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cluster - tilted algebras were introduced by buan , marsh and reiten @xcite and , independently in @xcite for type @xmath0 as a byproduct of the now extensive theory of cluster algebras of fomin and zelevinsky @xcite . since then , cluster - tilted algebras have been the subject of several investigations , see , for instance , @xcite .    in particular , in @xcite is given a construction procedure for cluster - tilted algebras : let @xmath1 be a triangular algebra of global dimension two over an algebraically closed field @xmath2 , and consider the @xmath1-@xmath1-bimodule @xmath3 , where @xmath4 is the standard duality , with its natural left and right @xmath1-actions . the trivial extension of @xmath1 by this bimodule is called the _ relation - extension _ @xmath5 of @xmath1 . it is shown there that , if @xmath1 is tilted , then its relation - extension is cluster - tilted , and every cluster - tilted algebra occurs in this way . our purpose in this paper is to study the relation - extensions of a wider class of triangular algebras of global dimension two , namely the class of quasi - tilted algebras , introduced by happel , reiten and smal   in @xcite . in general , the relation - extension of a quasi - tilted algebra is not cluster - tilted , however it is 2-calabi - yau tilted , see theorem [ thm 2.1 ] below . we then look more closely at those cluster - tilted algebras which are tame and representation - infinite . according to @xcite , these coincide exactly with the cluster - tilted algebras of euclidean type . we ask then the following question : given a cluster - tilted algebra @xmath6 of euclidean type , find all quasi - tilted algebras @xmath1 such that @xmath7 . a similar question has been asked ( and answered ) in @xcite , where , however , @xmath1 was assumed to be tilted .    for this purpose , we generalize the notion of reflections of @xcite . we prove that this operation allows to produce all tilted algebras @xmath1 such that @xmath7 , see theorem [ thm mainreflection ] . in @xcite this result was shown only for cluster - tilted algebras of tree type . we also prove that , unlike those of @xcite , reflections in the sense of the present paper are always defined , that the reflection of a tilted algebra is also tilted of the same type , and that they have the same relation - extension , see theorem [ thm reflection ] and proposition [ prop reflection ] below . because all tilted algebras having a given cluster - tilted algebra as relation - extension are given by iterated reflections , this gives an algorithmic answer to our question above . after that , we look at the tubes of a cluster - tilted algebra of euclidean type and give a procedure for constructing those tubes which contain a projective , see proposition [ prop5 ] . we then return to quasi - tilted algebras in our last section , namely we define a particular two - sided ideal of a cluster - tilted algebra , which we call the partition ideal . our first result ( theorem [ thm ctaqt ] ) shows that the quasi - tilted algebras which are not tilted but have a given cluster - tilted algebra @xmath6 of euclidean type as relation - extension are the quotients of @xmath6 by a partition ideal . we end the paper with the proof of our main result ( theorem [ thm ctaqt3 ] ) which says that if @xmath1 is quasi - tilted and such that @xmath7 , then either @xmath1 is the quotient of @xmath6 by the annihilator of a local slice ( and then @xmath1 is tilted ) or it is the quotient of @xmath6 by a partition ideal ( and then @xmath1 is not tilted except in two cases easy to characterize ) . throughout this paper , algebras are basic and connected finite dimensional algebras over a fixed algebraically closed field @xmath2 . for an algebra @xmath1 , we denote by @xmath8 the category of finitely generated right @xmath1-modules . all subcategories are full , and identified with their object classes . given a category @xmath9 , we sometimes write @xmath10 to express that @xmath11 is an object in @xmath9 . if @xmath9 is a full subcategory of @xmath8 , we denote by @xmath12 the full subcategory of @xmath8 having as objects the finite direct sums of summands of modules in @xmath9 .    for a point @xmath13 in the ordinary quiver of a given algebra @xmath1 , we denote by @xmath14 , @xmath15 , @xmath16 respectively , the indecomposable projective , injective and simple @xmath1-modules corresponding to @xmath13 . we denote by @xmath17 the auslander - reiten quiver of @xmath1 and by @xmath18 the auslander - reiten translations . for further definitions and facts , we refer the reader to @xcite . let @xmath19 be a finite connected and acyclic quiver . a module @xmath20 over the path algebra @xmath21 of @xmath19 is called _ tilting _ if @xmath22 and the number of isoclasses ( isomorphism classes ) of indecomposable summands of @xmath20 equals @xmath23 , see @xcite . an algebra @xmath1 is called _ tilted of type @xmath19 _ if there exists a tilting @xmath21-module @xmath20 such that @xmath24 . it is shown in @xcite that an algebra @xmath1 is tilted if and only if it contains a _ complete slice _ @xmath25 , that is , a finite set of indecomposable modules such that    * @xmath26 is a sincere @xmath1-module . * if @xmath27 is a sequence of nonzero morphisms between indecomposable modules with @xmath28 then @xmath29 for all @xmath30 ( _ convexity _ ) . * if @xmath31 is an almost split sequence in @xmath8 and at least one indecomposable summand of @xmath11 lies in @xmath25 , then exactly one of @xmath32 belongs to @xmath25 .    for more on tilting and tilted algebras , we refer the reader to @xcite . tilting can also be done within the framework of a hereditary category . let @xmath33 be an abelian @xmath2-category which is hom - finite , that is , such that , for all @xmath34 , the vector space @xmath35 is finite dimensional . we say that @xmath33 is _ hereditary _ if @xmath36 . an object @xmath37 is called a _ tilting object _ if @xmath38 and the number of isoclasses of indecomposable objects of @xmath20 is the rank of the grothendieck group @xmath39 . the endomorphism algebras of tilting objects in hereditary categories are called _ quasi - tilted algebras_. for instance , tilted algebras but also canonical algebras ( see @xcite ) are quasi - tilted . quasi - tilted algebras have attracted a lot of attention and played an important role in representation theory , see for instance @xcite . let @xmath19 be a finite , connected and acyclic quiver . the _ cluster category _ @xmath40 of @xmath19 is defined as follows , see @xcite . let @xmath41 denote the composition @xmath42 $ ] , where @xmath43 denotes the inverse auslander - reiten translation in the bounded derived category @xmath44 , and [ 1 ] denotes the shift of @xmath45 . then @xmath40 is the orbit category @xmath46 : its objects are the @xmath41-orbits @xmath47 of the objects @xmath48 , and the space of morphisms from @xmath47 to @xmath49 is @xmath50 then @xmath40 is a triangulated category with almost split triangles and , moreover , for @xmath51 we have a bifunctorial isomorphism @xmath52 . this is expressed by saying that the category @xmath40 is _ 2-calabi - yau_.    