 and is endowed with the involution @xmath428 . by schur s lemma @xmath83 is a division ring and , as we discuss , @xmath83 turns out to be of finite dimension over @xmath84 . we recall from the literature invariants of each group @xmath82 . in most cases some combination of dimension modulo @xmath111 , signatures , discriminant and hasse - witt invariant are sufficient to distinguish the elements of @xmath82 ( see the table  ( [ table_of_witt_invariants ] ) below ) . one class of division algebra with involution requires a secondary invariant , such as the lewis @xmath13 , which is defined only if all the other invariants vanish . let us make two remarks about the modules @xmath102 which appear in item 1 . firstly , every simple module @xmath417 such that @xmath429 is either @xmath5-self - dual or @xmath430-self - dual ( or both ) . secondly , @xmath102 is both @xmath5-self - dual and @xmath430-self - dual if and only if the involution @xmath428 induced on @xmath427}(m)$ ] is not the identity map for some ( and therefore every ) @xmath342-self - dual form @xmath420 . see lemmas  5.5 and  5.6 of  @xcite for details .    in the knot theory case @xmath56 we can add simplifying remarks to each of the three steps :    1 .   a simple self - dual module @xmath417 may be written @xmath431/(p)\ ] ] where @xmath87 is an irreducible polynomial and @xmath432 $ ] . 2 .   the endomorphism ring @xmath433}(m)$ ] may also be written as a quotient @xmath88/(p)$ ] and is an algebraic number field of finite dimension over @xmath84 . the involution on @xmath83 is given by @xmath434 and does not depend on the choice of form @xmath435 . + setting aside the case @xmath436/(1+z)$ ] , the involution on @xmath83 is not the identity so every self - dual @xmath102 is both @xmath5-self - dual and @xmath430-self - dual . the exceptional module @xmath436/(1+z)$ ] is only @xmath5-self - dual but plays little role since @xmath341 is not a root of any polynomial @xmath437 $ ] such that @xmath438 . in other words , the projection of @xmath439 on this exceptional summand of @xmath440 is zero . 3 .   as discussed in 2 . , one need only consider the witt groups @xmath82 of number fields with non - trivial involution , or in other words , hermitian forms over number fields . the dimension modulo @xmath111 , signatures and discriminant are sufficient to distinguish the elements of @xmath82 .    equation  ( [ iso_class_of_knot_cobordism_gp ] ) can be derived as a consequence of this analysis . returning to the general case @xmath95 and putting together steps 1 - 3 . we obtain the following restatement of theorem  [ bd_form_invariants_for_w_suffice : vague ] . [ bd_form_invariants_for_w_suffice : precise ] let @xmath340 or @xmath341 . an element @xmath441 is equal to zero if and only if for each finite - dimensional @xmath342-self - dual simple @xmath10-link module @xmath102 and non - singular @xmath342-hermitian form @xmath420 , the dimension modulo  @xmath111 , the signatures , the discriminant , the hasse - witt invariant and the lewis @xmath13-invariant of @xmath442}m)\ ] ] are trivial ( if defined ) . note that if the invariants corresponding to one form @xmath420 are trivial then @xmath443 so the invariants are trivial for any other choice @xmath444 . we now restate corollary  [ bd_form_invariants_for_flinks_suffice : vague ] :    [ bd_form_invariants_for_flinks_suffice : precise ] suppose @xmath38 and @xmath39 are @xmath1-dimensional @xmath10-links , where @xmath90 . let @xmath445 denote the free cover of the complement of @xmath446 , let @xmath447 and let @xmath448_\sigma/\z[f_\mu])$ ] denote the blanchfield - duval form for @xmath449 . the @xmath10-links @xmath38 and @xmath39 are cobordant if and only if for each finite - dimensional @xmath342-self - dual simple @xmath10-link module @xmath102 and each non - singular @xmath342-hermitian form @xmath420 , the dimension modulo  @xmath111 , the signatures , the discriminant , the hasse - witt invariant and the lewis @xmath13-invariant of @xmath450 \in w^1(\operatorname{end}_{\q[f_\mu]}m)\ ] ] are trivial ( if defined ) . we deduce corollary  [ bd_form_invariants_for_flinks_suffice : precise ] from theorem  [ bd_form_invariants_for_w_suffice : vague ] ( = theorem  [ bd_form_invariants_for_w_suffice : precise ] ) and theorem  [ main_covering_construction_theorem ] . as we remarked in the introduction , corollary  [ bd_form_invariants_for_flinks_suffice : precise ] also follows from theorem  [ theorem_relating_invariants : vague ] and theorem  b of  @xcite . proposition  [ b_respects_coefficient_change ] below says that the duality - preserving functor @xmath451 of section  [ section : covering_construction ] respects coefficient change so there is a commutative diagram @xmath452\ar[r]^b & w^{(-1)^{q+1}}(\bl(\z))\ar[d ] \\ w^{(-1)^{q}}(\sei(\q))\ar[r]_b & w^{(-1)^{q+1}}(\bl(\q ) ) . } \end{gathered}\ ] ] the category @xmath99 is defined in section  [ section : seifert_modules ] . the lower horizontal map in  ( [ b_respects_z_into_q ] ) is an isomorphism by theorem  [ main_covering_construction_theorem ] and the upper horizontal map is an isomorphism by ( [ identify_f_link_cobordism_with_w(seifert ) ] ) and ( [ identify_f_link_cobordism_with_w(flink ) ] ) above ( see also remark  [ remark_that_b(seifert_form)=bdform ] ) . it is easy to prove that the left hand vertical map is an injection ( see for example lemma  11.1 of  @xcite ) . thus the right - hand vertical map is also an injection . corollary  [ bd_form_invariants_for_flinks_suffice : precise ] therefore follows from theorem  [ bd_form_invariants_for_w_suffice : precise ]      let us briefly recall some definitions . if @xmath453 is an additive category and @xmath102 is an object in @xmath350 the symbol @xmath454 denotes the full subcategory such that @xmath455 if and only if @xmath188 is a summand of some finite direct sum of copies of @xmath102 . if @xmath350 is a hermitian category and @xmath102 is self - dual then @xmath454 is a hermitian subcategory . suppose now that @xmath350 is an abelian category . a non - zero module @xmath102 in @xmath350 is called simple ( or irreducible ) if there are no submodules of @xmath102 in @xmath350 other than @xmath127 and @xmath102 . the category @xmath350 has both ascending and descending chain conditions if and only if every module @xmath102 in @xmath350 has a finite composition series @xmath456 where @xmath457 is simple for @xmath458 . if @xmath102 is simple then every module in @xmath454 is a direct sum of copies of @xmath102 . let @xmath340 or @xmath341 . if @xmath459 is a hermitian category then a module @xmath102 is called @xmath342-self - dual if there is an isomorphism @xmath460 such that @xmath461 . the general decomposition theorem we need is the following :    [ hermitian_devissage ] suppose @xmath350 is an abelian hermitian category with ascending and descending chain conditions . there is an isomorphism of witt groups @xmath462 with one summand for each isomorphism class of simple @xmath342-self - dual modules in @xmath262 . see  ( * ? ? ? * theorem 5.3 ) or  @xcite . to prove equation  ( [ devissage_applied_to_flk(q ) ] ) it therefore suffices to show :    [ blanchfield_category_is_abelian_with_chain_conditions ] if @xmath110 is a ( commutative ) field then the category @xmath136 is an abelian category with ascending and descending chain conditions . we take coefficients in a field for simplicity . with little extra work one can show that proposition  [ blanchfield_category_is_abelian_with_chain_conditions ] holds when @xmath110 is replaced by any semi - simple artinian ring . see also remark  [ remark_on_nu_torsion ] below .    when @xmath55 a simple module in the category @xmath136 is not a simple module in the category of @xmath139$]-modules . we prove proposition  [ blanchfield_category_is_abelian_with_chain_conditions ] as follows : we first show that @xmath136 is a subcategory of the category @xmath463}$ ] of `` torsion '' modules which p.m.cohn introduced and proved to be an abelian category with ascending and descending chain conditions ( see proposition  [ torsion_category_is_abelian ] below ) . after giving details of cohn s work , we conclude the proof of proposition  [ blanchfield_category_is_abelian_with_chain_conditions ] by checking that @xmath136 is closed under direct sums and that the kernel and cokernel of every morphism in @xmath136 again lie in @xmath136 .      to describe cohn s results we must state some properties of the group ring @xmath139 $ ] . a ring @xmath279 has invariant basis number ( ibn ) if @xmath464 implies @xmath465 . in other words @xmath279 has ibn if every finitely generated free left @xmath279-module has unique rank . the existence of the augmentation @xmath466\to k$ ] implies that @xmath139 $ ] has ibn for if @xmath139^n\cong k[f_\mu]^m$ ] then @xmath467}k[f_\mu]^n\cong   k\otimes_{k[f_\mu]}k[f_\mu]^m\cong k^m\ ] ] so @xmath468 . note that if @xmath279 has ibn then one can use the duality functor @xmath469 to prove that finitely generated free right @xmath279-modules also have unique rank . an associative ring @xmath279 is called a _ free ideal ring ( fir ) _ if @xmath279 has ibn , every left ideal in @xmath279 is a free left @xmath279-module and every right ideal is a free right @xmath279-module . cohn showed in  ( * ? ? ? * corollary 3 ) that if @xmath110 is a field then the group ring @xmath139 $ ] of the free group @xmath10 is a fir . if @xmath279 is a fir then every submodule of a free @xmath279-module is free ( cohn  ( * ? ? ? * p71 ) ) . hence every @xmath279-module has a presentation @xmath470 where @xmath471 and @xmath472 are free . if an @xmath279-module @xmath102 has a presentation @xmath473 the _ euler characteristic _ of @xmath102 is @xmath474 . if a finite presentation exists the euler characteristic is independent of the choice of presentation by schanuel s lemma . note also that an exact sequence @xmath475 of finitely presented modules implies the equation @xmath476 ( compare the diagram ( [ horseshoe_diagram ] ) below ) . if @xmath279 is a fir then the category of finitely presented @xmath279-modules ( and @xmath279-module maps ) is an abelian category ; in other words direct sums of finitely presented modules are finitely presented and the cokernel and kernel of a map between finitely presented modules are finitely presented . in fact , if @xmath279 is any ring such that every finitely generated one - sided ideal is finitely related then the finitely presented @xmath279-modules form an abelian category ( e.g.  cohn  ( * ? ? ? * p554 - 556 ) ) . a morphism @xmath477 between free left @xmath279-modules is called _ full _ if every factorization @xmath478^\sigma \ar[r]_{\sigma_2 } & f \ar[r]_{\sigma_1 } & r^n } \ ] ] where @xmath130 is a free module has @xmath479 . [ characterize_torsion ] suppose @xmath279 is a fir and @xmath102 is a finitely presented @xmath279-module with @xmath480 . the following are equivalent :    1 . [ all_full ] in every finite presentation  ( [ present_torsion_module ] ) of @xmath102 , the map @xmath172 is full . [ exists_full ] there exists a presentation  ( [ present_torsion_module ] ) such that @xmath172 is full . [ euler_sub ] @xmath481 for all finitely generated submodules @xmath188 of @xmath102 . [ euler_quot ] @xmath482 for all finitely generated submodules @xmath188 of @xmath102 . the implication [ all_full]@xmath483  [ exists_full ] is immediate . to show  [ exists_full ] implies  [ euler_sub ] , suppose we are given a presentation  ( [ present_torsion_module ] ) such that @xmath468 and @xmath172 is full . if @xmath188 is a finitely generated submodule of @xmath102 then @xmath484 . now @xmath485 is a free module because @xmath279 is a fir and @xmath485 has rank at least @xmath52 since @xmath172 is full . the exact sequence @xmath486 implies that @xmath481 . this completes the proof that  [ exists_full ] implies  [ euler_sub ] . the equation @xmath487 implies that  [ euler_sub ] and  [ euler_quot ] are equivalent so we can conclude the proof of the lemma by showing that  [ euler_sub ] implies  [ all_full ] . the equation @xmath480 says that every finite presentation ( [ present_torsion_module ] ) has @xmath468 . we must prove that @xmath172 is full . suppose @xmath172 can be written as a composite @xmath488 . we aim to show @xmath489 . now @xmath490 and @xmath491 are free modules since @xmath279 is a fir and @xmath492 by statement  [ euler_sub ] , @xmath493 so @xmath494 . thus @xmath489 and hence @xmath172 is full .    