arXiv:1512.05105v1 [] 16 Dec 2015

INVARIANTS OF LINKAGE OF MODULES
TONY J. PUTHENPURAKAL
Abstract. Let (A, m) be a Gorenstein local ring and let M, N be two CohenMacaulay A-modules with M linked to N via a Gorenstein ideal q. Let L be
another finitely generated A-module. We show that ExtiA (L, M ) = 0 for all
i ≫ 0 if and only if TorA
i (L, N ) = 0 for all i ≫ 0. If D is Cohen-Macaulay then
we show that ExtiA (M, D) = 0 for all i ≫ 0 if and only if ExtiA (D † , N ) = 0 for
all i ≫ 0, where D † = ExtrA (D, A) and r = codim D. As a consequence we get
that ExtiA (M, M ) = 0 for all i ≫ 0 if and only if ExtiA (N, N ) = 0 for all i ≫ 0.
We also show that EndA (M )/ rad EndA (M ) ∼
= (EndA (N )/ rad EndA (N ))op .
We also give a negative answer to a question of Martsinkovsky and Strooker.

1. introduction
Let (A, m) be a Gorenstein local ring. Recall an ideal q in A is said to be a
Gorenstein ideal if q is perfect and A/q is a Gorenstein local ring. A special class
of Gorenstein ideals are CI(complete intersection) ideals, i.e., ideals generated by
an A-regular sequence. In this paper, a not necessarily perfect ideal q such that
A/q is a Gorenstein ring will be called a quasi-Gorenstein ideal.
Two ideals I and J are linked by a Gorenstein ideal q if q ⊆ I ∩ J; J = (q : I)
and I = (q : J). We write it as I ∼q J. If q is a complete intersection then we say I
is CI-linked to J via q. If q is a quasi-Gorenstein ideal then we say I is quasi-linked
to J via q. Note that traditionally only CI-linkage used to be considered. However
in recent times more general types of linkage are studied.
We say ideals I and J is in the same linkage class if there is a sequence of ideals
I0 , . . . , In in A and Gorenstein ideals q0 . . . , qn−1 such that
(i) Ij ∼qj Ij+1 , for j = 0, . . . , n − 1.
(ii) I0 = I and In = J.
If n is even then we say that I and J are evenly linked. We can analogously define
CI-linkage class, quasi-linkage class, even CI-linkage class and even quasi-linkage
class (of ideals).
A natural question is that if I and J are in the same linkage class then what
properties of I is shared by J. This was classically done when I and J are in
the same CI-linkage class (or even CI-linkage class). In their landmark paper [12],
Peskine and Szpiro proved that if I and J are in the same CI-linkage class and I
is a Cohen-Macaulay ideal (i.e., A/I is a Cohen-Macaulay ring) then so is J. This
can be proved more generally for ideals in a quasi-linkage class, see [10, Corollary
15, p. 616]. In another landmark paper [8, 1.14], Huneke proved that if I is in
the CI-linkage class of a complete intersection then the Koszul homology Hi (I) are
Date: March 1, 2018.
1991 Mathematics Subject Classification. Primary 13C40; Secondary 13D07.
Key words and phrases. liason of modules, Gorenstien ideals, vanishing of Ext, Tor.
1