an object @xmath53 is called _ tilting _ if @xmath54 and the number of isoclasses of indecomposable summands of @xmath55 equals @xmath23 . the endomorphism algebra @xmath56 is then called _ cluster - tilted _ of type @xmath19 . more generally , the endomorphism algebra @xmath57 of a tilting object @xmath55 in a @xmath58-calabi - yau category with finite dimensional hom - spaces is called a _ 2-calabi - yau tilted algebra _ , see @xcite . let now @xmath20 be a tilting @xmath21-module , and @xmath24 the corresponding tilted algebra . then it is shown in @xcite that the trivial extension @xmath59 of @xmath1 by the @xmath1-@xmath1-bimodule @xmath60 with the two natural actions of @xmath1 , the so - called _ relation - extension _ of @xmath1 , is cluster - tilted . conversely , if @xmath6 is cluster - tilted , then there exists a tilted algebra @xmath1 such that @xmath61 . let now @xmath6 be a cluster - tilted algebra , then a full subquiver @xmath25 of @xmath62 is a _ local slice _ , see @xcite , if :    * @xmath25 is a _ presection _ , that is , if @xmath63 is an arrow then : * * @xmath64 implies that either @xmath65 or @xmath66 * * @xmath65 implies that either @xmath67 or @xmath68 . * @xmath25 is _ sectionally convex _ , that is , if @xmath69 is a sectional path in @xmath62 then @xmath70 implies that @xmath71 for all @xmath30 . * @xmath72 . let @xmath1 be tilted , then , under the standard embedding @xmath73 , any complete slice in the tilted algebra @xmath1 embeds as a local slice in @xmath74 , and any local slice in @xmath75 occurs in this way . if @xmath6 is a cluster - tilted algebra , then a tilted algebra @xmath1 is such that @xmath61 if and only if there exists a local slice @xmath25 in @xmath62 such that @xmath76 , where @xmath77 , see @xcite . let @xmath25 be a local slice in the transjective component of @xmath62 having the property that all the sources in @xmath25 are injective @xmath6-modules . then @xmath25 is called a _ rightmost _ slice of @xmath6 . let @xmath13 be a point in the quiver of @xmath6 such that @xmath15 is an injective source of the rightmost slice @xmath25 . then @xmath13 is called a _ strong sink_. _ leftmost slices _ and _ strong sources _ are defined dually . we start with a motivating example . let @xmath1 be the tilted algebra of type @xmath78 given by the quiver @xmath79_{\beta}\\1&&4\ar[lu]_{\alpha}\ar[ld]^{\gamma}\\&3\ar[lu]^{\delta}}\ ] ] bound by @xmath80 , @xmath81 . its relation - extension is the cluster - tilted algebra @xmath6 given by the quiver @xmath82_{\beta}\\1\ar@<1.5pt>[rr]^{\lambda}\ar@<-1.5pt>[rr]_\mu&&4\ar[lu]_{\alpha}\ar[ld]^{\gamma}\\&3\ar[lu]^{\delta}}\ ] ] bound by @xmath80 , @xmath83 , @xmath84 , @xmath81 , @xmath85 , @xmath86 . however , @xmath6 is also the relation - extension of the algebra @xmath87 given by the quiver @xmath88_{\alpha}&1\ar@<1.5pt>[l ] _ { ^{\lambda}}\ar@<-1.5pt>[l]^{\ \atop\mu}&3\ar[l]_{\delta}}\ ] ] bound by @xmath84 , @xmath85 . this latter algebra @xmath87 is not tilted , but it is quasi - tilted . in particular , it is triangular of global dimension two . therefore , the question arises natrually whether the relation - extension of a quasi - tilted algebra is always cluster - tilted . this is certainly not true in general , for the relation - extension of a tubular algebra is not cluster - tilted . however , it is 2-calabi - yau tilted . in this section , we prove that the relation - extension of a quasi - tilted algebra is always 2-calabi - yau tilted . let @xmath89 be a hereditary category with tilting object @xmath20 . because of @xcite , there exist an algebra @xmath90 , which is hereditary or canonical , and a triangle equivalence @xmath91 . let @xmath92 denote the image of @xmath20 under this equivalence . because @xmath93 preserves the shift and the auslander - reiten translation , it induces an equivalence between the cluster categories @xmath94 and @xmath95 , see ( * ? ? ? * section 4.1 ) . indeed , because @xmath90 is canonical or hereditary , it follows that @xmath96 , where @xmath97.$ ] therefore , we have @xmath98 . we say that a 2-calabi - yau tilted algebra @xmath99 is of _ canonical type _ if the 2-calabi - yau category @xmath100 is the cluster category of a canonical algebra . the proof of the next theorem follows closely @xcite . [ thm 2.1 ] let @xmath1 be a quasi - tilted algebra . then its relation - extension @xmath59 is cluster - tilted or it is 2-calabi - yau titled of canonical type .    because @xmath1 is quasi - tilted , there exist a hereditary category @xmath89 and a tilting object @xmath20 in @xmath89 such that @xmath101 . as observed above , there exist an algebra @xmath90 , which is hereditary or canonical , and a triangle equivalence @xmath91 . let @xmath102.we have @xmath103 , and therefore @xmath104 , c[2 ] ) \\   & \cong & { \textup{hom}}_{{\mathcal{d}^b(\mathcal{h})}}(\tau t[1 ] , t[2 ] ) \\   & \cong & { \textup{hom}}_{{\mathcal{d}^b(\mathcal{h } ) } } ( t , \tau^{-1 } t[1 ] ) \\   & \cong & { \textup{hom}}_{{\mathcal{d}^b(\mathcal{h } ) } } ( t , f t ) .\\ \end{array}\ ] ] thus the additive structure of @xmath105 is that of @xmath106 then , we check exactly as in ( * ? ? ? * section 3.3 ) that the multiplicative structure is preserved . this completes the proof . let @xmath1 be a representation - infinite quasi - tilted algebra . then @xmath1 is derived equivalent to a hereditary or a canonical algebra @xmath90 . let @xmath107 denote the tubular type of @xmath90 . we then say that @xmath1 has canonical type @xmath108 . [ lem 1 ] let @xmath1 be a representation - infinite quasi - tilted . then its relation - extension @xmath5 is cluster - tilted of euclidean type if and only if @xmath109 is one of @xmath110    indeed , @xmath5 is cluster - tilted of euclidean type if and only if @xmath1 is derived equivalent to a tilted algebra of euclidean type , and this is the case if and only if @xmath109 belongs to the above list . [ rem 2 ] it is possible that @xmath1 is domestic , but yet @xmath5 is wild . indeed , we modify the example after corollary d in @xcite . recall from @xcite that there exists a tame concealed full convex subcategory @xmath111 such that @xmath1 is a semiregular branch enlargement of @xmath111 @xmath112k[f_j],\ ] ] where @xmath113 are ( truncated ) branches . then the representation theory of @xmath1 is determined by those of @xmath114k$ ] and @xmath115 $ ] . let @xmath1 be given by the quiver    @xmath116_{\delta}&11\ar[l]_\zeta\\ & 3\ar[ul]_{\alpha}\ar[dl]^{\beta}&4\ar[l]_{\gamma}\ar[d]^\nu&5\ar[l]_{\sigma}\\ 2&&8\ar[d]^\varphi & 9\ar[l]_\omega&7\ar[ul]_\rho \\&&10 } \ ] ] bound by the relations @xmath117 , @xmath118 , @xmath119 . here @xmath120 is the full subcategory generated by @xmath121 and @xmath122 the one generated by @xmath123 . then @xmath120 has domestic tubular type @xmath124 and @xmath122 has domestic tubular type @xmath125 . therefore @xmath1 is domestic . on the other hand , the canonical type of @xmath1 is @xmath126 , which is wild . in this example , the 2-calabi - yau tilted algebra @xmath5 is not cluster - tilted , because it is not of euclidean type , but the derived category of @xmath127 contains tubes , see @xcite . there clearly exist algebras which are not quasi - tilted but whose relation - extension is cluster - tilted of euclidean type . indeed , let @xmath1 be given by the quiver @xmath128^{\alpha}&5\ar[r]^{\beta}&4\ar[r]^{\gamma}&3\ar[r]^{\delta}&2\ar@<-2pt>[r]_\mu\ar@<2pt>[r]^{\lambda}&1}\ ] ] bound by @xmath129 . then @xmath1 is iterated tilted of type @xmath78 of global dimension 2 , see @xcite . its relation - extension is given by @xmath128^{\alpha}&5\ar[r]^{\beta}&4\ar@/_25pt/[ll]_{\sigma}\ar[r]^{\gamma}&3\ar[r]^{\delta}&2\ar@<-2pt>[r]_\mu\ar@<2pt>[r]^{\lambda}&1\ar@/_25pt/[ll]_\eta}\ ] ] bound by @xmath130 . this algebra is isomorphic to the relation - extension of the tilted algebra of type @xmath78 given by the quiver @xmath131^{\sigma}\ar[r]^{\gamma}&3\ar[r]^{\delta}&2\ar@<-2pt>[r]_\mu\ar@<2pt>[r]^{\lambda}&1\\ 5\ar[ru]^{\beta}}\ ] ] bound by @xmath132 , @xmath133 . therefore @xmath5 is cluster - tilted of euclidean type . on the other hand , @xmath1 is not quasi - tilted , because the uniserial module @xmath134 has both projective and injective dimension 2 . let @xmath1 be a tilted algebra . let @xmath135 be a rightmost slice , and let @xmath15 be an injective source of @xmath135 . thus @xmath13 is a strong sink in @xmath1 . we define the _ completion @xmath136 of @xmath13 _ by the following three conditions . * @xmath137 . * @xmath136 is closed under predecessors in @xmath135 . * if @xmath138 is an arrow in @xmath135 with @xmath139 having an injective successor in @xmath136 then @xmath140 .    observe that @xmath136 may be constructed inductively in the following way . we let @xmath141 , and @xmath142 be the closure of @xmath143 with respect to ( c ) ( that is , we simply add the direct successors of @xmath15 in @xmath135 , and if a direct successor of @xmath15 is injective , we also take its direct successor , etc . ) we then let @xmath144 be the closure of @xmath142 with respect to predecessors in @xmath135 . then we repeat the procedure ; given @xmath145 , we let @xmath146 be the closure of @xmath145 with respect to ( c ) and @xmath147 be the closure of @xmath146 with respect to predecessors . this procedure must stabilize , because the slice @xmath135 is finite . if @xmath148 with @xmath149 , we let @xmath150 .    we can decompose @xmath136 as the disjoint union of three sets as follows . let @xmath151 denote the set of injectives in @xmath136 , let @xmath152 be the set of non - injectives in @xmath136 which have an injective successor in @xmath136 , and let @xmath153 denote the complement of @xmath154 in @xmath136 . thus @xmath155 is a disjoint union . [ rem h ] if @xmath156 then @xmath136 reduces to the completion @xmath157 as defined in @xcite . recall that @xmath157 does not always exist , but , as seen above , @xmath136 does . conversely , if @xmath157 exists , then it follows from its construction in @xcite that @xmath156 . thus @xmath156 if and only if @xmath157 exists , and , in this case @xmath158 .    for every module @xmath11 over a cluster - tilted algebra @xmath6 , we can consider a lift @xmath159 in the cluster category @xmath100 . abusing notation , we sometimes write @xmath160 to denote the image of @xmath161 in @xmath162 , and say that the auslander - reiten translation is computed in the cluster category . let @xmath13 be a strong sink in @xmath1 and let @xmath135 be a rightmost local slice with injective source @xmath15 . recall that @xmath135 is also a local slice in @xmath162 . then the reflection of the slice @xmath135 in @xmath13 is @xmath163 where @xmath164 is computed in the cluster category . in a similar way , one defines the coreflection @xmath165 of leftmost slices with projective sink @xmath166 . [ thm reflection ] let @xmath13 be a strong sink in @xmath1 and let @xmath135 be a rightmost local slice in @xmath162 with injective source @xmath15 . then the reflection @xmath167 is a local slice as well . set @xmath168 and @xmath169 where again , @xmath170 and @xmath164 are computed in the cluster category @xmath100 . we claim that @xmath170 is a local slice in @xmath100 . notice that since @xmath136 is closed under predecessors in @xmath135 , then , if @xmath171 is a neighbor of @xmath172 , we must have an arrow @xmath173 in @xmath135 . this observation being made , @xmath170 is clearly obtained from @xmath135 by applying a sequence of apr - tilts . thus @xmath170 is a local slice in @xmath100 . we now claim that @xmath174 is closed under predecessors in @xmath170 . indeed , let @xmath175 and @xmath176 be such that we have an arrow @xmath173 . then , there exists an arrow @xmath177 in the cluster category . because @xmath178 , we have @xmath179 . now if @xmath180 , then the arrow @xmath177 would imply that @xmath172 , which is impossible , because @xmath176 and @xmath181 . thus @xmath182 , and therefore @xmath183 . hence @xmath184 . moreover , there is an arrow @xmath185 . using that @xmath179 , this implies that @xmath186 has an injective successor in @xmath136 and thus @xmath187 . this establishes our claim that @xmath174 is closed under predecessors in @xmath170 .    