a module which satisfies the equivalent conditions in lemma  [ characterize_torsion ] is called a _ torsion _ the symbol @xmath495 denote the category of torsion modules and @xmath279-module maps . ( cohn ( * ? ? ? * p166))[extension_of_torsion_modules ] if @xmath496 is an exact sequence and @xmath102 and @xmath497 are torsion modules then @xmath498 is a torsion module . in particular a direct sum of torsion modules is again a torsion module . if @xmath102 and @xmath497 are finitely presented then @xmath498 is also finitely presented ( compare  ( [ horseshoe_diagram ] ) below ) . since @xmath499 we have @xmath500 . now if @xmath501 is finitely generated it suffices by lemma  [ characterize_torsion ] to show that @xmath481 . note first that @xmath502 . now @xmath503 so @xmath504 and @xmath505 so @xmath506 . thus @xmath481 . ( cohn ( * ? ? ? * p167,234))[torsion_category_is_abelian ] suppose @xmath279 is a fir .    1 .   [ abelian_part ] the category @xmath495 of torsion modules is an abelian category . [ acc_dcc_part ] every module in @xmath495 has a finite composition series  ( [ composition_series ] ) in @xmath495 .    to establish statement  [ abelian_part ] , it suffices to show that if @xmath102 and @xmath498 lie in @xmath495 then @xmath507 and the kernel and cokernel of every morphism @xmath508 are in @xmath495 . lemma  [ extension_of_torsion_modules ] gives @xmath509 . suppose then that @xmath508 is an @xmath279-module morphism . since the finitely presented @xmath279-modules are an abelian category the kernel , image and cokernel of @xmath510 are finitely presented . now @xmath511 is a submodule of @xmath498 and a quotient module of @xmath102 so @xmath512 and @xmath513 by lemma  [ characterize_torsion ] . thus @xmath514 and it follows that @xmath515 and @xmath516 . every finitely generated submodule @xmath517 is a submodule of @xmath102 so @xmath481 and hence @xmath518 . similarly every quotient @xmath519 of @xmath520 is a quotient of @xmath498 and hence has @xmath521 . thus @xmath522 also .    to prove part 2 . of proposition  [ torsion_category_is_abelian ] we note first that it is sufficient to check the ascending chain condition . indeed , if @xmath523 then @xmath309 is a torsion right @xmath279-module and a descending chain @xmath524 in @xmath495 gives rise to an ascending chain @xmath525    in fact cohn showed that a larger class of modules , the finitely related bound modules , have the ascending chain condition . an @xmath279-module @xmath102 is _ bound _ if @xmath526 . [ torsion_implies_bound ] every torsion module over a fir is bound . if @xmath102 is an @xmath279-module and @xmath527 then @xmath528 is a free module so @xmath529 . now @xmath528 is a quotient module of @xmath102 so if @xmath102 is a torsion module then @xmath530 . it follows that @xmath531 so @xmath532 . thus @xmath533 . ( cohn  ( * ? ? ? * p231 ) ) if @xmath279 is a fir and @xmath102 is a finitely related @xmath279-module then every bound submodule of @xmath102 is finitely presented . there is an exact sequence @xmath534 where @xmath130 is a free @xmath279-module . a submodule @xmath535 has presentation @xmath536 and @xmath537 is a free module since @xmath279 is a fir . the image of @xmath538 is contained in a finitely generated summand of @xmath537 so @xmath4 is a direct sum of a free module and a finitely presented module . if @xmath4 is bound then @xmath4 does not have any non - zero free summand so @xmath4 itself is finitely presented . thus every torsion submodule of a torsion module @xmath102 is finitely generated so we have the ascending chain condition on torsion submodules . it follows that @xmath495 has both ascending and descending chain conditions and the proof of proposition  [ torsion_category_is_abelian ] is complete . we are now in a position to deduce proposition  [ blanchfield_category_is_abelian_with_chain_conditions ] . [ bl(k)_is_inside_t ] the category @xmath136 is a full subcategory of the category @xmath463}$ ] of torsion modules . it suffices to show that every module in @xmath136 is a torsion module . by definition , a module in @xmath136 has a presentation  ( [ presentation_of_f - link_module ] ) where @xmath171 is an isomorphism . since @xmath171 is full we can deduce that @xmath172 is full since each factorization of @xmath172 induces a corresponding factorization of @xmath171 .    since @xmath463}$ ] has ascending and descending chain conditions and @xmath539}$ ] we need only show that @xmath136 is an abelian category . suppose @xmath102 and @xmath498 are in @xmath136 . it follows directly from the definition of @xmath136 that @xmath540 . we must show that the kernel and cokernel of each map @xmath508 lie in @xmath136 . by proposition  [ torsion_category_is_abelian ] the kernel , image and cokernel all lie in @xmath463}$ ] so @xmath541 after choosing presentations @xmath172 and @xmath542 for @xmath543 and @xmath511 respectively one can fill in the dotted arrows below to obtain a commutative diagram with exact rows and exact columns : @xmath544 & 0\ar@{.>}[d ] & 0\ar[d ] \\ 0 \ar[r ] & k[f_\mu]^n \ar@{.>}[d ] \ar[r]^\sigma & k[f_\mu]^n\ar[r]\ar@{.>}[d ] & \operatorname{ker}(f ) \ar[r ] \ar[d ] & 0    \\ 0 \ar@{.>}[r ] & k[f_\mu]^{n+n ' } \ar@{.>}[r]^{\sigma''}\ar@{.>}[d ] & k[f_\mu]^{n+n ' } \ar@{.>}[r]\ar@{.>}[d ] & m \ar[r]\ar[d ] & 0 \\ 0 \ar[r ] & k[f_\mu]^{n ' } \ar[r]^{\sigma ' } \ar@{.>}[d ] & k[f_\mu]^{n ' } \ar[r ] \ar@{.>}[d ] & f(m)\ar[d ] \ar[r ] & 0   \\   & 0 & 0 & 0 } \end{gathered}\ ] ] the map @xmath545 is given by @xmath546 for some @xmath547 . since @xmath109 , the augmentation @xmath548 is an isomorphism by lemma  [ sato_condition ] above . it follows that @xmath171 and @xmath548 are isomorphisms and hence @xmath543 and @xmath511 are in @xmath136 . the same argument , applied to the exact sequence @xmath549 shows that @xmath520 is also in @xmath136 . [ remark_on_nu_torsion ] the arguments above can be adapted to generalize propositions  [ torsion_category_is_abelian ] and  [ blanchfield_category_is_abelian_with_chain_conditions ] as follows . suppose every finitely generated one - sided ideal of a ring @xmath279 is a projective module ( i.e.  @xmath279 is semi - hereditary ) . suppose @xmath550 is a ring with the property that @xmath551 implies @xmath552 ( i.e.  @xmath550 is weakly finite ) and @xmath289 is a ring homomorphism . the grothendieck group @xmath553 admits a partial order in which @xmath554 if and only @xmath555 lies in the positive cone @xmath556 \mid p\in s\proj\}\subset k_0(s ) .\ ] ] now the following category @xmath557 is an abelian category : an @xmath279-module @xmath102 lies in @xmath557 if @xmath102 has a finite presentation by projective modules @xmath558 such that @xmath559 is full with respect to the partial order on @xmath553 . the subcategory of modules for which @xmath234 is invertible is also an abelian category . under the additional hypotheses that all the one - sided ideals of @xmath279 are projective modules ( i.e.  @xmath279 is hereditary ) and that the equation @xmath560 implies @xmath552 for projective @xmath279-modules @xmath561 , one can also conclude that these abelian categories have ascending and descending chain conditions . one recovers proposition  [ torsion_category_is_abelian ] when @xmath279 is a fir by setting @xmath562 and one recovers proposition  [ blanchfield_category_is_abelian_with_chain_conditions ] by setting @xmath563\to k$ ] .      having reduced @xmath416 to a direct sum of witt groups @xmath424 where @xmath102 is simple and @xmath342-self - dual ( see equation  ( [ devissage_applied_to_flk(q ) ] ) ) we pass next from @xmath424 to the witt group @xmath564}m)$ ] of the endomorphism ring of @xmath102 .    recall that if @xmath262 is an additive category then an object in @xmath565 is a summand of a direct sum of copies of @xmath102 . the general theorem we employ in this section is the following :    [ hermitian_morita_equivalence ] let @xmath566 or @xmath341 . suppose that @xmath567 is a non - singular @xmath568-hermitian form in a hermitian category @xmath262 , and assume further that every idempotent endomorphism in the hermitian subcategory @xmath565 splits . let @xmath569 be endowed with the involution @xmath570 . then there is an equivalence of hermitian categories @xmath571 where for @xmath572 the map @xmath573 is the composite of natural isomorphisms @xmath574    see  ( * ? ? ? * theorem 4.7 ) ,  ( * ? ? ? * , ch.ii ) or  @xcite . the following is a corollary of theorem  [ hermitian_morita_equivalence ] and lemma  [ duality_preserving_functor_induces ] . [ apply_morita_eq_to_bl ] if @xmath102 is a simple module in @xmath136 and @xmath420 is a non - singular @xmath342-hermitian form then the duality - preserving functor @xmath575 of theorem  [ hermitian_morita_equivalence ] induces an isomorphism of witt groups @xmath576}m ) \\ [ n,\phi ] & \mapsto [ \operatorname{hom}(m , n),\omega^b_n\phi_*]\end{aligned}\ ] ] where @xmath577 and @xmath578}m\ ] ] for all @xmath579 . equation  ( [ devissage_applied_to_flk(q ) ] ) implies that @xmath580}m)~.\ ] ] with one summand for each isomorphism class of @xmath342-self - dual simple @xmath10-link modules @xmath102 .    every exact sequence in @xmath581 splits , so @xmath575 is exact and hence induces a morphism of witt groups @xmath582}(m)).\ ] ] since @xmath575 is an equivalence of hermitian categories it follows by lemma  [ nat_isomorphic_dpres_funct_give_same_witt_homo ] of appendix  [ section : naturality_of_b ] that the induced map of witt groups is an isomorphism . equation  ( [ hermitian_morita_equivalence_on_flk(q ) ] ) above is a special case of corollary  [ apply_morita_eq_to_bl ] so the following proposition completes step  2 :    [ bl_endo_rings_fin_diml ] the endomorphism ring of every module in @xmath136 is of finite dimension over @xmath110 . proposition  [ bl_endo_rings_fin_diml ] follows from theorem  [ bbar_is_equiv_on_fgproj_cats ] and lemma  [ fully_simplify_quotient ] below . theorem  [ bbar_is_equiv_on_fgproj_cats ] can be considered analogous to the geometric fact that one can choose a seifert surface for an @xmath10-link . since the homology of a seifert surface is finite - dimensional the endomorphism ring of the associated seifert module is finite - dimensional . using chapter  12 of  @xcite , part 2 .  of lemma  [ simple_goes_to_comes_from_simple ] and part 2 .  of lemma  [ quotient_lemma_with_duality ] we can also deduce that every division ring with involution which is finite - dimensional over @xmath84 arises as @xmath583}m\ , \ f\mapsto\overline f = b^{-1}f^\wedge b)\ ] ] for some pair @xmath584 where @xmath102 is a simple module in @xmath81 . the proofs of the results cited in the previous paragraph do not use proposition  [ bl_endo_rings_fin_diml ] ( i.e.  the arguments presented are not circular ) . however , the spirit of this section is to define invariants of @xmath10-links by studying the category @xmath81 directly so we desire a proof of theorem  [ bl_endo_rings_fin_diml ] which avoids any choice of seifert surface or seifert module . one such proof is due to lewin  @xcite . in a subsequent paper we shall give a proof which applies when @xmath139 $ ] is replaced by a wider class of rings .    