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TONY J. PUTHENPURAKAL

Cohen-Macaulay for all i ≥ 0. It is known that this result is not true in if I is linked
to a complete intersection ( via Gorenstein ideals and not-necessarily CI-ideals). If
A = k[[X1 , . . . , Xn ]] (where k is a field or a complete discrete valuation ring) then
Huneke defined some invariants of even CI-linkage class of equidimensional unmixed
ideals, see [9, 3.2]. Again these invariants are not stable under Gorenstein (even)liason.
In a remarkable paper Martsinkovsky and Strooker, [10], introduced liason for
modules. See section two for definition. We note here that ideals I and J are linked
as ideals if and only if A/I is linked to A/J as modules. One can analogously define
linkage class of modules, even linkage of modules etc. We can also define CI linkage
of modules, quasi-linkage of modules etc.
Thus a natural question arises: If M, N are in the same linkage class of modules
(or same even linkage class of modules) then what properties of M are shared by N .
The generalization of Peskine and Szpiro’s result holds. If M is Cohen-Macaulay
and N is quasi-linked to M then N is also Cohen-Macaulay, see [10, Corollary
15, p. 616]. To state another property which is preserved under linkage first let us
recall the definition of Cohen-Macaulay approximation from [1]. A Cohen-Macaulay
approximation of a finitely generated A-module M is a exact sequence
0 −→ Y −→ X −→ M −→ 0
where X is a maximal Cohen-Macaulay A-module and Y has finite projective dimension. Such a sequence is not unique but X is known to unique up to a free
summand and so is well defined in the stable category CM(A) of maximal CohenMacaulay A-modules. We denote by X(M ) the maximal Cohen-Macaulay approximation of M . In [10, Theorem 13, p. 620], Martsinkovsky and Strooker proved
that if M is evenly linked to N then X(M ) ∼
= X(N ) in CM(A). They also asked
if this result holds for M and N are in the same even quasi-linkage class, see [10,
Question 3, p. 623]. A motivation for this paper was to try and solve this question.
We answer this question in the negative. We prove
Theorem 1.1. There exists a complete intersection A of dimension one and finite
length modules M, N such that M is evenly quasi-linked to N but X(M ) ≇ X(N ).
To prove our next result we introduce a construction (essentially due to Ferrand)
which is very useful, see section three. Let CMg (A) be the full subcategory of
Cohen-Macaulay A-modules of codimension g. We prove:
Theorem 1.2. Let (A, m) be a Gorenstein local ring of dimension d. Let M ∈
CMg (A). Assume M ∼q N where q is a Gorenstien ideal in A. Let L be a finitely
generated A-module and let D ∈ CMr (A). Set D† = ExtrA (D, A). Then
(1) ExtiA (L, M ) = 0 for all i ≫ 0 if and only if TorA
i (L, N ) = 0 for all i ≫ 0.
(2) ExtiA (M, D) = 0 for all i ≫ 0 if and only if ExtiA (D† , N ) = 0 for all i ≫ 0.
We should note that this result is new even in the case for cyclic modules. A
remarkable consequence of Theorem 1.2 is the following result
Corollary 1.3. Let (A, m) be a Gorenstein local ring of dimension d. Let M ∈
CMg (A) and C ∈ CMr (A). Assume M ∼q N and C ∼n D where q, n are Gorenstein ideals in A. Then
ExtiA (M, C) = 0 for all i ≫ 0 ⇐⇒ ExtiA (D, N ) = 0 for all i ≫ 0.

INVARIANTS OF LINKAGE OF MODULES

3

In particular
ExtiA (M, M ) = 0 for all i ≫ 0 ⇐⇒ ExtiA (N, N ) = 0 for all i ≫ 0.
We also prove the following surprising invariant of quasi-linkage.
Theorem 1.4. Let (A, m) be a Gorenstein local ring and let M, N, N ′ ∈ CMg (A).
Assume M is quasi-evenly linked to N and that it is quasi-oddly linked to N ′ . Then
(1) End(M )/ rad End(M ) ∼
= End(N )/ rad End(N ).
op
(2) End(M )/ rad End(M ) ∼
= (End(N ′ )/ rad End(N ′ )) .
Here if Γ is a ring then Γop is its opposite ring.
We now describe in brief the contents of this paper. In section two we recall some
preliminaries regarding linkage of modules as given in [10]. In section three we give
a construction which is needed to prove our resuts. We prove Theorem 1.2(1) in
section four and Theorem 1.2(2) in section five. We recall some facts regarding
cohomological operators in section six. This is needed in section seven where we
prove Theorem 1.1. Finally in section eight we prove Theorem 1.4.
2. Some preliminaries on Liason of Modules
In this section we recall the definition of linkage of modules as given in [10].
Throughout (A, m) is a Gorenstein local ring of dimension d.
φ