thus applying the same reasoning as before , we get that @xmath188 is a local slice in @xmath100 . now we claim that @xmath189 first , because @xmath190 , we have @xmath191 . next , @xmath192 contains no injectives , by definition . thus @xmath193 assume now that @xmath194 belongs to @xmath195 . then @xmath196 and there exists an injective predecessor @xmath197 of @xmath198 in @xmath136 , and since @xmath136 is part of the local slice @xmath135 , there exists a sectional path from @xmath197 to @xmath198 . applying @xmath199 , we get a sectional path from @xmath200 to @xmath201 in the cluster category . but this means @xmath202 , which is a contradiction to the hypothesis that @xmath194 . finally , if @xmath203 then @xmath201 is a summand of @xmath20 , which , again , is contradicting the hypothesis that @xmath194 . following @xcite , let @xmath204 be the full subcategory of @xmath1 consisting of those @xmath205 such that @xmath206 . [ lem s ]    * @xmath204 is hereditary . * @xmath204 is closed under successors in @xmath1 . * @xmath1 can be written in the form @xmath207 , \ ] ] where @xmath208 is hereditary , @xmath87 is tilted and @xmath11 is a @xmath87-@xmath208-bimodule . \(a ) let @xmath209 then @xmath208 is a full subcategory of the hereditary endomorphism algebra of @xmath135 . therefore @xmath208 is also hereditary , and so @xmath204 is hereditary . \(b ) let @xmath210 and @xmath211 in @xmath1 . then there exists a morphism @xmath212 . because @xmath213 is an injective @xmath1-module and @xmath135 is sincere , there exist a module @xmath214 and a non - zero morphism @xmath215 . then we have a path @xmath216 , and since @xmath217 , we get that @xmath218 by convexity of the slice @xmath135 in @xmath127 . moreover , since @xmath206 and @xmath136 is closed under predecessors in @xmath135 , it follows that @xmath219 . thus @xmath220 and this shows ( b ) . \(c ) this follows from ( a ) and ( b ) . we recall that the cluster duplicated algebra was introduced in @xcite . [ cor dup ] the cluster duplicated algebra @xmath221 of @xmath1 is of the form @xmath222\ ] ] where @xmath223 and @xmath224 . we start by writing @xmath1 in the matrix form of the lemma . by definition , @xmath208 consists of those @xmath225 such that the corresponding injective @xmath226 lies in @xmath227 inside the slice @xmath135 . in particular , the projective dimension of these injectives is at most 1 , hence @xmath228 . the result now follows upon multiplying by idempotents . let @xmath13 be a strong sink in @xmath1 . the reflection at @xmath13 of the algebra @xmath1 is @xmath229\ ] ] where @xmath223 . [ prop reflection ] the reflection @xmath230 of @xmath1 is a tilted algebra having @xmath167 as a complete slice . moreover the relation - extensions of @xmath1 and @xmath167 are isomorphic . we first claim that the support @xmath231 of @xmath167 is contained in @xmath232 . let @xmath233 . recall that @xmath234 . if @xmath203 , then @xmath235 is projective corresponding to a point @xmath236 . thus @xmath206 and the radical of @xmath237 has no non - zero morphism into @xmath226 . therefore @xmath238 . assume next that @xmath239 , that is , @xmath240 , where @xmath241 has an injective successor @xmath213 in @xmath136 . because all sources in @xmath135 are injective , there is an injective @xmath242 and a sectional path @xmath243 . applying @xmath199 , we obtain a sectional path @xmath244 . in particular the point @xmath245 belongs to the support of @xmath201 . assume that there is a point @xmath246 in @xmath208 that is in the support of @xmath201 . then there exists a nonzero morphism @xmath247 . but @xmath248 and there is no morphism from @xmath249 to @xmath135 . therefore @xmath250 . by the same argument , we show that if @xmath251 , then @xmath250 .    finally , all modules of @xmath252 are supported in @xmath87 . this establishes our claim . now , by theorem [ thm reflection ] , @xmath253 is a local slice in @xmath254 therefore @xmath255 is a tilted algebra in which @xmath256 is a complete slice . since the support of @xmath256 is the same as the support of @xmath257 , we are done .    we now come to the main result of this section , which states that any two tilted algebras that have the same relation - extension are linked to each other by a sequence of reflections and coreflections . let @xmath6 be a cluster - tilted algebra and let @xmath25 and @xmath258 be two local slices in @xmath259 . we write @xmath260 whenever @xmath261 . [ lem 310 ] let @xmath6 be a cluster - tilted algebra , and @xmath262 be two local slices in @xmath162 . then there exists a sequence of reflections and coreflections @xmath263 such that @xmath264    given a local slice @xmath25 in @xmath259 such that @xmath25 has injective successors in the transjective component @xmath265 of @xmath62 , let @xmath266 be the rightmost local slice such that @xmath267 . then @xmath266 contains a strong sink @xmath13 , thus reflecting in @xmath13 we obtain a local slice @xmath268 that has fewer injective successors in @xmath265 than @xmath25 . to simplify the notation we define @xmath269 . similarly , we define @xmath270 , where @xmath271 is the leftmost local slice containing a strong source @xmath205 and @xmath272 . since we can always reflect in a strong sink , there exist sequences of reflections such that @xmath273 @xmath274 and @xmath275 have no injective successors in @xmath265 . this implies that @xmath276 . let @xmath277 thus @xmath278 .    [ thm mainreflection ] let @xmath279 and @xmath280 be two tilted algebras that have the same relation - extension . then there exists a sequence of reflections and coreflections @xmath263 such that @xmath281 . let @xmath6 be the common relation - extension of the tilted algebras @xmath279 and @xmath280 . by @xcite , there exist local slices @xmath282 in @xmath162 such that @xmath283 , for @xmath284 . now the result follows from lemma [ lem 310 ] and theorem  [ thm reflection ] . let @xmath90 be the path algebra of the quiver @xmath285\ar[ld]\\   & 2\ar[ld ] \\ 3\\   & 4\ar[lu]\\   & & 5\ar@<1pt>[lu]\ar@<-1pt>[lu]\ar[ld]\\   & 6      } \ ] ] mutating at the vertices 4,5 , and 2 yields the cluster - tilted algebra @xmath6 with quiver    @xmath286\ar@<-2pt>@/_10pt/[lldd]\\   & 2\ar[ur ] \\   3\ar[ur]\ar@<0pt>@/^15pt/[rrdd]\ar@<-2pt>@/^15pt/[rrdd]\\   & 4\ar@<2pt>[lu]\ar@<-2pt>[lu]\ar[ul]\ar@<1pt>[dd]\ar@<-1pt>[dd]\\   & & 5\ar@<1pt>[lu]\ar@<-1pt>[lu]\\   & 6\ar[ur ]    } \ ] ] in the auslander - reiten quiver of @xmath162 we have the following local configuration . @xmath287 & & \circ & & p(1 ) \\    & 1\ar[ru ] & & 2\ar[rd ] & & { \begin{smallmatrix } 3\\ 5\,5\\4\end{smallmatrix}}\ar[ru ] \\ i(3)\ar[ru]\ar@/^15pt/[rruu ] \ar[rd ] & & \circ & & p(3)\ar[ru]\ar@/^15pt/[rruu]\\ &   { \begin{smallmatrix } 5555\\444 \end{smallmatrix}}\ar@<1pt>[rd]\ar@<-1pt>[rd ] & &    { \begin{smallmatrix } 55\\4 \end{smallmatrix}}\ar@<1pt>[rd]\ar@<-1pt>[rd ] \ar[ru]\\    & & { \begin{smallmatrix } 555\\44 \end{smallmatrix}}\ar@<1pt>[ru]\ar@<-1pt>[ru ] & & 5 \ar[rd ] \\    & i(6 ) \ar[ru ] & & \circ & & p(6 )    } \ ] ] where @xmath288    the 6 modules on the left form a rightmost local slice @xmath135 in which both @xmath289 and @xmath290 are sources , so 3 and 6 are strong sinks . for both strong sinks the subset @xmath152 of the completion consists of the simple module @xmath291 . the simple module @xmath292 does not lie on a local slice . the completion @xmath293 is the whole local slice @xmath135 and therefore the reflection @xmath294 is the local slice consisting of the 6 modules on the right containing both @xmath295 and @xmath296 .    