before leaving the subject of morita equivalence we pause to note the following `` naturality '' statement which we will need in section  [ section : proof_that_b_induces_witt_isomorphism ] to prove theorem  [ theorem_relating_invariants : vague ] . the reader may refer to equation  ( [ how2_compose_dpres_functors ] ) for the definition of composition for duality - preserving functors . [ naturality_of_h_morita_eq ] suppose @xmath585 is a duality - preserving functor between hermitian categories and @xmath567 is an @xmath568-hermitian form in  @xmath262 . let @xmath586 and let @xmath587 . the following diagram of duality - preserving functors commutes up to natural isomorphism : @xmath588^{(g\,,\,\psi\,,\,\eta ' ) } \ar[d]_{(\operatorname{hom}(m,\functor)\ , , \,\omega^b\ , , \,\eta ) } & { \cy{d}|_{g(m ) } } \ar[d]^{(\operatorname{hom}(g(m),\functor)\ , , \ , \omega^{\eta'\psi g(b)}\ , , \,\eta\eta ' ) } \\ { e\proj } \ar[r]_{(e'\otimes_e\functor\,,\ , \pi\,,\,1 ) } & { e'\proj}. } \ ] ]    see section  [ section : natuality_of_h_morita_eq ] of appendix  [ section : naturality_of_b ] . [ witt_naturality_of_h_morita_eq ] if @xmath382 is exact then the following square also commutes : @xmath589^{g } \ar[d]^\cong_{\theta_{m , b } } & w^{\zeta\eta'}(\cy{d}|_{g(m ) } ) \ar[d]_\cong^{\theta_{g(m),\eta'\psi g(b)}}\\ w^{\zeta\eta}(e ) \ar[r]_g & w^{\zeta\eta}(e ' ) . } \ ] ]      equation  ( [ devissage_applied_to_flk(q ) ] ) leads one to consider invariants to distinguish witt classes of forms over division algebras @xmath83 of finite dimension over @xmath84 . such division algebras are well understood ( see albert  ( * ? ? ? * p149,p161 ) , scharlau  @xcite or our earlier summary in  @xcite ) . one considers five distinct classes of division algebras with involution . firstly a division algebra @xmath83 may be commutative or non - commutative . secondly , if @xmath590 is an involution on @xmath83 let @xmath591 . the involution is said to be `` of the first kind '' if @xmath592 contains the center @xmath593 of @xmath83 . otherwise , the involution is `` of the second kind '' . finally , one of these four classes is further partitioned . a non - commutative division algebra with involution of the first kind is necessarily a quaternion algebra , with presentation @xmath594 for some number field @xmath595 and some elements @xmath596 . if @xmath597 then the involution is called `` standard '' . on the other hand if @xmath592 strictly larger than @xmath595 then the involution is called `` non - standard '' .    table  ( [ table_of_witt_invariants ] ) below lists sufficient invariants to distinguish the witt classes of forms over each class of division algebras with involution . the symbol @xmath598 denotes dimension modulo @xmath111 . the letter @xmath172 signifies all signature invariants ( if any ) each of which takes values in @xmath60 . the discriminant @xmath599 is the determinant with a possible sign adjustment and takes values in the group of `` square classes '' @xmath600 where @xmath595 is the center of @xmath83 and @xmath601 . @xmath602 two symbols in the table have not yet been mentioned . the hasse - witt invariant , @xmath603 , which appears in the first row takes values in a direct sum of copies of @xmath604 , one copy for each prime of the number field @xmath595 . finally , if @xmath83 is a quaternion algebra with non - standard involution of the first kind then the local - global principle fails and one requires a secondary invariant such as the lewis @xmath13 which is defined if all the other invariants vanish . the value group for @xmath13 is the quotient @xmath605 where @xmath550 is the set of primes @xmath606 of @xmath595 such that the completion @xmath607 is a division algebra and the relation @xmath608 identifies each element @xmath609 with its antipode @xmath610 . in this section we describe algebraic structures arising in the study of a seifert surface of an @xmath10-link . we define in section  [ section : seifert_modules ] a category @xmath98 of `` seifert modules '' and a full subcategory @xmath611 in which the objects are finitely generated and projective as @xmath96-modules . the category @xmath99 was denoted @xmath612 in  @xcite .    in section  [ blanchfield_module_has_seifert_structure ] we prove that every @xmath10-link module is also a seifert module in a canonical way ( cf farber  @xcite ) and obtain a `` forgetful '' functor @xmath613 the image of @xmath101 is usually not contained in @xmath99 which explains our motivation for introducing @xmath98 . we construct later ( section  [ section : covering_construction ] ) a functor @xmath4 from @xmath99 to @xmath101 which extends in an obvious way to a functor @xmath151 and is left adjoint to  @xmath113 . in section  [ section : cobordism_invariants_seifert ] we put hermitian structure on @xmath99 . we will see in section  [ section : proof_that_b_induces_witt_isomorphism ] below that the functor @xmath4 induces an isomorphism of witt groups @xmath614 when @xmath96 is semi - simple and artinian and , in particular , when @xmath148 . suppose @xmath61 is a finite - dimensional vector space over a field @xmath110 and @xmath615 is an endomorphism . a time - honoured technique in linear algebra regards the pair @xmath616 as a module over a polynomial ring @xmath617 $ ] in which the action of  @xmath133 on  @xmath61 is given by  @xmath20 . equivalently , @xmath616 is a representation of @xmath618 $ ] in the category of finite - dimensional vector spaces over @xmath110 . we shall use the words `` module '' and `` representation '' interchangeably .    given a seifert surface @xmath619 for a knot @xmath7 , small translations in the directions normal to @xmath113 induce homomorphisms @xmath620 using alexander duality one finds that @xmath621 is an isomorphism for @xmath622 , so @xmath623 is endowed with an endomorphism @xmath624 and may therefore be regarded as a representation of a polynomial ring @xmath618 $ ] . the homology of a seifert surface for a @xmath2-component boundary link has , in addition to the endomorphism @xmath624 , a system of @xmath2 orthogonal idempotents which express the component structure of the seifert surface . following farber  @xcite ( see also  @xcite ) we regard @xmath623 as a representation of the ring @xmath625 * _ \z \left(\prod_\mu \z\right).\end{aligned}\ ] ] @xmath626 is also the path ring of a quiver which we illustrate in the case @xmath627 : @xmath628 \ar@(ul , dl ) [ ] & \bullet \ar@/^/[l ] \ar@(ur , dr ) [ ] } \ ] ]    [ sei_notation ] let @xmath98 denote the category of representations of @xmath626 by @xmath96-modules . a module in @xmath98 is a pair @xmath629 where @xmath61 is an @xmath96-module and @xmath630 is a ring homomorphism . let @xmath99 denote the category of representations of @xmath626 by finitely generated projective @xmath96-modules . in other words , @xmath99 is the full subcategory of pairs @xmath629 such that @xmath61 is finitely generated and projective . we sometimes omit @xmath631 , confusing an element of @xmath626 with its image in @xmath632 . every @xmath10-link module has a canonical seifert module structure which we describe next . in fact @xmath114 can be regarded as a full subcategory of @xmath98 . note however that a module in @xmath101 is in general neither finitely generated nor projective as an @xmath96-module ( e.g.  see lemma  [ sato_condition ] above ) and therefore does not lie in @xmath99 . if @xmath228 then lemma  [ sato_condition ] implies that the @xmath96-module map @xmath633 is an isomorphism . let @xmath634 denote the projection of @xmath635 onto its @xmath16th component and let @xmath636 denote addition . define @xmath637 by @xmath638 we denote by @xmath106 the @xmath96-module @xmath102 with the seifert module structure @xmath631 . as remarked in the introduction , in the case @xmath56 the seifert module structure @xmath639 can be described more simply by the equation @xmath640 the following lemma says that @xmath152 is a full and faithful functor , so @xmath114 can be regarded as a full subcategory of @xmath98 .    [ bl_infty_is_full_subcat_of_sei_infty ] an @xmath96-module morphism @xmath508 between @xmath10-link modules @xmath102 and @xmath498 is an @xmath156$]-module morphism if and only if @xmath510 is a morphism of seifert modules . in other words , @xmath641 if and only if @xmath642 . if @xmath508 is an @xmath156$]-module morphism then the diagram @xmath643^{f^{\oplus\mu}}\ar[d]_\gamma & { m'}^{\oplus\mu}\ar[d]^\gamma \\ m\ar[r]_f & m'}\ ] ] is commutative . conversely , if the diagram  ( [ f_commute_with_gamma ] ) commutes then the equation @xmath644 holds for each @xmath645 and @xmath646 , so @xmath647 for each @xmath16 and hence @xmath510 is an @xmath156$]-module morphism . it remains to show that  ( [ f_commute_with_gamma ] ) commutes if and only if @xmath510 is a seifert morphism . if  ( [ f_commute_with_gamma ] ) commutes then @xmath648 for each @xmath645 so @xmath510 is a seifert morphism . conversely , suppose @xmath510 is a seifert morphism . now @xmath649 so the diagram  ( [ f_commute_with_gamma ] ) commutes . let us make @xmath99 a hermitian category , assuming @xmath96 is a ring with involution . recall that if @xmath61 is a finitely generated projective left @xmath96-module then @xmath650 is a left @xmath96-module with @xmath651 for all @xmath652 , @xmath389 and @xmath252 . if @xmath629 is an object in @xmath99 define @xmath653 where @xmath654 it is easy to see that if @xmath655 is a morphism in @xmath99 then the dual @xmath656 again lies in @xmath99 . equivalently , if one gives the ring @xmath626 the involution defined by @xmath657 and @xmath658 for each @xmath16 then @xmath659 is given by @xmath660 for all @xmath252 , @xmath389 and @xmath661 .    if @xmath662 then the intersection form @xmath65 is a morphism in @xmath58 . in other words , the intersection form respects the projections @xmath663 and respects the endomorphism @xmath624 ( see section  [ section : seifert_modules ] ) in the sense that @xmath664 for all @xmath665 . furthermore , @xmath77 is an isomorphism by poincar duality and is @xmath67-hermitian . this form @xmath77 will be called the seifert form associated to a seifert surface . kervaire  ( * ? ? ? * p94 ) or lemma  3.31 of  @xcite shows that by an elementary change of variables , this form @xmath77 is equivalent to the seifert matrix of linking numbers more commonly encountered in knot theory .    by ko  @xcite and mio  @xcite ( see also lemma  3.31 of  @xcite ) , the association of this form @xmath77 to a seifert surface induces the isomorphism @xmath666 mentioned in the introduction . every @xmath67-hermitian seifert form is associated to some @xmath667-dimensional seifert surface . although there are many possible seifert surfaces for a given @xmath10-link , all are cobordant and the corresponding seifert forms lie in the same witt class . in  @xcite the author applied to @xmath668 the steps 1 - 3 .  