→ F0 → M → 0
2.1. Let us recall the definition of transpose of a module. Let F1 −
be a minimal presentation of M . Let (−)∗ = Hom(−, A). The transpose Tr(M ) is
defined by the exact sequence
φ∗

0 → M ∗ → F0∗ −→ F1∗ → Tr(M ) → 0.
Also let Ω(M ) be the first syzygy of M .
Definition 2.2. Two A-modules M and N are said to be horizontally linked if
M∼
= Ω(Tr(M )).
= Ω(Tr(N )) and N ∼
Next we define linkage in general.
Definition 2.3. Two A-modules M and N are said to be linked via a Gorenstein
ideal q if
(1) q ⊆ ann M ∩ ann N , and
(2) M and N are horizontally linked as A/q-modules.
We write it as M ∼q N .
If q is a complete intersection we say M is CI-linked to N via q. If q is a quasi
Gorenstein ideal then we say M is quasi-linked to N via q.
Remark 2.4. It can be shown that ideals I and J are linked by a quasi-Gorenstein
ideal q (definition as in the introduction) if and only if the module A/I is quasilinked to A/J by q, see [10, Proposition 1, p. 592].
2.5. We say M, N are in same linkage class of modules if there is a sequence of
A-modules M0 , . . . , Mn and Gorenstein ideals q0 . . . , qn−1 such that
(i) Mj ∼qj Mj+1 , for j = 0, . . . , n − 1.
(ii) M0 = M and Mn = N .
If n is even then we say that M and N are evenly linked. Analogously we can define
the notion of CI-linkage class, quasi-linkage class, even CI-linkage class and even
quasi-linkage class (of modules).

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TONY J. PUTHENPURAKAL

3. A Construction
In this section we describe a construction essentially due to Ferrand. Throughout
(A, m) is a Gorenstein local ring of dimension d.
3.1. We note the following well-known result, see [4, 3.3.10]. Let D ∈ CMg (A).
Then ExtiA (D, A) = 0 for i 6= g. Set D† = ExtgA (D, A). Then D† ∈ CMg (A).
Furthermore (D† )† ∼
= D.
The following result is well-known. However we give a proof as we do not have
a reference.
Lemma 3.2. Let (A, m) be a Gorenstein local ring of dimension d and let M ∈
CMg (A). Let q be a quasi-Gorenstein ideal of grade g contained in ann M . Set
B = A/q. Then ExtgA (M, A) ∼
= HomB (M, B).
Proof. Let y = y1 , . . . , yg ∈ q be a regular sequence. Set C = A/(y). Then we
have a natural ring homomorphism C → B. As C, B are Gorenstein rings we have
HomC (B, C) ∼
= B, see [4, 3.3.7]. We now note that
Extg (M, A) ∼
see [4, 3.1.16]
= HomC (M, C),
A

= HomC (M ⊗B B, C),
∼
= HomB (M, HomC (B, C)),
= HomB (M, B).

3.3. Construction: Let M ∈ CMg (A) and let q be a quasi-Gorenstein ideal in A of
codimension g contained in ann M . Let M ∼q N . Let P be minimal free resolution
of M and let Q be a minimal free resolution of P0 /qP0 . We have a natural map
P0 /qP0 → M → 0. We lift this to a chain map φ : Q → P. We then dualize this
map to get a chain map φ∗ : P∗ → Q∗ . Let C = cone(φ∗ ).
Lemma 3.4.
i

H (C) =

(

N,
0,

if i = g,
if i =
6 g.

Proof. We have an exact sequence of complexes
0 → Q∗ → C → P∗ (−1) → 0.
Notice
i

∗

H (P ) =

ExtiA (M, A)

=

(

M † = ExtgA (M, A),
0,

if i = g,
if i =
6 g.

We also have
H i (Q∗ ) = ExtiA (P0 /qP0 , A) = 0 if i 6= g.
It is now immediate that
H i (C) = 0

for i 6= g − 1, g.