on the other hand , the completion @xmath297 consists of the four modules @xmath289 , @xmath298 , @xmath299 and @xmath300 , and therefore the reflection @xmath301 is the local slice consisting of the 6 modules on the straight line from @xmath290 to @xmath295 . this local slice admits the strong sink @xmath302 and the completion @xmath303 in @xmath304 consists of the two modules @xmath290 and @xmath305 . therefore the reflection @xmath306 is equal to @xmath294 . thus @xmath307    this example raises the question which indecomposable modules over a cluster - tilted algebra do not lie on a local slice . we answer this question in a forthcoming publication @xcite . the objective of this section is to show how to construct those tubes of a tame cluster - tilted algebra which contain projectives . let @xmath6 be a cluster - tilted algebra of euclidean type , and let @xmath265 be a tube in @xmath62 containing at least one projective . first , consider the transjective component of @xmath62 . denote by @xmath308 a local slice in the transjective component that precedes all indecomposable injective @xmath6-modules lying in the transjective component . then @xmath309 is a tilted algebra having a complete slice in the preinjective component . define @xmath310 to be a local slice which is a successor of all indecomposable projectives lying in the transjective component . then @xmath311 is a tilted algebra having a complete slice in the postprojective component . also , @xmath279 ( respectively , @xmath312 has a tube @xmath313 ( respectively , @xmath314 ) containing the indecomposable projective @xmath279-modules ( respectively , injective @xmath280-modules ) corresponding to the projective @xmath6-modules in @xmath265 ( respectively , injective @xmath6-modules in @xmath265 ) .    an indecomposable projective @xmath14 ( respectively , injective @xmath15 ) @xmath6-module that lies in a tube , is said to be a _ root projective _ ( respectively , a _ root injective _ ) if there exists an arrow in @xmath6 between @xmath13 and @xmath205 , where the corresponding indecomposable projective @xmath237 lies in the transjective component of @xmath62 . let @xmath315 be the coray in @xmath313 passing through the projective @xmath279-module that corresponds to the root projective @xmath316 in @xmath265 . similarly , let @xmath317 be the ray in @xmath314 passing through the injective that corresponds to the root injective @xmath318 in @xmath265 .    recall that if @xmath90 is hereditary and @xmath319 is a tilting module , then there exists an associated torsion pair @xmath320 in @xmath321 , where    @xmath322    [ lem tube1 ] with the above notation    * @xmath323 is a coray in @xmath265 passing through @xmath316 . * @xmath324 is a ray in @xmath265 passing through @xmath318 . since @xmath279 is tilted , we have @xmath325 where @xmath20 is a tilting module over a hereditary algebra @xmath90 . as seen in the proof of theorem 5.1 in @xcite , we have a commutative diagram    @xmath326\ar[r]^{\text{hom}_a(t,-)}&\mathcal{y}(t)\ar[d]^{-\otimes_{c_1 } b}\\ \mathcal{c}_a\ar[r]^{\text{hom}_{\mathcal{c}_a}(t,-)}&\text{mod}\,b\;\;}\ ] ] where @xmath327 . let @xmath328 be the tube in @xmath321 corresponding to the tube @xmath265 in @xmath259 . by what has been seen above , we have a commutative diagram    @xmath329^{\text{hom}_a(t,-)}\ar[dr]_-{\text{hom}_{\mathcal{c}_a}(t,-)\;\;\;\;\;}&\mathcal{t}_1\ar[d]^{-\otimes_{c_1 } b}\\ & \mathcal{t}_1\otimes_{c_1 } b \subset \mathcal{t}\;\;.}\ ] ]    let @xmath330 be any coray in @xmath313 , so it can be lifted to a coray @xmath331 in @xmath332 via the functor @xmath333 . if we apply @xmath334 to this lift , we obtain a coray in @xmath335 . thus , any coray in @xmath313 induces a coray in @xmath265 . let @xmath315 be the coray passing through the root projective @xmath336 . then @xmath337 is the coray passing through @xmath338 . this proves ( a ) and part ( b ) is proved dually . however , we must still justify that the ray @xmath337 and the coray @xmath339 actually intersect ( and thus lie in the same tube of @xmath62 ) . because @xmath340 , we have @xmath341 , and @xmath316 lies in a tube @xmath265 . it is well - known that the injective @xmath318 also lies in @xmath265 . in particular , we have the following local configuration in @xmath265 , where @xmath342 is an indecomposable summand of the radical of @xmath316 and @xmath343 an indecomposable summand of the quotient of @xmath318 by its socle . @xmath344 & & \circ \ar@{-->}[dr ] & & p_b(i)\\ & j\ar[dr]\ar@{-->}[ur]&&r\ar[ur]\\ & & n\ar[ur]}\ ] ]    now @xmath345 is coinduced , and we have shown above that the ray containing it is also coinduced . because @xmath346 , this is the ray @xmath347 . therefore , this ray and this coray lie in the same tube , so must intersect in a module @xmath348 , where there exists an almost split sequence    @xmath349&j\ar[r ] & n \ar[r ] & r \ar[r ] & 0.}\ ] ]    knowing the ray @xmath347 and the coray @xmath337 for every root projective @xmath316 in @xmath265 , one may apply the knitting procedure to construct the whole of @xmath265 . in this way , @xmath265 can be determined completely . next we show that all modules over a tilted algebra lying on the same coray change in the same way under the induction functor . [ lem tube2 ] let @xmath90 be a hereditary algebra of euclidean type , @xmath20 a tilting @xmath90-module without preinjective summands and let @xmath350 be the corresponding tilted algebra . let @xmath328 be a tube in @xmath351 and @xmath352 an indecomposable summand of @xmath20 , such that @xmath353 . then there exists an @xmath90-module @xmath11 on the mouth of @xmath328 such that we have @xmath354 in @xmath127 . in particular , the module @xmath355 lies on the mouth of the tube @xmath356 in @xmath127 . the injective @xmath1-module @xmath357 is given by @xmath358 where the first identity holds by ( * ? ? ? * proposition vi 5.8 ) and the second identity is the auslander - reiten formula . moreover , since @xmath359 lies in the tube @xmath328 and @xmath20 has no preinjective summands , we have @xmath360 only if @xmath200 lies in the hammock starting at @xmath359 . furthermore , if @xmath200 is a summand of @xmath20 then it must lie on a sectional path starting from @xmath359 because @xmath361 . this shows that a point @xmath362 is in the support of @xmath357 if and only if there is a sectional path @xmath363 in @xmath328 . we shall distinguish two cases . if @xmath359 lies on the mouth of @xmath328 then let @xmath364 be the ray starting at @xmath359 and denote by @xmath365 the last summand of @xmath20 on this ray . let @xmath366 be the direct predecessor of @xmath365 