described in section  [ section : overview_of_bd_form_invariants ] obtaining explicit invariants to distinguish @xmath10-link cobordism classes . although the blanchfield - duval form is more intrinsic , the advantage of the seifert form is that it is easier to compute the numerical invariants . for illustration , we treat a worked example :    [ worked_example_seifert ] setting @xmath627 , consider the seifert module @xmath669 with the endomorphism @xmath133 and @xmath430-hermitian form @xmath77 given by @xmath670 the horizontal and vertical lines indicate the component structure of the seifert form . in other words , @xmath671 projects onto the span of the first four basis elements while @xmath672 projects onto the span of the last two . the corresponding seifert matrix of linking numbers is @xmath673 but we shall work directly with @xmath133 and @xmath77 . the first step is to pass from @xmath60 to @xmath84 so we regard the entries in the matrices as rational numbers . devissage is next ; let @xmath674 denote the standard basis of @xmath675 . now @xmath676 is @xmath133-invariant and @xmath677 so our seifert form is witt - equivalent to the induced form on @xmath678 . we have reduced @xmath133 and @xmath77 to @xmath679 the two - dimensional representation ( over @xmath84 ) of @xmath618 $ ] given by the matrix @xmath680 is simple ( = irreducible ) since there do not exist eigenvalues in @xmath84 . it follows that the seifert module @xmath681 with the action @xmath682 and @xmath683 shown is simple . the devissage process is therefore complete .    turning to the morita equivalence step , the endomorphism ring of this module @xmath178 has @xmath84-basis consisting of the identity and @xmath684 . the minimum polynomial of @xmath685 is @xmath686 so @xmath687 is isomorphic to @xmath688 . we may choose @xmath689 . it is easy to verify that the involution @xmath690 is not the identity map so it must send @xmath691 to @xmath692 . morita equivalence sends the form @xmath693 to the composite @xmath694 which is given by @xmath695 for @xmath696 . this form may be written @xmath697 as a form over @xmath688 .    reading the fourth line of the table ( [ table_of_witt_invariants ] ) , an element of the witt group @xmath698 for non - trivial involution is determined by signatures and discriminant ( and rank modulo @xmath111 if there are no signatures ) . up to complex conjugation there is precisely one embedding of @xmath688 in @xmath699 ( with the complex conjugate involution ) , so there is in fact one signature , which takes value @xmath700 with our choice of @xmath435 . the discriminant is @xmath701 in this section we introduce a functor @xmath129 which is the algebraic analogue of the geometric construction of the free cover of an @xmath10-link complement from a seifert surface . ( illustrated in figure  1 on page  ) . the restriction of @xmath4 to @xmath99 takes values in @xmath101 and extends to a duality - preserving functor @xmath451 which is natural in @xmath96 ( see propositions  [ extend_b_to_duality_pres_functor ] and  [ b_respects_coefficient_change ] ) . we show that @xmath129 is left adjoint to the full and faithful functor @xmath150 ; in other words , among functors @xmath702 , the geometrically motivated functor @xmath4 satisfies a universal property with respect to @xmath113 ( see definition  [ define_left_adjoint ] ) . in particular there is a natural transformation @xmath703 for @xmath704 and a natural isomorphism @xmath705 for @xmath228 . the reader is referred to chapter 3 of borceux  @xcite or chapter iv of mac lane  @xcite for a detailed treatment of adjoint functors . we use the adjunction in section  [ localization_of_cat ] to show that the covering construction @xmath129 is equivalent to a universal localization @xmath706 of categories . we describe the structure of the `` primitive '' modules @xmath144 in section  [ section : structure_of_primitives ] and outline a construction of the quotient category in section  [ section : construction_of_quotient ] . we show in sections  [ localization_of_cat ] and  [ section : duality_on_quotient_arbitrary_a ] that @xmath707 is equivalent to a localization @xmath708 of hermitian categories . in the case where @xmath96 is a semi - simple artinian ring we simplify the descriptions of the quotient and primitive modules in section  [ section : global_dim_zero ] . to simplify notation in this section and section  [ section : adjunction ] , we suppress the symbol @xmath631 which appears in the definition of a seifert module @xmath629 , identifying an element @xmath661 with @xmath709 . we shall extend seifert structure from an @xmath96-module @xmath61 to the induced module @xmath156\otimes_a v$ ] by @xmath710 and @xmath711 for @xmath712 $ ] . recall that @xmath176 are distinguished generators of @xmath10 ; let us now write @xmath713 . [ define_covering_construction ] if @xmath61 is a module in @xmath98 let @xmath714\to v[f_\mu]\right).\ ] ]    since @xmath715 , it is clear that @xmath116 lies in @xmath114 . to achieve more symmetric notation we write @xmath716 and deduce @xmath717 where @xmath718 . we write @xmath719 when we wish to indicate that elements of @xmath10 are to be written as words in the symbols @xmath720 . the @xmath156$]-module @xmath721}$ ] is isomorphic to @xmath160 $ ] and will be written @xmath722 $ ] for brevity . now we have @xmath723\to      v[f_\mu(y^2)]\right ) \\   & = \operatorname{coker}\left((1-s)y^{-1 } + sy : v[f_\mu(y^2)y]\to v[f_\mu(y^2)]\right ) . \end{aligned}\ ] ] in detail , @xmath724 for @xmath725 , @xmath726 and @xmath727 . a morphism @xmath655 induces a commutative diagram @xmath728 & v[f_\mu ] \ar[r]^\sigma\ar[d]_f & v[f_\mu ] \ar[r]^q\ar[d]_f & b(v ) \ar[r]\ar@{.>}[d]^{b(f ) } & 0 \\    0 \ar[r ] & v'[f_\mu ] \ar[r]^{\sigma ' } & v'[f_\mu ] \ar[r]^{q ' } & b(v ' ) \ar[r ] & 0 } \end{gathered}\ ] ] where @xmath729 and hence induces an @xmath156$]-module map @xmath730 as shown . [ exactness_of_b ] the functor @xmath4 is exact . in other words , if @xmath731 is an exact sequence in @xmath98 then the sequence @xmath732 induced is also exact . it suffices to show that @xmath4 preserves short exact sequences . suppose @xmath733 is exact . there is a commutative diagram @xmath734 & 0 \ar[d ] & 0 \ar[d ] & \\ 0 \ar[r ] & v[f_\mu ] \ar[r]\ar[d ] & v[f_\mu ] \ar[r]\ar[d ] & b(v ) \ar[r]\ar[d ] & 0 \\    0 \ar[r ] & v'[f_\mu ] \ar[r]\ar[d ] & v'[f_\mu]\ar[d ] \ar[r ] & b(v')\ar[d ] \ar[r ] & 0 \\ 0 \ar[r ] & v''[f_\mu ] \ar[r]\ar[d ] & v''[f_\mu ] \ar[r]\ar[d ] & b(v '' ) \ar[r]\ar[d ] & 0 \\    & 0 & 0 & 0 &   } \end{gathered}\ ] ] in which the rows and the two left - most columns are exact . it follows that the right - hand column is also exact .    the category @xmath98 has limits and colimits . for example , the coproduct of a system of modules is the direct sum . since @xmath4 is exact and respects arbitrary direct sums @xmath4 respects all colimits :    [ b_is_cocontinuous ] the functor @xmath4 is cocontinuous . in particular if @xmath61 is a direct limit @xmath735 then @xmath736 . on the other hand @xmath4 does not respect infinite limits . for example one finds @xmath737 because @xmath738\ncong \prod ( v_i[f_\mu])$ ] . however , @xmath4 does respect finite limits . the idea behind the proof of the following proposition is due to a.ranicki . [ extend_b_to_duality_pres_functor ] the functor @xmath4 extends to a duality - preserving functor @xmath131 . for each finitely generated projective @xmath96-module @xmath61 there is a natural isomorphism @xmath739\to ( v[f_\mu])^*$ ] by example  [ nat_iso_in_coeff_change_dpres_funct ] . naturality asserts that for each morphism @xmath740 the diagram @xmath741\ar[r]^{\alpha^*}\ar[d]_{\pi_w } & v^*[f_\mu ] \ar[d]^{\pi_v } \\ ( w[f_\mu])^*\ar[r]^{\alpha^ * } & ( v[f_\mu])^ * } \ ] ] is commutative . moreover one can check commutativity of @xmath742\ar[r]^{z_i^{-1}}\ar[d]_{\pi_v } & v^*[f_\mu ] \ar[d]^{\pi_v } \\ ( v[f_\mu])^*\ar[r]^{z_i^ * } & ( v[f_\mu])^ * } \ ] ] where , as usual , @xmath743\to v[f_\mu]$ ] and @xmath744\to v^*[f_\mu]$ ] denote multiplication on the right by @xmath745 .    now if @xmath704 then there is a commutative diagram : @xmath746 & v^*[f_\mu]\ar[d]_{-\pi_v(1-z)}\ar[r]^{\sigma(v^ * ) } & v^*[f_\mu ] \ar[r]\ar[d]^{\pi_v(1-z^{-1 } ) } & b(v^*)\ar@{.>}[d]^{\phi_v } \ar[r ] & 0 \\ 0 \ar[r ] & ( v[f_\mu])^*\ar[r]^{\sigma(v)^ * } & ( v[f_\mu])^ * \ar[r ] & b(v)^\wedge \ar[r ] & 0 } \end{gathered}\ ] ] where @xmath747 and @xmath748 . by definition , @xmath749 is the induced morphism . plainly @xmath750 is a natural transformation . the duality - preserving functor @xmath403 has the property @xmath751 } = \pi_{v^*}i_v : v[f_\mu]\to ( v^*[f_\mu])^*\ ] ] ( indeed , this equation features in definition  [ define_dpres_funct ] ) . the equations @xmath752 } = \pi_{v^*}(1-z ) i_v \\ ( \pi_v(1-z))^ * i_{v[f_\mu ] } = \pi_{v^*}(1-z^{-1 } ) i_v\end{aligned}\ ] ] imply that @xmath753 to show that @xmath121 is a duality - preserving functor it remains to check that @xmath754 is an isomorphism . there is a commutative diagram @xmath755 & ( v[f_\mu])^*\ar[d]_{-(1-s^*)\pi_v^{-1}}\ar[r]^{\sigma(v)^ * } & ( v[f_\mu])^ * \ar[r]\ar[d]^{-(1-s^*)z\pi_v^{-1 } } & b(v)^\wedge\ar@{.>}[d ] \ar[r ] & 0 \\ 0 \ar[r ] & v^*[f_\mu]\ar[r]^{\sigma(v^ * ) } & v^*[f_\mu ] \ar[r ] & b(v^ * ) \ar[r ] & 0 } \ ] ] and the composite morphisms of chain complexes @xmath756\ar[d]_{(1-s^*)(1-z)}\ar[r]^{\sigma(v^ * ) } & v^*[f_\mu]\ar@{.>}[dl]_1 \ar[d]^{(1-s^*)(1-z ) } \\ v^*[f_\mu]\ar[r]_{\sigma(v^ * ) } & v^*[f_\mu ] } \\ \intertext{and } & \xymatrix@c=4ex@r=3ex{(v[f_\mu])^*\ar[d]_{(1-z^{-*})(1-s^*)}\ar[r]^{\sigma(v)^ * } & ( v[f_\mu])^*\ar@{.>}[dl]_{z^{- * } } \ar[d]^{-(1-z^*)(1-s^*)z^{- * } }   \\ ( v[f_\mu])^*\ar[r]_{\sigma(v)^ * } & ( v[f_\mu])^ * } \end{aligned}\ ] ] are chain homotopic to the identity by the indicated chain homotopies . [ chain complexes are drawn horizontally and morphisms of chain complexes are given by vertical arrows . the symbol @xmath757 is shorthand for @xmath758 . these composite chain maps therefore induce the identity on @xmath759 and @xmath760 respectively so @xmath750 is an isomorphism and @xmath121 is a duality - preserving functor . since @xmath4 is an exact functor we have :    the functor @xmath121 induces a homomorphism of witt groups @xmath761    if @xmath344 is a @xmath342-hermitian form in @xmath99 then the covering construction gives @xmath762 ( lemma  [ duality_preserving_functor_induces ] ) which can be described explicitly as follows . the morphism @xmath65 induces @xmath763\to v^*[f_\mu]$ ] . let @xmath764\to \operatorname{hom}_{a[f_\mu]_\sigma}(v[f_\mu]_\sigma , a[f_\mu]_\sigma)$ ] denote the composition of @xmath77 with @xmath765\to ( v[f_\mu])^*=\operatorname{hom}_{a[f_\mu]}(v[f_\mu],a[f_\mu])\ ] ] ( see example  [ simplest_ring_change_example ] ) and the localization @xmath766}(v[f_\mu],a[f_\mu])\to \operatorname{hom}_{a[f_\mu]_\sigma}(v[f_\mu]_\sigma , a[f_\mu]_\sigma).\ ] ] if @xmath767 we may write @xmath768 , @xmath769 for some @xmath770 $ ] . it follows from remark  [ explicit_formula_for_mwedge_pres ] that in @xmath771_\sigma}{a[f_\mu]}}$ ] we have @xmath772    [ remark_that_b(seifert_form)=bdform ] if @xmath773 and @xmath65 is the seifert form corresponding to a seifert surface for an @xmath10-link then by  ( [ formula_for_bd_form ] ) @xmath774 is the corresponding blanchfield - duval form for the @xmath10-link ; compare kearton  @xcite , levine  ( * ? ? * prop 14.3 ) , cochran and orr  ( * ? ? ? * thm4.2 ) and ranicki  ( * ? ? ? * defn32.7 ) . for example , setting @xmath775 , @xmath776 , @xmath777 , @xmath778 , and @xmath779 one obtains from  ( [ formula_for_bd_form ] ) the equations appearing immediately prior to theorem 4.2 in  @xcite .    