Set B = A/q and P = P0 /qP0 . Then by 3.2 we have
Extg (M, A) ∼
= HomB (M, B), and
A
ExtgA (P , A)

∼
= HomB (P , B).

INVARIANTS OF LINKAGE OF MODULES

5

We have an exact sequence of MCM B-modules
ǫ

→ M → 0.
0→K→P −
Dualizing with respect to B we get an exact sequence
ǫ∗

0 → HomB (M, B) −→ HomB (P , B) → K ∗ → 0.
∼ K ∗ . We note that the map H g (P∗ ) → H g (Q∗ ) is ǫ∗ .
As M ∼q N we get that N =
As a consequence we obtain that
H g−1 (C) = 0 and H g (C) = N.

Remark 3.5.
(1) If A is regular local, M = A/I, q is a complete intersection
and I ∼q J, then this construction was used by Ferrand to give a projective
resolution of A/J, see [12, Proposition 2.6].
(2) If M is perfect A-module of codimension g and q is a Gorenstein ideal
then this construction was used by Martsinkovsky and Strooker to give a
projective resolution of N , see [10, Proposition 10, p. 597].
Our interest in this construction is due to the following:
3.6. Observation: Let B g (C) be the module of g-boundaries of C and Z g (C) be
the module of g-cocycles of C. Then projdimA B g (C) is finite and Z g (C) is a
maximal Cohen-Macaulay A-module. Thus the sequence
0 → B g (C) → Z g (C) → N → 0,
is a maximal Cohen-Macaulay approximation of N .
Proof This follows from Lemma 3.4.
4. Proof of Theorem 1.2(1)
In this section we prove Theorem 1.2(1). It is an easy consequence of the following result:
Theorem 4.1. Let (A, m) be a Gorenstein local ring of dimension d. Let M ∈
CMg (A). Assume M ∼q N where q is a Gorenstien ideal in A. Let L be a maximal
Cohen-Macaulay A-module. Let P be minimal free resolution of N and let Q be
a minimal free resolution of P0 /qP0 . We do the construction as in 3.3 and let
C = cone(φ∗ ). Set X = Z g (C) and Y = B g (C). For s ≥ 1, let Xs be the image
of the map P∗s−1 → P∗s . Let x = x1 , . . . , xd be a maximal regular A-sequence. Set
A = A/(x) and L = L/xL. Then the following assertions are equivalent:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)

ExtiA (L, M ) = 0 for all i ≫ 0.
ExtiA (L, X) = 0 for all i ≫ 0.
For all s ≫ 0 and for all i ≥ 1 we have ExtiA (L, Xs ) = 0.
H i (HomA (L, P∗ )) = 0 for all i ≫ 0.
H i (HomA (L ⊗A P, A)) = 0 for all i ≫ 0.
H i (HomA (L ⊗A P, A)) = 0 for all i ≫ 0.
TorA
i (L, N ) = 0 for i ≫ 0.
TorA
i (L, N ) = 0 for i ≫ 0.

We need a few preliminary results before we are able to prove Theorem 4.1.

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TONY J. PUTHENPURAKAL
d

n
Kn+1 → · · · be a co-chain complex. Let s be an integer.
4.2. Let K : · · · → Kn −→
By K≥s we mean the co-chain complex