not on the ray @xmath364 . thus we have the following local configuration in @xmath328 . @xmath367&&\tau t_i \ar[dr ] & & t_i \ar[dr]\\ & \ar[ur]\ar[dr]&&\ar[ur]\ar[dr]&&\ar@{ .. }[dr]\\ & & \ar[ur]\ar@{ .. }[dr]&&\ar@{ .. }[dr]&&\ar[dr]\\ & & & \ar[dr]&&\tau t_1\ar[dr]\ar[ur]&&t_1\ar[dr]&\\ & & & & \tau l_1\ar[ur]\ar[dr]&&l_1\ar[ur]&&\tau^{-1 } l_1 \\ & & & & & e_1\ar[ur]}\ ] ]    then @xmath357 is uniserial with simple top @xmath298 . moreover there is a short exact sequence    @xmath349&\tau t_i\ar[r ] & l_1\ar[r ] & t_1\ar[r ] & 0}\ ] ] and applying @xmath368 yields    @xmath369&\text{hom}_a(t , l_1)\ar[r ] & p_c(1 ) \ar[r]^{f } & i_c(i ) \ar[r ] & \text{ext}^1(t , l_1)\ar[r]&0}\ ] ]    by the auslander - reiten formula , we have @xmath370 and this is zero because @xmath365 is the last summand of @xmath20 on the ray @xmath364 . thus the sequence ( [ s1 ] ) is short exact , the morphism @xmath371 is a projective cover , because @xmath357 is uniserial , and hence    @xmath372 applying @xmath373 yields    @xmath374    let @xmath375 be the indecomposable direct predecessor of @xmath366 such that the almost split sequence ending at @xmath366 is of the form    @xmath376&\tau l_1 \ar[r ] & e_1\oplus \tau t_1 \ar[r ] & l_1 \ar[r ] & 0}\ ] ]    we claim that @xmath377 . recall that @xmath366 is not a summand of @xmath20 because @xmath378 is non projective . also , recall that @xmath365 is the last summand of @xmath20 on the ray @xmath364 . suppose @xmath379 , thus @xmath380 . then it follows that there is a summand of @xmath20 on the ray @xmath381 that is a successor of @xmath382 . let @xmath383 denote the first such indecomposable summand . @xmath367&&\tau t_1\ar[dr ] & & t_1\ar[dr]\\ & \tau l_1\ar[ur]\ar[dr ] & & l_1 \ar[dr]\ar[ur]&&\tau^{-1}l_1\ar[dr]\\ & & e_1 \ar[ur ] & & \tau^{-1 } e_1\ar[dr ] \ar[ur ] & & \ar@{ .. }[dr]\\ & & & & & \ar@{ .. }[dr ] & & n \ar[dr]\\ & & & & & & t^1\ar[ur]\ar[dr ] & & \ar@{ .. }[dr]\\ & & & & & & & \ar@{ .. }[dr ] & & \omega   \\ & & & & & & & & \tau\omega & & & \\ } \ ] ]    then we have a short exact sequence    @xmath349 & l_1\ar[r]^-{h}&t_1\oplus t^1 \ar[r ] & n \ar[r ] & 0}\ ] ] with @xmath246 an @xmath384-approximation . applying @xmath385 yields    @xmath386&\text{hom}_a(n , t)\ar[r ] & \text{hom}_a(t_1\oplus t^1 , t)\ar[r]^-{h^ * } & \text{hom}_a(l_1 , t)\\ & \hspace{2cm}\ar[r]&\text{ext}^1_a(n , t)\ar[r]&0\hspace{1.5cm}}\ ] ] and since @xmath246 is an @xmath384-approximation , the morphism @xmath387 is surjective . thus @xmath388 . on the other hand , @xmath389 generates @xmath348 , so @xmath390 , and thus @xmath391 . but then both @xmath392 and we see that @xmath348 is a summand of @xmath20 . this is a contradiction to the assumption that @xmath365 is the last summand of @xmath20 on the ray @xmath364 . thus @xmath393 .    therefore , in the almost split sequence ( [ s2 ] ) , we have @xmath394 and @xmath395 . moreover , all predecessors of @xmath396 on the ray @xmath397 are also in @xmath398 because the morphisms on the ray are injective . since @xmath399 is an equivalence of categories , it follows that @xmath400 has only one direct predecessor @xmath401 in @xmath8 and this irreducible morphism is surjective . the kernel of this morphism is @xmath402 where @xmath403 is the torsion radical . thus we get @xmath404    we will show that @xmath405 lies on the mouth of @xmath328 and this will complete the proof in case 1 . let @xmath11 be the indecomposable @xmath90-module on the mouth of @xmath328 such that the ray starting at @xmath11 passes through @xmath406 . thus @xmath11 is the starting point of the ray @xmath407 . then there is a short exact sequence of the form    @xmath408 & m \ar[r ] & \tau_a l_1 \ar[r ] & \tau_a t_1 \ar[r ] & 0}\ ] ]    with @xmath409 . we claim that @xmath410 . suppose to the contrary that @xmath411 . since @xmath412 lies on the mouth of @xmath328 , this implies that there is a direct summand @xmath383 of @xmath20 which lies on the ray @xmath397 starting at @xmath412 . since @xmath20 is tilting , @xmath383 can not be a predecessor of @xmath396 on this ray and since @xmath366 is not a summand of @xmath20 , we also have @xmath413 . thus @xmath383 is a successor of @xmath366 on the ray @xmath381 . this is impossible since such a @xmath383 would satisfy @xmath414 contradicting the fact that @xmath377 . therefore , @xmath415 and the sequence ( [ s3 ] ) is the canonical sequence for @xmath406 in the torsion pair @xmath320 . this shows that @xmath416 and hence @xmath417 as desired . now suppose that @xmath359 does not lie on the mouth of @xmath328 . let @xmath418 denote the ray passing through @xmath359 and @xmath419 the coray passing through @xmath359 . denote by @xmath365 the last summand of @xmath20 on @xmath418 , by @xmath420 the last summand of @xmath20 on @xmath419 , and by @xmath421 the direct predecessor of @xmath200 which does not lie on @xmath422 . note that @xmath423 does not exist if @xmath420 lies on the mouth of @xmath328 , and in this case we let @xmath424 . thus we have the following local configuration in @xmath328 . @xmath425 & & & & & & & & &   & l_2 \ar[dr ] & & \tau^{-1}l_2\\ & \ar@{ .. }[dr]&&&&&&&&\tau t_2\ar[ur]\ar[dr]\ar@{ .. }[dl ] & & t_2\ar[ur]&&\\ & & \ar@{ .. }[dr]&&&&&&&&\ar[ur]\ar@{ .. }[dl]\\ & & & \ar@{ .. }[dr]&&\ar[dr]\ar@{ .. }[ur]&&\ar[dr]\ar@{ .. }[ur]&&&&&&&&\\ & & & & \tau^2 t_i\ar[ur]\ar[dr ] & & \tau t_i \ar[dr]\ar[ur ] & & t_i \ar[ur]\ar[dr]\\ & & & & & \ar[ur]\ar@{ .. }[dr]&&\ar[ur]\ar@{ .. }[dr]&&\ar@{ .. }[dr]&&\\ & & & & & & \ar[dr]&&\tau t_1\ar[dr ] & & t_1\ar[dr ] \\ & & & & & & & \tau l_1 \ar[ur ] & & l_1\ar[ur ] & & \tau^{-1}l_1}\ ] ]    the injective @xmath1-module @xmath426 is biserial with top @xmath427 . moreover , there is a short exact sequence @xmath349&\tau t_i \ar[r ] & l_1\oplus l_2 \oplus t_i \ar[r ] & t_1\oplus t_2 \ar[r ] & 0.}\ ] ] applying @xmath333 yields the following exact sequence .    @xmath428&\text{hom}_a(t , l_1\oplus l_2)\oplus p_c(i ) \ar[r ] & p_c(1)\oplus p_c(2 ) \ar[r]^-{f}&i_c(i)\\ & \hspace{3cm}\ar[r]&\text{ext}^1_a(t , l_1\oplus l_2)\ar[r]&0.}\ ] ]    by the same argument as in case 1 , using that @xmath365 and @xmath420 are the last summands of @xmath20 on @xmath418 and @xmath419 respectively , we see that @xmath429 . therefore , the sequence ( [ s4 ] ) is short exact . moreover , the morphism @xmath371 is a projective cover and thus @xmath430 applying @xmath373 yields @xmath431    by the same argument as in case 1 we see that    @xmath432 where @xmath11 is the indecomposable @xmath90-module on the mouth of @xmath328 such that the ray starting at @xmath11 passes through @xmath433 . in other words , @xmath11 is the starting point of the ray @xmath434 . therefore , it only remains to show that @xmath435 . to do so , it suffices to show that @xmath423 is a summand of @xmath20 . we have already seen that @xmath436 . we show now that we also have @xmath437 . suppose the contrary . then there exists a non - zero morphism @xmath438 . composing it with the irreducible injective morphism @xmath439 yields a non - zero morphism in @xmath440 . but this is impossible since @xmath20 is tilting .    