for the proof of theorem  [ theorem_relating_invariants : vague ] in section  [ section : proof_that_b_induces_witt_isomorphism ] we will need the observation that @xmath121 respects a change of coefficients from @xmath60 to @xmath84 . let us make a more general statement . recall from equation  ( [ how2_compose_dpres_functors ] ) the definition of composition for duality - preserving functors . [ b_respects_coefficient_change ] a ring homomorphism @xmath226 induces a diagram of duality - preserving functors @xmath780_{(b,\phi,-1 ) } \ar[r]^{(a'\otimes_a\functor,\pi,1 ) } & { \sei(a')}\ar[d]^{(b,\phi,-1 ) } \\ { \bl(a)}\ar[r]_{(a'[f_\mu]\otimes_{a[f_\mu]}\functor,\upsilon,1 ) } & { \bl(a')}. } \ ] ] which commutes up to natural isomorphism . consequently , there is a commutative diagram of witt groups @xmath781_b \ar[r ] & w^\zeta(\sei(a'))\ar[d]^b \\ w^{-\zeta}(\bl(a))\ar[r ] & w^{-\zeta}(\bl(a ' ) ) . } \end{gathered}\ ] ]    see appendix  [ section : naturality_of_b ] . we leave duality structures behind for the present and prove that the functor @xmath151 is left adjoint to @xmath150 . [ define_left_adjoint ] suppose @xmath782 is a functor . a functor @xmath783 is called _ left adjoint _ to @xmath130 if there exists a natural transformation @xmath784 such that for every object @xmath785 the morphism @xmath786 has the following universal property : for every morphism @xmath787 in @xmath788 of the form @xmath789 there is a unique morphism @xmath790 in @xmath262 such that @xmath791 . @xmath792^d\ar[dr]_{\theta_d } & & f(c ) \\ & fg(d ) \ar@{.>}[ur]_{f(c ) } & } \end{gathered}\ ] ]    let us recall a few examples : 1 ) the inclusion of the category of abelian groups in the category of groups has left adjoint known as `` abelianization '' which sends a group @xmath382 to @xmath793 $ ] . 2 ) the inclusion of the category of compact hausdorff topological spaces in the category of ( all ) topological spaces has a left adjoint known as `` stone - ech compactification '' . 3 ) colimit constructions ( e.g. direct limit or coproduct ) can be expressed via a left adjoint as follows . suppose @xmath262 is a category , @xmath794 is a small category and @xmath795 denotes the category of functors @xmath796 . if there is a left adjoint to the constant functor @xmath797 then that left adjoint sends each functor @xmath796 to its colimit in @xmath262 ( and the colimit exists ) . [ b_is_left_adjoint_to_u ] the functor @xmath129 is left adjoint to @xmath150 .    the required map @xmath798 is the restriction of the map @xmath799 in the diagram ( [ b_of_morphism ] ) above . in symbols @xmath800 . during the proof of proposition  [ b_is_left_adjoint_to_u ] below we show that @xmath801 is a morphism of seifert modules . it follows from the diagram  ( [ b_of_morphism ] ) that @xmath802 is a natural transformation .    before proving proposition  [ b_is_left_adjoint_to_u ] , we note some consequences :    [ deductions_from_left_adjointness ] let @xmath122 and @xmath228 . there is a natural isomorphism @xmath705 and the composites @xmath803 are identity morphisms . in particular @xmath804 and @xmath805 are isomorphisms . the existence of a natural transformation @xmath705 follows from proposition  [ b_is_left_adjoint_to_u ] alone . to prove that @xmath806 is an isomorphism one requires the additional information that @xmath113 is full and faithful . we are not claiming that @xmath798 is an isomorphism . indeed , @xmath113 and @xmath4 are not equivalences of categories . let @xmath705 be the unique morphism such that @xmath807 . one can check that @xmath808 is a natural transformation and that @xmath809 ( see for example theorem 3.1.5 of  @xcite ) . the functor @xmath150 is full and faithful by lemma  [ bl_infty_is_full_subcat_of_sei_infty ] so @xmath808 is an isomorphism ( see theorem 3.4.1 of  @xcite ) . it follows that @xmath810 and @xmath805 are isomorphisms .    by definition  [ define_left_adjoint ] there are two statements to prove :    1 . the map @xmath798 is a morphism of seifert modules . if @xmath228 and @xmath811 is a morphism in @xmath98 then there is a unique morphism @xmath812 such that @xmath813 .    as we remarked above , it follows from the diagram  ( [ b_of_morphism ] ) that @xmath802 is a natural transformation . we shall need the following lemma which is proved shortly below :    [ relating_seifert_structures ] suppose @xmath122 , @xmath228 and @xmath814 is an @xmath96-module morphism . let @xmath815\to m$ ] denote the induced @xmath156$]-module morphism . the map @xmath811 is a morphism in @xmath98 if and only if @xmath816 for all @xmath252 . let us deduce statement 1 . above . by the definition of @xmath116 there is an exact sequence @xmath817\xrightarrow{1-s(1-z)}v[f_\mu]\xrightarrow{q } b(v)\to 0\ ] ] so @xmath818 for all @xmath252 . by lemma  [ relating_seifert_structures ] , @xmath819 is a morphism of seifert modules . we turn now to statement 2 . since @xmath61 generates @xmath116 as an @xmath156$]-module , an @xmath156$]-module morphism @xmath812 satisfies @xmath813 if and only if @xmath820 fits into the diagram @xmath821 & v[f_\mu ] \ar[r]^{1-s(1-z ) } & v[f_\mu ] \ar[r]^q   \ar[dr]_{\widetilde{f } }   & b(v )   \ar@{.>}[d]^g   \ar[r ] & 0   \\ & & & m & } \ ] ] since @xmath510 is a morphism of seifert modules we have @xmath816 for all @xmath252 by lemma  [ relating_seifert_structures ] . so @xmath822 , and therefore there exists unique @xmath812 such that @xmath823 . it follows that there exists unique @xmath820 such that @xmath813 . thus we have established both 1 .  and 2 . assuming lemma  [ relating_seifert_structures ] . the seifert module structure on @xmath106 is given by  ( [ seifert_structure_on_f - link_module ] ) above so @xmath510 is a seifert morphism if and only if + a ) @xmath824  and + b ) @xmath825 for each @xmath252 .    to prove the ` if ' part of lemma  [ relating_seifert_structures ] , suppose @xmath816 . + a ) the equations @xmath826 and @xmath827 imply that @xmath828 . + b ) observe that @xmath829 while @xmath830 thus @xmath831 .    to prove the `` only if '' part of lemma  [ relating_seifert_structures ] , suppose we have a ) and b ) above . now @xmath832    this completes the proof of proposition  [ b_is_left_adjoint_to_u ] .      when one passes from seifert modules to @xmath10-link modules , certain seifert modules disappear altogether ; following farber we shall call such modules primitive . [ define_primitive ] let @xmath124 denote the full subcategory of @xmath98 containing precisely the modules @xmath61 such that @xmath833 . modules in @xmath124 will be called primitive . for example , if @xmath834 or @xmath835 then @xmath836 \to v[f_\mu(y^2)]\ ] ] is an isomorphism and therefore has zero cokernel . a module in @xmath98 with @xmath834 or @xmath5 will be called _ trivially primitive_. we show in section  [ section : structure_of_primitives ] that all the primitive seifert modules in @xmath98 can be `` built '' from trivially primitive modules . if @xmath96 is semi - simple artinian then a similar result applies when one restricts attention to the category @xmath99 of representations of @xmath626 by finitely generated projective @xmath96-modules : every primitive in @xmath99 can be `` built '' from a finite number of trivially primitive modules in @xmath99 ( see proposition  [ structure_of_artinian_finite_primitives ] ) . this statement is not true for all rings @xmath96 ; one must consider primitives which exhibit a kind of nilpotence . such primitives were described by bass , heller and swan when @xmath56 ( see also ranicki  @xcite ) . the general case @xmath95 will be analyzed in a subsequent paper ( joint work with a.ranicki ) .    in the present section we construct an equivalence between @xmath114 and a quotient category @xmath128 . this quotient is an example of universal localization for categories ; the objects in @xmath128 are the same as the objects in @xmath98 but the morphisms are different . the universal property is that a morphism in @xmath98 whose kernel and cokernel are in @xmath124 has an inverse in @xmath128 . a more detailed construction of the quotient appears in section  [ section : construction_of_quotient ] . we proceed to derive an equivalence between @xmath101 and a corresponding quotient of @xmath99 . [ universal_localization_of_sei_infty ] the functor @xmath837 is the universal functor which makes invertible all morphisms whose kernel and cokernel are primitive . in other words , any functor which makes these morphisms invertible factors uniquely through @xmath130 . we outline in section  [ section : construction_of_quotient ] one construction of @xmath130 which will be convenient for our purposes ; see gabriel  @xcite or srinivas  ( * ? ? ? * appendix b.3 ) for further details . a more general construction can be found in gabriel and zisman  @xcite or borceux  ( * ? ? ? it follows directly from the definition that the localization @xmath130 is unique ( up to unique isomorphism ) . applying definition  [ universal_localization_of_sei_infty ] to the functor @xmath129 , there is a unique functor @xmath838 such that @xmath839 : @xmath840^b \ar[r]_f & { \displaystyle{\frac{\sei_\infty(a)}{\prim_\infty(a ) } } } \ar[r]_{\overline{b } } & { \bl_\infty(a ) } \ar@/^2pc/[ll]^u } \ ] ]    proposition  [ b_is_left_adjoint_to_u ] stated that @xmath129 is left adjoint to the forgetful functor @xmath113 . we deduce in the next proposition that @xmath4 satisfies the same universal property as @xmath130 , but only `` up to natural isomorphism '' . if @xmath510 is a morphism in @xmath98 let us write @xmath841 if the kernel and cokernel of @xmath510 both lie in @xmath124 . [ b_is_universal_localization_up_to_equivalence ] if @xmath842 is a functor which sends every morphism in @xmath843 to an invertible morphism then there is a functor @xmath844 such that @xmath845 is naturally isomorphic to @xmath382 . the functor @xmath846 is unique up to natural isomorphism . we prove uniqueness first . if there is a natural isomorphism @xmath847 then @xmath848 by corollary  [ deductions_from_left_adjointness ] .    to prove existence we must show that if @xmath849 then @xmath850 . indeed , by corollary  [ deductions_from_left_adjointness ] @xmath851 is an isomorphism for each @xmath122 . since @xmath4 respects exact sequences we have @xmath852 . it follows that @xmath853 is a natural isomorphism . the following is an immediate consequence of the fact that @xmath130 and @xmath4 have the same universal property ( up to natural isomorphism ) :    [ bbar_is_equiv_on_infty_cats ] the functor @xmath854 is an equivalence . we turn now to the categories @xmath99 and @xmath101 . let @xmath855 denote the full subcategory whose objects are precisely the modules in @xmath99 ( i.e.  the modules which are finitely generated and projective as @xmath96-modules ) . there is a commutative diagram of functors @xmath856^b \ar[r]_f & { \displaystyle{\frac{\sei_\infty(a)}{\prim_\infty(a ) } } } \ar[r]_{\overline{b } } & { \bl_\infty(a ) } \\   { \sei(a ) }   \ar[u ] \ar@/_2pc/[rr]_b \ar[r]_f & { \displaystyle{\frac{\sei(a)}{\prim_\infty(a ) } } } \ar[u]\ar[r]_{\overline{b } } & { \bl(a ) } \ar[u ] } \end{gathered}\ ] ] in which all the vertical arrows are inclusions of full subcategories . [ bbar_is_equiv_on_fgproj_cats ] the functor @xmath857 is an equivalence of categories .    we will use the following general lemma in the proof of theorem  [ bbar_is_equiv_on_fgproj_cats ] . recall that a functor @xmath377 is called full and faithful if it induces an isomorphism @xmath858 for every pair of objects @xmath859 . [ describe_cat_equivalence ] a functor @xmath377 is an equivalence of categories if and only if @xmath382 is full and faithful and every object in @xmath788 is isomorphic to @xmath860 for some @xmath266 .    see for example borceux  ( * ? ? ? * prop 3.4.3 ) . it follows from corollary  [ bbar_is_equiv_on_infty_cats ] and lemma  [ describe_cat_equivalence ] that @xmath861 is full and faithful . theorem  [ bbar_is_equiv_on_fgproj_cats ] is therefore a consequence of the following proposition :    [ b_is_surjective ] every module in @xmath101 is isomorphic to @xmath116 for some @xmath704 .    