d

d

n
s
Kn+1 → · · ·
Ks+1 → · · · → Kn −→
0 → Ks −→

Clearly H i (K≥s ) = H i (K) for all i ≥ s + 1.
4.3. Let P, Q, C be as in Theorem 4.1. As q is a Gorenstein ideal, it has in
particular finite projective dimension. It follows that for i ≥ d+1, we get Ci = P∗i+1
and the map Ci → Ci+1 is same as the map P∗i+1 → P∗i+2 . In particular if s ≥ d+1
then C≥s = P∗≥s+1 .
We need the following:
Lemma 4.4. Let (A, m) be a Gorenstein local ring and let D be a chain-complex
with Dn = 0 for n ≤ −1. Assume that Dn is a maximal Cohen-Macaulay Amodule for all n ≥ 0. Let x ∈ m be A-regular. Let A = A/(x) and let D be the the
∗
complex D ⊗A A. Let D∗ be the complex HomA (D, A) and let D be the complex
HomA (D, A). Then the following are equivalent:
(i) H i (D∗ ) = 0 for i ≫ 0.
∗
(ii) H i (D ) = 0 for i ≫ 0.
Proof. Let E be a maximal Cohen-Macaulay A-module. Notice x is E-regular. Furthermore HomA (E, A) is a maximal Cohen-Macaulay A-module and ExtiA (M, A) =
0 for i ≥ 1. Set E = E/xE.
x
The exact sequence 0 → A −
→ A → A → 0 induces an exact sequence
x

0 → HomA (E, A) −
→ HomA (E, A) → HomA (E, A) → 0.
Thus we have an exact sequence of co-chain complexes of A-modules
x

∗

0 → D∗ −
→ D∗ → D → 0.
This in turn induces a long-exact sequence in cohomology
(4.4.1)

x

∗

· · · → H i (D∗ ) −
→ H i (D∗ ) → H i (D ) → H i+1 (D∗ ) → · · ·

We now prove (i) =⇒ (ii). This follows from (4.4.1).
x
(ii) =⇒ (i). From (4.4.1) we get that for all i ≫ 0 the map H i (D∗ ) −
→ H i (D∗ )
is surjective. The result now follows from Nakayama’s Lemma.

As an easy consequence of 4.4 we get the following:
Corollary 4.5. (with same hypotheses as in Lemma 4.4). Let x = x1 , . . . , xd be
a maximal regular sequence in A. Set B = A/(x), K = D ⊗A B, Let K∗ be the
complex HomB (K, B). Then the following are equivalent:
(i) H i (D∗ ) = 0 for i ≫ 0.
(ii) H i (K∗ ) = 0 for i ≫ 0.
We now give
Proof of Theorem 4.1. (i) ⇐⇒ (ii). By 3.6 we get that X is a maximal CohenMacaulay A-module, projdimA Y is finite and we have an exact sequence
0 → Y → X → M → 0.
As A is Gorenstein we have that injdimA Y is finite. It follows that ExtiA (L, X) ∼
=
ExtiA (L, M ) for all i ≥ d + 1. The result follows.

INVARIANTS OF LINKAGE OF MODULES

7

fs = image(Cs → Cs+1 ). For s ≥ g + 1 by Lemma
(ii) ⇐⇒ (iii). For s ≥ 1 let X
fs is maximal Cohen-Macaulay. By Lemma 3.4 we also have an
3.4 we get that X
exact sequence
fs → 0.
0 → X → Cg → Cg+1 → · · · → Cs → X

It follows that if s ≫ 0 then ExtiA (L, X) = 0 for i ≫ 0 is equivalent to
fs ) = 0 for i ≥ 1. The result now follows as q is a Gorenstein ideal; so we
ExtiA (L, X
fs = Xs+1 for s ≥ d + 1, see 4.3.
have X
(iii) =⇒ (iv). Suppose ExtiA (L, Xs ) = 0 for all i ≥ 1 and all s ≥ c. Set
a = max{g + 1, c}. As H i (P∗ ) = ExtA
i (N, A) = 0 for i ≥ g + 1 we have an exact
sequence
0 → Xa → P∗a+1 → P∗a+2 → · · · → P∗n → P∗n+1 → · · ·
As ExtiA (L, Xs ) = 0 for all i ≥ 1 and all s ≥ a we get that the induced sequence
0 → HomA (L, Xa ) → HomA (L, P∗a+1 ) → · · ·
is exact. It follows that H i (L, P∗≥a+1 ) = 0 for i ≥ a + 2. Thus by 4.2 we get that
H i (L, P∗ ) = 0 for i ≥ a + 2.
(iv) =⇒ (iii). Suppose H i (L, P∗ ) = 0 for i ≥ r. If s ≥ r then H i (L, P∗≥s ) = 0
for all i ≥ s + 1. Now let a = max{g + 1, r}. Let s ≥ a. As argued before we have
an exact sequence
0 → Xs → P∗s+1 → P∗s+2 → · · · → P∗n → P∗n+1 → · · ·
As H i (L, P∗≥s ) = 0 for all i ≥ s + 1 we get that the induced sequence
0 → HomA (L, Xs ) → HomA (L, P∗s+1 ) → · · ·
2
1
∼
is exact. In particular ExtA
1 (L, Xs ) = 0. Notice ExtA (L, Xs ) = ExtA (L, Xs+1 ).
i
The latter module is zero by the same argument. Iterating we get ExtA (L, Xs ) = 0
for all i ≥ 1.
(iv) ⇐⇒ (v). We have an isomorphism of complexes
HomA (L, P∗ ) ∼
= HomA (L ⊗A P, A).