thus we have @xmath441 and thus @xmath423 is a summand of @xmath20 , the module @xmath442 is projective and @xmath443 . this completes the proof . the module @xmath11 in the statement of the lemma is the starting point of the ray passing through @xmath444 . [ cor5 ] let @xmath445 be as in lemma [ lem tube2 ] , and let @xmath446 , with @xmath447 . let @xmath448 be two modules lying on the same coray in the tube @xmath356 in @xmath127 . then @xmath449 and thus the two projections @xmath450 and @xmath451 have isomorphic kernels .    for all @xmath1-modules @xmath201 we have @xmath452 where the first isomorphism is ( * ? ? ? * proposition 3.3 ) and the second is ( * ? ? ? * proposition 4.1 ) . since @xmath20 has no preinjective summands , and @xmath201 is regular , the only summand of @xmath453 for which @xmath454 can be nonzero , must lie in the same tube as @xmath201 . by the lemma , the only summands of @xmath453 in the tube lie on the mouth of the tube . let @xmath11 denote an indecomposable @xmath1-module on the mouth of a tube . then    @xmath455    we summarize the results of this section in the following proposition . [ prop5 ]    * let @xmath456 be the coray in @xmath457 passing through the projective @xmath279-module corresponding to the root projective @xmath316 then @xmath458 is a coray in @xmath459 passing through @xmath316 . furthermore all modules in @xmath460 are extensions of modules of @xmath456 by the same module @xmath461 . * let @xmath462 be the ray in @xmath463 passing through the injective @xmath280-module corresponding to the root injective @xmath318 then @xmath464 is a ray in @xmath459 passing through @xmath318 . furthermore all modules in @xmath464 are extensions of modules of @xmath462 by the same module @xmath465 . \(a ) the first statement is lemma [ lem tube1 ] , and the second statement is a restatement of corollary [ cor5 ] . let @xmath6 be the cluster - tilted algebra given by the quiver    @xmath466^{\lambda}\ar@<-2pt>[rr]_{\beta}&&5\ar[dl]^{\epsilon}\\ & 3\ar[ul]^{\alpha}\ar[dr]^{\delta}\\ 2\ar[ur]^{\gamma}&&4\ar[ll]^{\sigma}}\ ] ] bound by @xmath467 . the algebras @xmath279 and @xmath280 are respectively given by the quivers    @xmath466^{\lambda}\ar@<-2pt>[rr]_{\beta}&&5 & & & & 1\ar@<2pt>[rr]^{\lambda}\ar@<-2pt>[rr]_{\beta}&&5\ar[dl]^{\epsilon }   \\ & 3\ar[dr]^{\delta } \ar[ul]^{\alpha}&&&\text{and } & & & 3\ar[dr]^{\delta}\\ 2\ar[ur]^{\gamma}&&4 & & & & 2\ar[ur]^{\gamma}&&4}\ ] ] with the inherited relations . we can see the tube in @xmath468 below and the coray passing through the root projective @xmath469 is given by    @xmath470 &   { \begin{smallmatrix}1\\5\end{smallmatrix}}\ar[r]&{\begin{smallmatrix}3\\4\;1\\\;\;\;5\end{smallmatrix}}\ar[r ] & { \begin{smallmatrix}3\\1\\5\end{smallmatrix}}\ar[r]&{\begin{smallmatrix}2\\3\\1\\5\end{smallmatrix}}.}\ ] ]    @xmath471\ar@{--}[dd]\\ & & & & & & & { \begin{smallmatrix}3\\4\;1\\\;\;\;5\end{smallmatrix}}\\ & & & & { \begin{smallmatrix}2\\3\\1\\5\end{smallmatrix}}\ar[dr]&&{\begin{smallmatrix}1\\5\end{smallmatrix}}\ar[ur]\ar@{--}[d]\\ & { \begin{smallmatrix}4\end{smallmatrix}}\ar[dr]\ar@{--}[dd]&&{\begin{smallmatrix}3\\1\\5\end{smallmatrix}}\ar[ur]\ar[dr]&&{\begin{smallmatrix}2\;\;\\3\;\;\\1\;1\\\;5\;5\end{smallmatrix}}\ar[ur]&&\\ & & { \begin{smallmatrix}3\\4\;1\\\;\;\;5\end{smallmatrix}}\ar[ur]\ar[dr]&&{\begin{smallmatrix}3\\1\;1\\\;5\;5\end{smallmatrix}}\ar[ur]&&\\ & { \begin{smallmatrix}1\\5\end{smallmatrix}}\ar[ur]\ar[dr]\ar@{--}[d]&&{\begin{smallmatrix}3\\4\;1\;1\\\;\;\;\;5\;5\end{smallmatrix}}\ar[ur]\\ \ar[ur]&&{\begin{smallmatrix}1\;1\\\;5\;5\end{smallmatrix}}\ar[ur]}\ ] ]    dually , the ray in @xmath472 passing through the root injective @xmath473 is given by    @xmath474&{\begin{smallmatrix}1\\5\\3\end{smallmatrix}}\ar[r]&{\begin{smallmatrix}1\;\;\;\\5\;2\\3\end{smallmatrix}}\ar[r]&{\begin{smallmatrix}1\\5\end{smallmatrix}}\ar[r]&\dots } \ ] ]    the root projective @xmath475 lies on the coray    @xmath476 &   { \begin{smallmatrix}1\\5\\3\\4\end{smallmatrix}}\ar[r]&{\begin{smallmatrix}3\\4\;1\\\;\;\;5\\\;\;\;3\\\;\;\;4\end{smallmatrix}}\ar[r ] & { \begin{smallmatrix}3\\1\\5\\3\\4\end{smallmatrix}}\ar[r]&{\begin{smallmatrix}2\\3\\1\\5\\3\\4\end{smallmatrix}}}\ ] ]    and the root injective @xmath477 lies on the ray    @xmath478&{\begin{smallmatrix}2\\3\\1\\5\\3\end{smallmatrix}}\ar[r]&{\begin{smallmatrix}2\;\;\;\\3\;\;\;\\1\;\;\;\\5\;2\\3\end{smallmatrix}}\ar[r]&{\begin{smallmatrix}2\\3\\1\\5\end{smallmatrix}}\ar[r]&\dots } \ ] ]    note that by proposition  [ prop5 ] , every module in @xmath479 is an extension of a module in @xmath315 by @xmath480 . similarly , every module in @xmath324 is an extension of a module in @xmath317 by @xmath481 .    applying the knitting algorithm we obtain the tube in @xmath482 containing both @xmath337 and @xmath324 . @xmath483\ar@{--}[dd]&&\circ&&{\begin{smallmatrix}2\\3\\1\\5\\3\\4\end{smallmatrix}}\ar[dr]&&\circ&&{\begin{smallmatrix}4\\2\end{smallmatrix}}\ar@{--}[dd]\\ & { \begin{smallmatrix}4\end{smallmatrix}}\ar[dr]&&{\begin{smallmatrix}3\\1\\5\\3\\4\end{smallmatrix}}\ar[dr]\ar[ur]&&{\begin{smallmatrix}2\\3\\1\\5\\3\end{smallmatrix}}\ar[dr]&&{\begin{smallmatrix}2\end{smallmatrix}}\ar[ur]\\ \circ \ar@{--}[dd]&&{\begin{smallmatrix}3\\4\;1\\\;\;\;5\\\;\;\;3\\\;\;\;4\end{smallmatrix}}\ar[ur]\ar[dr]&&{\begin{smallmatrix}3\\1\\5\\3\end{smallmatrix}}\ar[ur]\ar[dr]&&{\begin{smallmatrix}2\;\;\;\\3\;\;\;\\1\;\;\;\\5\;2\\3\end{smallmatrix}}\ar[ur]\ar[dr]&&\circ\ar@{--}[dd]\\ & { \begin{smallmatrix}1\\5\\3\\4\end{smallmatrix}}\ar[ur]\ar[dr]&&{\begin{smallmatrix}3\\4\;1\\\;\;\;\;5\\\;\;\;3\end{smallmatrix}}\ar[ur]\ar[dr]&&{\begin{smallmatrix}3\;\;\;\\1\;\;\;\\5\;2\\3\end{smallmatrix}}\ar[ur]\ar[dr]&&{\begin{smallmatrix}2\\3\\1\\5\end{smallmatrix}}\ar[dr]\\ { \begin{smallmatrix}2\;\;\;\\3\;\;\;\\\;1\;1\\\;\;5\;5\\\;\;\;\;\;3\\\;\;\;\;\;4\end{smallmatrix}}\ar[ur]\ar[dr]\ar@{--}[d]&&{\begin{smallmatrix}1\\5\\3\end{smallmatrix}}\ar[ur]\ar[dr]&&{\begin{smallmatrix}3\;\;\;\\4\;1\\\;\;\;5\;2\\\;\;\;\;\;\;3\end{smallmatrix}}\ar[ur]\ar[dr]&&{\begin{smallmatrix}3\\1\\5\end{smallmatrix}}\ar[ur]\ar[dr]&&{\begin{smallmatrix}2\;\;\;\\3\;\;\;\\\;1\;1\\\;\;5\;5\\\;\;\;\;\;3\\\;\;\;\;\;4\end{smallmatrix}}\ar@{--}[d]\\ & \ar[ur]&&\ar[ur]&&\ar[ur]&&\ar[ur]&}\ ] ] let @xmath6 be cluster - tilted of euclidean type @xmath19 and let @xmath484 . then there exists @xmath485 tilting such that @xmath486 .    