by definition , every module @xmath97 has presentation @xmath817\xrightarrow{\sigma } v[f_\mu]\to m \to 0\ ] ] where @xmath61 is a finitely generated projective @xmath96-module and @xmath170 is an isomorphism . given any @xmath96-module @xmath862 there is a canonical isomorphism @xmath766}(v[f_\mu],w[f_\mu])\cong \operatorname{hom}_a(v , w)[f_\mu]\ ] ] and in particular @xmath172 can be expressed uniquely as a sum @xmath863 with each @xmath864 . [ linearize ] every @xmath97 is isomorphic to the cokernel of an endomorphism @xmath218\to v[f_\mu]$ ] of the form @xmath865 where @xmath61 is finitely generated and projective and @xmath866 .    by the definition of @xmath101 , the module @xmath102 is isomorphic to the cokernel of some map @xmath218\to v[f_\mu]$ ] where @xmath61 is finitely generated and projective . the idea of this proof is to reduce the support of @xmath172 to @xmath867 at the expense of replacing @xmath61 by a larger finitely generated projective module . note first that @xmath868\to ( v\oplus v')[f_\mu]\ ] ] where @xmath178 is any @xmath96-module and @xmath5 denotes the identity morphism . the equation @xmath869 therefore implies that @xmath870 is isomorphic to @xmath871 . repeated application of equation  ( [ gaussian_elimination ] ) implies that @xmath102 is isomorphic to the cokernel of an endomorphism @xmath872 with @xmath873 , @xmath874 and @xmath875 in @xmath876 for some @xmath61 . for each of the indices @xmath646 in turn , one can apply the identity @xmath877 followed by further equations  ( [ gaussian_elimination ] ) . one obtains an identity @xmath878 where @xmath879 and @xmath880 for some finitely generated projective module @xmath61 over @xmath96 . by lemma  [ sato_condition ] , @xmath881 is an isomorphism . let @xmath882 . now @xmath883 and so @xmath884 for some @xmath866 . this completes the proof of lemma  [ linearize ] . we may now finish the proof of proposition  [ b_is_surjective ] . if @xmath885 then the equation @xmath886 implies that @xmath887 \to v^{\oplus\mu}[f_\mu]\right)\ ] ] where @xmath663 acts as projection on the @xmath16th component of @xmath888 and @xmath133 acts as @xmath889 thus @xmath890 . this completes the proof of theorem  [ bbar_is_equiv_on_fgproj_cats ] . having established that @xmath891 is an equivalence , we may use @xmath838 to give duality structure to @xmath892 and make the lower part of  ( [ factor_b_through_quotient_diagram ] ) a commutative diagram of duality - preserving functors . since the objects in @xmath892 coincide with those in @xmath99 we define @xmath893 and @xmath894 where @xmath895 is the canonical functor . if @xmath655 is a morphism in @xmath892 let @xmath896 it is easy to see that @xmath267 is a contravariant functor and that @xmath265 for all @xmath61 so @xmath892 is a hermitian category . recall that the composite of duality - preserving functors is defined by @xmath897    [ composite_of_d_pres_functors ] the duality - preserving functor @xmath898 coincides with the composite @xmath899 . it follows from equation  ( [ duality_on_morphisms_in_sei / prim_infty ] ) and proposition  [ extend_b_to_duality_pres_functor ] that @xmath900 is a duality - preserving functor .    by definition @xmath893 and @xmath901 ; to show that @xmath902 is a duality - preserving functor we must check that @xmath903 for each morphism @xmath655 in @xmath99 . indeed , @xmath904 it is easy to verify that @xmath905 . [ bbar_is_eq_of_hermitian_cats ] let @xmath340 or @xmath341 . the duality - preserving functor @xmath906 is an equivalence of hermitian categories and induces an isomorphism of witt groups @xmath907    since @xmath838 is an equivalence of categories ( by theorem  [ bbar_is_equiv_on_fgproj_cats ] above ) it follows that @xmath900 is an equivalence of hermitian categories ( see proposition ii.7 of  @xcite ) . it also follows that @xmath838 preserves limits and colimits so @xmath838 preserves exact sequences and hence induces a homomorphism  ( [ isomorphism_of_witt_groups_induced_by_bbar ] ) of witt groups . by lemma  [ nat_isomorphic_dpres_funct_give_same_witt_homo ] of appendix  [ section : naturality_of_b ] this homomorphism is an isomorphism  ( [ isomorphism_of_witt_groups_induced_by_bbar ] ) . recall that a module @xmath629 in @xmath98 is called trivially primitive if @xmath834 or @xmath835 . in this section we prove that every primitive module in @xmath98 is composed of trivially primitive modules . [ s=0_submodule_lemma ] if @xmath908 and there exists a non - zero element @xmath252 such that @xmath909 for all @xmath16 then @xmath61 has a non - zero submodule @xmath910 such that @xmath911 . note that @xmath912 and at least one of the terms @xmath913 must be non - zero . choose non - zero @xmath914 and define @xmath915 by @xmath916 now @xmath910 is the required non - zero submodule of @xmath629 . [ primitive_has_s=1_or_s=0_submodule ] if @xmath629 is primitive and non - zero then there exists a non - zero trivially primitive submodule @xmath910 . ( compare lemma  7.10c in farber  @xcite ) since @xmath629 is primitive , @xmath917\to v[f_\mu(y^2)]\ ] ] is an isomorphism with inverse @xmath20 say . now @xmath20 can be written as a finite sum @xmath918 where @xmath550 is a finite subset of @xmath919 and @xmath920 has non - zero image in @xmath921 for each @xmath922 . choose an element @xmath922 whose expression in reduced form as a product of letters @xmath923 is of maximal length . we consider two cases : + * case 1 : @xmath924 * for some @xmath925 and some @xmath16 . the equation @xmath926 implies that @xmath927 for each @xmath158 . any element @xmath928 in the image of @xmath929 satisfies the conditions of lemma  [ s=0_submodule_lemma ] so @xmath629 has a non - zero submodule @xmath910 with @xmath911 . + * case 2 : @xmath930 * for some @xmath925 . the equation  ( [ linear_sigma_inverted ] ) implies that @xmath931 for each @xmath16 . by lemma  [ s=0_submodule_lemma ] there is a non - zero submodule @xmath910 with @xmath932 or in other words @xmath933 . recall that a module @xmath61 is called simple if there are no submodules other than @xmath127 and @xmath61 . the following remark is a consequence of lemma  [ primitive_has_s=1_or_s=0_submodule ] . [ every_simple_primitive_module_is_trivially_primitive ] every simple primitive module is trivially primitive .    if @xmath350 is an abelian category then a non - empty full subcategory @xmath934 is called a _ serre subcategory _ if for every exact sequence @xmath935 in @xmath350 one has @xmath936    note that every serre subcategory of an abelian category is again an abelian category . since @xmath4 preserves exact sequences and arbitrary direct sums @xmath124 is a serre subcategory of @xmath98 and is closed under direct sums .    [ maximal_primitive_submodule ] suppose @xmath937 is a serre subcategory of an abelian category and @xmath938 is closed under arbitrary direct sums . every module @xmath939 contains a unique maximal submodule @xmath940 such that @xmath941 . if @xmath942 and @xmath943 then @xmath944 . let @xmath113 be the sum in @xmath61 of all the submodules @xmath945 with @xmath946 . since @xmath113 is a factor module of @xmath947 , one finds @xmath941 . [ identify_prim_infty ] the category @xmath124 is the smallest serre subcategory of @xmath98 which a ) contains the trivially primitive modules and b ) is closed under arbitrary direct sums . let @xmath948 denote the smallest full subcategory of @xmath98 satisfying the conditions of the proposition . now @xmath124 satisfies these conditions so @xmath949 . conversely , we must show that @xmath950 . suppose @xmath833 . let @xmath951 be the maximal submodule such that @xmath952 ( the module @xmath862 exists by lemma  [ maximal_primitive_submodule ] ) . now @xmath953 since @xmath4 respects exact sequences so lemma  [ primitive_has_s=1_or_s=0_submodule ] implies that either @xmath954 or there is a non - zero submodule @xmath178 of @xmath955 which lies in @xmath948 . in the latter case , let @xmath956 denote the projection and note the exact sequence @xmath957 since @xmath952 and @xmath958 we have @xmath959 which contradicts the maximality of @xmath862 . thus @xmath954 and hence @xmath960 , so @xmath61 lies in @xmath948 . we outline next a construction of @xmath128 . we will use this construction in section  [ section : proof_that_b_induces_witt_isomorphism ] to show that @xmath961 is an isomorphism when @xmath96 is a semi - simple artinian ring . the notion of serre subcategory was defined in the preceding section . let us note some basic properties :    [ add_and_coadd_in_serre_subcat ] suppose @xmath350 is an abelian category and @xmath938 is a serre subcategory . suppose @xmath939 , @xmath951 and @xmath962 .    1 .   [ add_in_serre_subcat ] if @xmath963 and @xmath964 then @xmath965 . [ coadd_in_serre_subcat ] if @xmath966 and @xmath967 then @xmath968 . [ add_in_serre_subcat ] . there is an exact sequence @xmath969 and @xmath970 is isomorphic to @xmath971 . hence @xmath972 . + [ coadd_in_serre_subcat ] . there is an exact sequence @xmath973 now @xmath974 is isomorphic to @xmath975 which is contained in @xmath976 and so @xmath977 and hence @xmath978 . we may now recall a construction for the quotient of an abelian category by a serre subcategory . see gabriel  @xcite or srinivas  ( * ? ? ? * appendix b.3 ) for further details . suppose @xmath350 is an abelian category and @xmath938 is a serre subcategory . the symbol @xmath979 will denote a category with the same objects as @xmath350 but different groups of morphisms . to define @xmath980 , consider the pairs @xmath981 where @xmath951 , @xmath982 , @xmath966 and @xmath983 . one says that @xmath984 if @xmath985 and @xmath986 [ note the directions of inclusion ] . lemma  [ add_and_coadd_in_serre_subcat ] above implies that these pairs are a directed set . indeed , given pairs @xmath987 and @xmath988 one finds @xmath989 and @xmath990 . the following definition can now be made : @xmath991 we leave to the reader the definition of composition of morphisms and the canonical functor @xmath992 . proofs of the following statements can be found in the references cited above :    1 .   the quotient category @xmath979 is an abelian category and @xmath130 is an exact additive functor . [ morphisms_become_isomorphisms ] if @xmath510 is a morphism in @xmath350 then @xmath993 is an isomorphism if and only if @xmath994 and @xmath995 . in particular if @xmath61 is a module in @xmath350 then @xmath996 if and only if @xmath997 . as we indicated in earlier sections , the functor @xmath998 is universal with respect to property  ( b ) . in detail , if @xmath999 makes invertible every morphism whose kernel and cokernel lie in @xmath938 then there is a unique functor @xmath1000 such that @xmath1001 . in particular the functor @xmath1002 satisfies definition  [ universal_localization_of_sei_infty ] . let us be explicit about @xmath846 :    if @xmath61 is an module in @xmath979 then one writes @xmath1003 . every morphism @xmath1004 is represented by some @xmath1005 with @xmath983 and @xmath966 if @xmath1006 and @xmath1007 denote the canonical monomorphism and epimorphism respectively one must define @xmath1008    in our particular example lemma  [ maximal_primitive_submodule ] provides one simplification in our description of the quotient category @xmath128 . if @xmath1009 let us call a submodule @xmath951 _ coprimitive _ if @xmath1010 . [ half_simplify_quotient ] if @xmath1011 and @xmath1012 denotes the maximal primitive submodule of @xmath178 then @xmath1013 where the direct limit is over coprimitive submodules @xmath862 of @xmath61 . note that there is not in general a minimal coprimitive in @xmath61 ; the functor @xmath4 does not respect infinite limits and an infinite intersection of coprimitives is not in general coprimitive ( but see lemma  [ max_prim_min_coprim_assuming_finiteness ] below ) . _ in this section the ring @xmath96 will be assumed semi - simple and artinian or , in other words , a finite product of matrix rings over division rings_. the basic theory of semi - simple artinian rings can be found in many algebra textbooks ( e.g.  lam  @xcite or lang  ( * ? ? ? * ch.xvii ) ) . in particular , all @xmath96-modules are projective and @xmath99 is an abelian category with ascending and descending chain conditions ; these facts lead to simplifications of results in sections  [ section : structure_of_primitives ] and  [ section : construction_of_quotient ] above . we show that the primitive modules in @xmath99 are composed of a finite number of simple trivially primitive modules ( proposition  [ structure_of_artinian_finite_primitives ] ) and give a simplified description of the hermitian category @xmath892 . we shall consider semi - simple artinian rings again in section  [ section : proof_that_b_induces_witt_isomorphism ] but it is not essential to read the present section before section  [ section : proof_that_b_induces_witt_isomorphism ] .    