The result follows.
(v) ⇐⇒ (vi). D = L⊗A P is a complex of maximal Cohen-Macaulay A-modules
since L is maximal Cohen-Macaulay and Pn is a finitely generated free A-module.
Also Dn = 0 for n ≤ −1. Set K = D ⊗A A = L ⊗A P. The result now follows from
Corollary 4.5.
(vi) ⇐⇒ (vii). We note that A is a zero-dimensional Gorenstein local ring and
so is injective as an A-module. Furthermore A is the injective hull of its residue
field. It follows that
H i (Hom (L ⊗A P, A)) ∼
= Hom (Hi (L ⊗A P), A) = Hom (TorA (L, N ), A).
A

A

A

i

Therefore by [4, 3.2.12] we get the result.
(vii) ⇐⇒ (viii). As L is a maximal Cohen-Macaulay A-module we get that x
is an L-regular sequence. Set Lj = L/(x1 , . . . , xj ) for j = 1, . . . , d. Note Ld = L.
x1
L → L1 → 0. This induces a long exact
We have an exact sequence 0 → L −→
sequence
x

1
A
A
A
· · · → TorA
i (L, N ) −→ Tori (L, N ) → Tori (L1 , N ) → Tori−1 (L, N ) → · · ·

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TONY J. PUTHENPURAKAL

A
Clearly if TorA
i (L, N ) = 0 for i ≫ 0 then Tori (L1 , N ) = 0 for all i ≫ 0. Conversely
x1
A
A
if Tori (L1 , N ) = 0 for all i ≫ 0 then for the map TorA
i (L, N ) −→ Tori (L, N ) is
A
surjective for all i ≫ 0. By Nakayama’s Lemma we get Tori (L, N ) = 0 for i ≫ 0.
x2
L1 → L2 → 0. A similar argument
We also have an exact sequence 0 → L1 −→
A
gives that TorA
i (L1 , N ) = 0 for i ≫ 0 if and only if Tori (L2 , N ) = 0 for all i ≫ 0.
A
Combining with the previous result we get that Tori (L, N ) = 0 for i ≫ 0 if and
only if TorA
i (L2 , N ) = 0 for all i ≫ 0.
Iterating this argument yields the result.


We now give
Proof of Theorem 1.2(1). Let 0 → W → E → L → 0 be a maximal CohenMacaulay approximation of L. As projdimA W is finite we get that ExtiA (L, M ) ∼
=
A
∼
ExtiA (E, M ) for i ≥ d + 2 and TorA
i (L, N ) = Tori (E, N ) for i ≥ d + 2. The result
now follows from Theorem 4.1.