because @xmath19 is euclidean , @xmath95 contains at most 3 exceptional tubes . denote by @xmath487 the direct sums of those summands of @xmath20 that respectively lie in the transjective component and in the three exceptional tubes . let @xmath89 be a hereditary category that is derived equivalent to @xmath351 and such that @xmath89 is not the module category of a hereditary algebra . then @xmath89 is of the form @xmath489 , where @xmath490 consist of tubes , and @xmath100 is a transjective component , see @xcite . let @xmath491 , @xmath492 be the direct sum of all indecomposable summands of @xmath20 lying in @xmath493 , @xmath494 respectively . we define two subspaces @xmath495 and @xmath342 of @xmath6 as follows . @xmath496    the transjective component of @xmath162 contains a left section @xmath497 and a right section @xmath498 , see @xcite . thus @xmath499 are local slices , @xmath497 has no projective predecessors , and @xmath500 has no projective successors in the transjective component . define @xmath111 to be the two - sided ideal of @xmath6 generated by @xmath501 and the two subspaces @xmath495 and @xmath342 . thus @xmath502 we call @xmath111 the _ partition ideal _ induced by the partition @xmath503 . we have @xmath507 , where the last equality is as @xmath2-vector spaces . using the decomposition @xmath508 , we see that @xmath6 is equal to @xmath509 where all hom spaces are taken in @xmath488 . on the other hand , @xmath510 is a quasi - tilted algebra . thus in order to prove that @xmath1 is quasi - tilted it suffices to show that @xmath111 is the ideal generated by @xmath511 but this follows from the definition of @xmath495 and @xmath342 and the fact that the annihilators of the local slices @xmath497 and @xmath498 are given by the morphisms in @xmath512 that factor through the lifts of the corresponding local slice in the cluster category . more precisely , @xmath513 and thus @xmath514 where we used the fact that @xmath515 . this completes the proof that @xmath1 is quasi - tilted . now assume that @xmath506 . then @xmath517 and thus @xmath111 is generated by @xmath518 , and this is equal to @xmath519 on the other hand , @xmath517 implies that @xmath520 and since @xmath521 , this implies that @xmath522 is the annihilator of a local slice . therefore @xmath504 is tilted by @xcite . the case where @xmath505 is proved in a similar way .    conversely , assume @xmath1 is tilted . then @xmath523 for some local slice @xmath304 in @xmath162 . we show that @xmath522 or @xmath524 . suppose to the contrary that @xmath304 has both a predecessor and a successor in @xmath525 . then there exists an arrow @xmath526 in the quiver of @xmath6 such that @xmath527 and @xmath528 . but by definition of @xmath529 and @xmath342 , we see that this is impossible . [ thm ctaqt2 ] if @xmath1 is quasi - tilted of euclidean type and @xmath7 then @xmath530 where @xmath531 is a right section in the postprojective component of @xmath1 and @xmath532 is a left section in the preinjective component . @xmath1 being quasi - tilted implies that there is a hereditary category @xmath89 with a tilting object @xmath20 such that @xmath101 . moreover , @xmath533 is the corresponding cluster - tilted algebra . as before we use the decomposition @xmath534 . then the algebras @xmath535 are tilted . let @xmath536 and @xmath537 be complete slices in @xmath538 and @xmath539 respectively . note that @xmath536 lies in the postprojective component and @xmath532 lies in the preinjective component of their respective module categories . then @xmath1 is a branch extension of @xmath120 by the module @xmath540 similarly @xmath1 is a branch coextension of @xmath122 by the module @xmath541 observe that the postprojective component of @xmath120 does not change under the branch extension , and the preinjective component of @xmath122 does not change under the branch coextension . therefore @xmath536 is a right section in the postprojective component of @xmath1 and @xmath532 is a left section in the preinjective component . moreover , by construction , we have @xmath542 and therefore @xmath543 this completes the proof . assume first that @xmath1 is tilted . then , because of @xcite , there exists a local slice @xmath135 in the transjective component of @xmath459 such that @xmath545 . otherwise , assume that @xmath1 is quasi - tilted but not tilted . then , because of @xcite , there exists a hereditary category @xmath89 of the form @xmath546 and a tilting object @xmath20 in @xmath89 such that @xmath101 . because of theorem  [ thm ctaqt ] we get @xmath504 where @xmath111 is the partition ideal induced by the given partition of @xmath89 . let @xmath6 be the cluster - tilted algebra of type @xmath547 given by the quiver @xmath548 & & 7\ar[ll]_{\epsilon}\\   & 6\ar[ru]\ar[rdd]^(0.33){{{\beta}_3 } } \\   & 5\ar[rd]^(0.4){{\beta}_2\quad}\\   1\ar[ruu]^(0.65 ) { { \alpha}_3}\ar[ru]^(0.6 ) { { \alpha}_2}\ar[rd]_(0.6){{\alpha}_1 } & & 2\ar@<1.5pt>[ll]\ar@<-1.5pt>[ll]\\   & 3\ar[ru]_(0.4){{\beta}_1}\ar[rd ] \\ & & 4   } \ ] ] as usual let @xmath359 denote the indecomposable summand of @xmath20 corresponding to the vertex @xmath30 of the quiver . in this example @xmath20 has two transjective summands @xmath549 , and the other summands lie in three different tubes . @xmath550 lie in a tube @xmath551 , @xmath552 lies in a tube @xmath553 and @xmath554 lie in a tube @xmath555 .    choosing a partition ideal corresponds to choosing a subset of tubes to be predecessors of the transjective component . thus there are 8 different partition ideals corresponding to the 8 subsets of @xmath556 . if the tube @xmath557 is chosen to be a predecessor of the transjective component , then the arrow @xmath558 is in the partition ideal . and if @xmath559 is not chosen to be a predecessor of the transjective component , then it is a successor and consequently the arrow @xmath560 is in the partition ideal . the arrow @xmath561 is always in the partition ideal since it corresponds to a morphim from @xmath562 to @xmath563 in the derived category . the quiver of the corresponding quasi - 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in this paper , we prove that relation - extensions of quasi - tilted algebras are 2-calabi - yau tilted . with the objective of describing the module category of a cluster - tilted algebra of euclidean type , 
 we define the notion of reflection so that any two local slices can be reached one from the other by a sequence of reflections and coreflections . 
 we then give an algorithmic procedure for constructing the tubes of a cluster - tilted algebra of euclidean type . 
 our main result characterizes quasi - tilted algebras whose relation - extensions are cluster - tilted of euclidean type .