the key lemma we will need is the following :    [ max_prim_min_coprim_assuming_finiteness ] suppose @xmath350 is an abelian category with ascending and descending chain conditions and @xmath938 is a serre subcategory .    1 . every module @xmath939 contains a unique maximal submodule in @xmath938 which contains all others in @xmath938 . 2 .   every module @xmath939 contains a unique submodule @xmath951 which is minimal with respect to the property @xmath966 . if @xmath967 then @xmath1014 .    1 . since @xmath350 has the ascending chain condition there is a submodule @xmath940 which is maximal with respect to the property @xmath941 . in other words , if @xmath1015 and @xmath983 then @xmath1016 . if @xmath1012 is any other submodule in @xmath938 then @xmath1017 by lemma  [ add_and_coadd_in_serre_subcat ] so @xmath1018 and hence @xmath944 . + 2 .  since @xmath350 has the descending chain condition there is a submodule @xmath951 which is minimal with respect to the property @xmath966 ( i.e.  if @xmath1019 and @xmath967 then @xmath1020 ) . if @xmath967 then @xmath968 by lemma  [ add_and_coadd_in_serre_subcat ] so @xmath1021 and hence @xmath1014 . recall that if @xmath704 , a submodule @xmath951 is called coprimitive if @xmath955 is primitive . since @xmath96 is artinian and noetherian , lemma  [ max_prim_min_coprim_assuming_finiteness ] implies that there is a maximal primitive submodule @xmath940 and a minimal coprimitive submodule @xmath951 for each @xmath704 . let @xmath1022 denote the intersection of @xmath124 and @xmath99 . in other words , @xmath1023 is the full subcategory containing those modules @xmath61 such that @xmath833 . note that @xmath1022 is both a serre subcategory and a hermitian subcategory of @xmath99 . moreover , @xmath1024 . [ structure_of_artinian_finite_primitives ] the category @xmath1022 is the smallest serre subcategory of @xmath99 which contains the trivially primitive modules in @xmath99 . we proceed as in the proof of proposition  [ identify_prim_infty ] , using lemma  [ max_prim_min_coprim_assuming_finiteness ] in place of lemma  [ maximal_primitive_submodule ] . let @xmath1025 denote the smallest serre subcategory of @xmath99 which contains all the trivially primitive modules . to show that @xmath1026 it suffices to observe that @xmath1022 is a serre subcategory which contains these modules . conversely , to show @xmath1027 suppose @xmath1028 . there exists , by lemma  [ max_prim_min_coprim_assuming_finiteness ] , a maximal submodule @xmath940 such that @xmath1029 . now @xmath1030 so either @xmath1031 or by lemma  [ primitive_has_s=1_or_s=0_submodule ] @xmath1032 has a non - zero trivially primitive submodule @xmath1012 . if @xmath1033 is the canonical map then the exact sequence @xmath1034 implies that @xmath1035 contradicting the maximality of @xmath113 . thus @xmath1031 and @xmath1036 .      with the benefit of lemma  [ max_prim_min_coprim_assuming_finiteness ] we can give a simpler description of the quotient category than lemma  [ half_simplify_quotient ] . [ fully_simplify_quotient ] the morphisms in @xmath1037 are @xmath1038 where @xmath951 is the minimal coprimitive and @xmath982 is the maximal primitive . we simplify next the hermitian structure on @xmath1037 . we have seen that the duality - preserving functor @xmath131 factors through an equivalence of hermitian categories @xmath900 : @xmath1039^{(b,\phi,-1 ) } \ar[r]_{(f,\operatorname{id},1 ) } & { \frac{\sei(a)}{\prim(a ) } } \ar[r]_{(\overline{b},\phi,-1 ) } & { \bl(a ) } } \ ] ] ( theorem  [ bbar_is_equiv_on_fgproj_cats ] and proposition  [ composite_of_d_pres_functors ] above ) . the duality functor on the quotient @xmath1037 was defined in section  [ section : duality_on_quotient_arbitrary_a ] above by @xmath893 and by equation  ( [ duality_on_morphisms_in_sei / prim_infty ] ) . using the assumption that @xmath96 is artinian we can re - interpret equation  ( [ duality_on_morphisms_in_sei / prim_infty ] ) . suppose that @xmath1040 . as usual , let @xmath862 denote the minimal coprimitive submodule of @xmath61 and let @xmath1012 denote the maximal primitive submodule of @xmath178 . the morphism @xmath510 is identified with some @xmath1041 . since @xmath267 preserves exact sequences the following are exact @xmath1042 now @xmath1043 is a duality - preserving functor , so for each @xmath704 one has @xmath833 if and only if @xmath1044 . thus @xmath1022 is a hermitian subcategory of @xmath99 and in particular @xmath1045 and @xmath1046 are primitive . it follows that @xmath1046 is the maximal primitive in @xmath349 and @xmath1047 is the minimal coprimitive in @xmath1048 . since @xmath1049 is a duality - preserving functor , @xmath1050 is represented by @xmath1051 cobordism invariants of @xmath10-links have been defined in two different ways in sections  [ section : blanchfield_form_invariants ] and  @xcite . in this section we use the duality - preserving functor @xmath121 which was studied in section  [ section : covering_construction ] to relate the two approaches , proving theorems  [ main_covering_construction_theorem ] and  [ theorem_relating_invariants : vague ] . to prove theorem  [ theorem_relating_invariants : vague ] we show that the functor @xmath4 respects each of the three steps laid out in sections  [ section : overview_of_bd_form_invariants ] and [ section : cobordism_invariants_seifert ] . a more detailed version of theorem  [ theorem_relating_invariants : vague ] is set out in theorem  [ theorem_relating_invariants : precise ] below . suppose @xmath350 is an abelian category with ascending and descending chain conditions and @xmath938 is a serre subcategory . let @xmath992 denote the quotient functor . recall that a module @xmath61 in @xmath350 is called simple if @xmath61 is not isomorphic to @xmath127 and @xmath61 does not have submodules other than @xmath127 and @xmath61 . [ simple_goes_to_comes_from_simple ]    1 . if @xmath939 is simple then either @xmath1052 or @xmath1053 is simple . every simple module in @xmath979 is isomorphic to @xmath1053 for some simple module @xmath939 which does not lie in @xmath938 .    1 . suppose @xmath939 is simple , @xmath1054 and @xmath1055 is the inclusion of a submodule in @xmath979 . now @xmath16 is represented by some morphism @xmath1056 where @xmath1057 and @xmath1058 . either @xmath1059 in which case @xmath1060 in @xmath979 or @xmath1061 is an epimorphism which implies that @xmath1062 ( recall that @xmath130 is exact ) . every module in @xmath979 is @xmath1053 for some module @xmath939 . suppose @xmath1053 is simple . now @xmath61 has a finite filtration @xmath1063 where each quotient @xmath1064 is a simple module . since @xmath130 respects exact sequences @xmath1065 for all @xmath1066 except one , for which there is an isomorphism @xmath1067 . this module @xmath1064 does not lie in @xmath938 . suppose now that @xmath350 and @xmath979 are hermitian categories and the quotient functor extends to a duality - preserving functor @xmath1068    [ quotient_lemma_with_duality ]    1 . the serre subcategory @xmath938 is a hermitian subcategory . 2 .   let @xmath340 or @xmath341 . if @xmath939 is simple and @xmath1054 then @xmath61 is @xmath342-self - dual if and only if @xmath1053 is @xmath342-self - dual .    1 . if @xmath1052 then @xmath1069 so @xmath1070 . + 2 .  to prove the `` only if '' part it suffices to recall that for @xmath1071 one has @xmath1072 . for the `` if '' part , note also that @xmath1073 is an isomorphism . [ witt_isomorphism_for_hermitian_quotient ] suppose @xmath350 and @xmath979 are hermitian categories and @xmath1074 is a duality - preserving functor . for each @xmath342-self - dual simple module @xmath939 such that @xmath1054 there is a canonical isomorphism @xmath1075 if @xmath350 has ascending and descending chain conditions then there is a canonical isomorphism @xmath1076    if @xmath939 is a simple module and @xmath1054 then @xmath1053 is simple by part 1 .  of lemma  [ simple_goes_to_comes_from_simple ] . every module in @xmath1077 is a direct sum of copies of @xmath61 so by equation  ( [ define_morphisms_in_quotient ] ) the restriction @xmath1078 is a full and faithful functor and hence an equivalence of categories .    by part 2 .  of lemma  [ quotient_lemma_with_duality ] , @xmath61 is @xmath342-self - dual if and only if @xmath1053 is @xmath342-self - dual , in which case @xmath1079 is an equivalence of hermitian categories and induces an isomorphism @xmath1080    to prove the last sentence of the lemma , note first that by part 1 .  of lemma  [ quotient_lemma_with_duality ] , @xmath938 is a hermitian subcategory of @xmath350 . theorem  [ hermitian_devissage ] provides canonical decompositions @xmath1081 where the right hand side of each identity has one summand for each isomorphism class of @xmath342-self - dual simple modules @xmath61 .    by part 2 .  of lemma  [ simple_goes_to_comes_from_simple ] and part 2 . of lemma  [ quotient_lemma_with_duality ] every summand of @xmath1082 is the isomorphic image of @xmath1083 for some simple @xmath342-self - dual module @xmath61 in @xmath350 .    on the other hand , if @xmath1052 is simple and @xmath342-self - dual then @xmath1049 sends @xmath1083 to zero . the last sentence of the lemma follows .    in our application , we set @xmath1084 and @xmath1085 where @xmath96 is semi - simple artinian . recall that @xmath1086 is an abelian category with ascending and descending chain conditions and @xmath1024 . [ witt_group_of_primitives_is_zero ]    1 . none of the simple primitive modules in @xmath1022 are self - dual . 2 .   @xmath1087 .    by remark  [ every_simple_primitive_module_is_trivially_primitive ] above , every simple primitive module is trivially primitive . if @xmath1088 then @xmath834 if and only if @xmath1089 so none of the simple trivially primitive modules are self - dual . thus part 1 .  is proved , and part 2 .  follows immediately from theorem  [ hermitian_devissage ] .    by proposition  [ composite_of_d_pres_functors ] the duality - preserving functor @xmath121 is the composite @xmath899 . setting @xmath1084 and @xmath1085 in proposition  [ witt_isomorphism_for_hermitian_quotient ] , and invoking also lemma  [ witt_group_of_primitives_is_zero ] , we learn that @xmath1049 induces an isomorphism @xmath1090 by theorem  [ bbar_is_equiv_on_fgproj_cats ] , @xmath1091 is an equivalence and hence induces an isomorphism @xmath1092 ( see lemma  [ nat_isomorphic_dpres_funct_give_same_witt_homo ] in appendix  [ section : naturality_of_b ] ) . thus @xmath121 induces an isomorphism @xmath1093 this completes the proof of theorem  [ main_covering_construction_theorem ] . we prove in this section that the functor @xmath121 identifies the invariants defined in  @xcite with those of section  [ section : blanchfield_form_invariants ] . more precisely , we prove the following theorem :    [ theorem_relating_invariants : precise ]    1 . if @xmath1094 is simple and @xmath67-self - dual then @xmath1095 is simple and @xmath76-self - dual . every simple @xmath76-self - dual module @xmath417 is isomorphic to @xmath116 for some simple @xmath67-self - dual module @xmath1094 . if @xmath1094 and @xmath1095 are simple then the functor @xmath4 induces an isomorphism of rings @xmath1096 . 4 .   suppose @xmath1094 is simple and @xmath1097 is a @xmath342-hermitian form . the ring isomorphism in part 3 . respects involutions . explicitly , if @xmath1098 then @xmath1099 . suppose @xmath1100 and @xmath1101 is a @xmath67-hermitian form . the dimension modulo @xmath111 , signatures , discriminant , hasse - witt invariant and lewis @xmath13-invariant of @xmath1102\in w^1(\operatorname{end}_{\sei(\q)}(v))\ ] ] coincide ( if defined ) with the corresponding invariants of @xmath1103\in w^1(\operatorname{end}_{\bl(\q)}(b(v))).\ ] ]    recall that @xmath1104 and @xmath1105 is an equivalence of categories . in parts 1 .  through 3 .  of theorem  [ theorem_relating_invariants : precise ] it therefore suffices to prove corresponding statements with the functor @xmath130 in place of @xmath4 and @xmath67 in place of @xmath76 : + 1 . the statement follows from part 1 .  of lemma  [ simple_goes_to_comes_from_simple ] , part 2 . of lemma  [ quotient_lemma_with_duality ] and part 1 .  of lemma  [ witt_group_of_primitives_is_zero ] . the statement follows from part 2 .  of lemma  [ simple_goes_to_comes_from_simple ] and part 2 .  of lemma  [ quotient_lemma_with_duality ] . this is a consequence of equation  ( [ morphisms_in_sei / prim ] ) . + 4 . since @xmath1106 it suffices to prove that @xmath1107 this equation is a consequence of the fact that @xmath754 is a natural isomorphism . the proof of part 5 .  of theorem  [ theorem_relating_invariants : precise ] is slightly more involved . recall from proposition  [ b_respects_coefficient_change ] that @xmath4 respects changes of coefficients and , in particular , that the inclusion of @xmath60 in @xmath84 induces the commutative diagram ( [ b_respects_z_into_q ] ) . one must check that @xmath121 respects each of the three steps in the definitions of the @xmath10-link invariants ( see section  [ section : overview_of_bd_form_invariants ] ) . * devissage : * let @xmath1094 be a @xmath342-self - dual simple module . if @xmath862 is isomorphic to a direct sum of copies of @xmath61 then @xmath1108 is isomorphic to a direct sum of copies of @xmath116 . hence the image of @xmath1109 under @xmath4 lies in @xmath1110 and there is a commutative diagram of isomorphisms @xmath1111^b\ar@{<->}[d ] & w^{-\zeta}(\bl(\q))\ar@{<->}[d ] \\ { \displaystyle{\bigoplus_v w^\zeta(\sei(\q)|_v ) } } \ar[r]_(0.45){\bigoplus b| } & { \displaystyle{\bigoplus_{v } w^{-\zeta}(\bl(\q)|_{b(v)})}}.   } \end{gathered}\ ] ] where the direct sums are indexed by the isomorphism classes of simple @xmath342-self - dual modules in @xmath668 .    * morita equivalence : * suppose @xmath1094 is a simple module and @xmath1112 is a non - singular @xmath342-hermitian form . let us denote the endomorphism rings @xmath1113 and @xmath1114 . by corollary  [ witt_naturality_of_h_morita_eq ] above the duality - preserving functor @xmath121 induces a commutative diagram @xmath1115_{\theta_{v , b } } \ar[r]^(.45)b & w^{-\zeta}(\bl(\q)|_{b(v ) } )   \ar[d]^{\theta_{b(v),-\phi_v b(b ) } } \\ w^1(e ) \ar[r]_b & w^1(e ' ) . } \end{gathered}\ ] ] * invariants : * the isomorphism @xmath1116 in part 4 .  induces isomorphisms between the target groups for the invariants in part 5 . for example , if @xmath83 and @xmath1117 are commutative with trivial involution then the discriminant @xmath599 of @xmath1118 $ ] lies in @xmath1119 and the functor @xmath4 induces an isomorphism @xmath1120 . the word `` coincide '' in part 5 . is understood to mean that the image of @xmath599 in @xmath1121 is equal to the discriminant of @xmath1122 $ ] . the isomorphism @xmath1123 of rings with involution induces an isomorphism @xmath1124 . we leave it to the reader to check that if @xmath1125 then all the listed invariants of @xmath20 coincide ( in this sense ) with the corresponding invariants of @xmath1126 . further details of the invariants can be found in chapter 11 of  @xcite . this completes the proof of part 5 .  and hence of theorems  [ theorem_relating_invariants : vague ] and  [ theorem_relating_invariants : precise ] . in this appendix we prove naturality theorems for the covering construction @xmath121 and for hermitian morita equivalence , proving propositions  [ b_respects_coefficient_change ] and  [ naturality_of_h_morita_eq ] above . suppose @xmath409 and @xmath1127 are duality - preserving functors between hermitian categories @xmath262 and @xmath788 . a natural transformation @xmath1128 is a natural transformation between the underlying functors , @xmath1129 , such that @xmath1130 for each object @xmath266 . if @xmath1129 is a natural isomorphism between the underlying functors and @xmath20 satisfies  ( [ condition_for_nat_trans ] ) then @xmath1131 also satisfies  ( [ condition_for_nat_trans ] ) so @xmath20 is in fact a natural isomorphism of duality - preserving functors . we noted in lemma  [ duality_preserving_functor_induces ] that an exact duality - preserving functor induces a homomorphism of witt groups . the following lemma says that naturally isomorphic duality - preserving functors induce the same homomorphism on witt groups . [ nat_isomorphic_dpres_funct_give_same_witt_homo ] suppose @xmath1132 are duality - preserving functors which respect exact sequences and @xmath1133 is a natural isomorphism . if @xmath1134 is a hermitian form in @xmath262 then there is a natural isomorphism between the induced hermitian forms @xmath1135 . let @xmath340 or @xmath341 . the duality - preserving functors @xmath376 and @xmath1136 induce the same homomorphism of witt groups @xmath1137 .    in the diagram @xmath1138_{\alpha_v } \ar[r]^{g(\phi ) } & g(v^ * ) \ar[d]_{\alpha_{v^*}}\ar[r]^{\psi_v } & g(v)^ * \\ g'(v ) \ar[r]_{g'(\phi ) } & g'(v^ * ) \ar[r]_{\psi'_v } & g'(v)^ * \ar[u]_{\alpha_v^ * } } \ ] ] the left - hand square commutes by the naturality of @xmath20 while the right - hand square commutes because @xmath20 satisfies equation  ( [ condition_for_nat_trans ] ) . the lemma follows easily .        in this section we prove that the covering construction @xmath4 respects changes to coefficients ( proposition  [ b_respects_coefficient_change ] ) . we need one more observation which is straightforward to verify :    [ nat_isos_for_composed_ring_homos ] if @xmath1139 are ring homomorphisms then the diagram @xmath1140 \ar[r ] & a''\otimes_{a'}(a'\otimes_a v^ * ) \ar[dr ] & \\ & &   a''\otimes_{a'}(a'\otimes_a v)^ * \ar[dl ]   \\ ( a''\otimes_a v)^ * & ( a''\otimes_{a'}(a'\otimes_a v))^*\ar[l]}\ ] ] of natural isomorphisms is commutative . suppose @xmath226 is a ring homomorphism and @xmath61 is a module in @xmath99 . the natural isomorphism @xmath1141\to a'[f_\mu]\otimes_{a[f_\mu ] } ( v[f_\mu])\ ] ] induces a natural isomorphism ( see lemma  [ bl(functor)_is_functor ] ) @xmath1142\otimes_{a[f_\mu ] } b(v ) .\ ] ]    we aim to show that @xmath20 is a natural isomorphism between duality - preserving functors @xmath1143\otimes_{a[f_\mu]}\functor,\upsilon,1)\circ ( b,\phi,-1).\ ] ] applying lemma  [ nat_isos_for_composed_ring_homos ] to both composites in the commutative square @xmath1144\ar[d ] & a[f_\mu ] \ar[d ] \\ a'_{\phantom{\mu}}\ar[r ] & a'[f_\mu]}\ ] ] of ring homomorphisms one obtains commutative diagrams @xmath1145 \ar[r]\ar[d ] & ( a'\otimes_a v)^*[f_\mu ] \ar[r]^{\gamma } & ( a'\otimes_a v[f_\mu])^ * \\ a'[f_\mu]\otimes_{a[f_\mu]}(v^*[f_\mu ] ) \ar[r]_{\delta } & a'[f_\mu]\otimes_{a[f_\mu]}(v[f_\mu])^ * \ar[r ] & ( a'[f_\mu ] \otimes_{a[f_\mu ] } v[f_\mu])^ * \ar[u ] } \ ] ] where @xmath1146 and @xmath1147 and hence the commutative diagram @xmath1148^{b(\pi_v)}\ar[d]_{\alpha_{v^ * } } & b((a'\otimes_a v)^ * ) \ar[r]^{\phi_{a'\otimes v } } & b(a'\otimes v)^\wedge \\ a'[f_\mu]\otimes_{a[f_\mu ] } b(v^ * ) \ar[r]_{1\otimes\phi_v } & a'[f_\mu]\otimes_{a[f_\mu ] } b(v)^\wedge \ar[r]_{\upsilon_{b(v ) } } & ( a'[f_\mu]\otimes_{a[f_\mu ] } b(v))^\wedge \ar[u]_{(\alpha_v)^\wedge~. } } \ ] ] thus @xmath20 is a natural transformation between duality - preserving functors as claimed . it follows by lemma  [ nat_isomorphic_dpres_funct_give_same_witt_homo ] that the diagram  ( [ witt_diagram : b_respects_coeff_change ] ) of witt group homomorphisms commutes . the proof of proposition  [ b_respects_coefficient_change ] is complete .      in this section we prove that hermitian morita equivalence respects duality - preserving functors ( proposition  [ naturality_of_h_morita_eq ] ) . let @xmath585 denote a duality - preserving functor and let @xmath1149 , @xmath569 and @xmath1150 . we shall define a natural isomorphism between the composite functors @xmath1151 if @xmath572 then @xmath1152 is a left @xmath83-module for the action @xmath1153 where @xmath1154 and @xmath1155 . the group @xmath1156 is regarded as a left @xmath1117-module in the same way . define @xmath1157 since @xmath1158 is an isomorphism in the case @xmath1159 it follows that @xmath1158 is an isomorphism for all @xmath1160 . it is easy to see that @xmath1161 is a natural transformation @xmath1162 one must check that @xmath20 is a natural transformation of duality - preserving functors . by equations  ( [ how2_compose_dpres_functors ] ) and  ( [ condition_for_nat_trans ] ) one must show that @xmath1163 this equation can be checked by direct calculation , substituting the formulae  ( [ nat_iso_for_naturality_of_h_morita_eq ] ) , ( [ natural_isomorphism_phi ] ) and  ( [ nat_iso_in_coeff_change_dpres_funct ] ) for @xmath20 , @xmath1164 and @xmath403 respectively and applying the naturality of @xmath754 and the equation  ( [ duality_functor ] ) . this completes the proof of proposition  [ naturality_of_h_morita_eq ] . p.  m. cohn . localization in general rings , a historical survey . to appear in the proceedings of the conference on noncommutative localization in algebra and topology , icms , edinburgh , 29 - 30 april , 2002 . a.  a. ranicki . noncommutative localization in topology . to appear in the proceedings of the conference on noncommutative localization in algebra and topology , icms , edinburgh , 29 - 30 april , 2002 . arxiv : math.at/0303046 .
we use the blanchfield - duval form to define complete invariants for the cobordism group @xmath0 of @xmath1-dimensional @xmath2-component boundary links ( for @xmath3 ) . 
 the author solved the same problem in earlier work via seifert forms . 
 although seifert forms are convenient in explicit computations , the blanchfield - duval form is more intrinsic and appears naturally in homology surgery theory .    the free cover of the complement of a link is constructed by pasting together infinitely many copies of the complement of a @xmath2-component seifert surface . 
 we prove that the algebraic analogue of this construction , a functor denoted @xmath4 , identifies the author s earlier invariants with those defined here . 
 we show that @xmath4 is equivalent to a universal localization of categories and describe the structure of the modules sent to zero . 
 taking coefficients in a semi - simple artinian ring , we deduce that the witt group of seifert forms is isomorphic to the witt group of blanchfield - duval forms . 
 invariants of boundary link cobordism ii . 
 + the blanchfield - duval form +   desmond sheiham