5. Proof of Theorem 1.2(2) and Corollary 1.3
In this section we prove Theorem 1.2(2). We need a few preliminary facts to
prove this result.
5.1. Let (A, m) be a Noetherian local ring and let E be the injective hull of its
residue field. If G is an A-module then set G∨ = HomA (G, E). Let ℓ(G) denote
the length of G. The following result is known. We give a proof as we are unable
to find a reference.
Lemma 5.2. Let (A, m) be a Noetherian local ring and let E be the injective hull
of its residue field. Let M be finitely generated A-module. Let L be an A-module
with ℓ(L) < ∞. Then for all i ≥ 0 we have an isomorphism
Exti (M, L) ∼
= (TorA (M, L∨ ))∨ .
A

i

Proof. We note that (L ) ∼
= L, see [4, 3.2.12]. Let P be a minimal projective
resolution of M . We have the following isomorphism of complexes:
HomA (P ⊗A L∨ , E) ∼
= HomA (P, HomA (L∨ , E)) ∼
= HomA (P, L).
∨ ∨

We have H i (HomA (P, L)) = ExtA
i (M, L). Notice as E is an injective A-module we
have
H i (HomA (P ⊗A L∨ , E)) ∼
= HomA (Hi (P ⊗A L∨ ), E)
∨
∼
= HomA (TorA
i (M, L ), E)
∨ ∨
= (TorA
i (M, L )) .


The following result is also known. We give a proof as we are unable to find a
reference.
Lemma 5.3. Let (A, m) be a Gorenstein local ring of dimension d and let E be the
injective hull of its residue field. Let D ∈ CMr (A) and let x = x1 , . . . , xc ∈ m be a
D-regular sequence (note c ≤ d − r). Then
(1) x is a D† -regular sequence.
(2) (D/xD)† ∼
= D† /xD† .

INVARIANTS OF LINKAGE OF MODULES

9

(3) If r = d (and so ℓ(D) < ∞) then
D† ∼
= D∨ (= HomA (D, E)).
(4) If T is another finitely generated D-module then
(a) ExtiA (T, D) = 0 for i ≫ 0 if and only if ExtiA (T, D/xD) = 0 for i ≫ 0.
(b) ExtiA (D, T ) = 0 for i ≫ 0 if and only if ExtiA (D/xD, T ) = 0 for i ≫ 0.
Proof. (1) and (2) : Let x be D-regular. Then notice D/xD is a Cohen-Macaulay
x
A-module of codimension r + 1. We have a short-exact sequence 0 → D −
→D→
D/xD → 0. Applying the functor HomA (−, A) yields a long exact sequence which
after applying 3.1 reduces to a short-exact sequence
x

0 → ExtrA (D, A) −
→ ExtrA (D, A) → Extr+1
A (D/xD, A) → 0.
It follows that x is D† -regular and D† /xD† ∼
= (D/xD)† .
Iterating this argument yields (1) and (2).
(3): Let y = y1 , . . . , yd ⊂ annA D be a maximal A-regular sequence. Set B =
A/(y). Then
D† = ExtnA (D, A) ∼
= HomB (D, B).
We have
D∨ = HomA (D, E) = HomA (D ⊗A B, E) ∼
= HomB (M, HomA (B, E)).
Now HomA (B, E) is an injective B-module of finite length. As B is an Artin Gorenstein local ring we get that HomA (B, E) is free as a B-module. As ℓ(HomA (B, E)) =
ℓ(B) (see [4, 3.2.12]) it follows that HomA (B, E) = B. The result follows.
(4): For i = 1, . . . , c set Di = D/(x1 , . . . , xi )D.
x1
D → D1 → 0 induces a long exact sequence
4(a): The exact sequence 0 → D −→
(5.3.2)

x

1
ExtiA (T, D) → ExtiA (T, D1 ) → Exti+1
· · · ExtiA (T, D) −→
A (T, D) → · · ·

If Exti (T, D) = 0 for all i ≫ 0 then by (5.3.2) we get ExtiA (T, D1 ) = 0 for all i ≫ 0.
Conversely if ExtiA (T, D1 ) = 0 for all i ≫ 0 then by (5.3.2) we get that the map
x1
ExtiA (T, D) −→
ExtiA (T, D) is surjective for all i ≫ 0. So by Nakayama’s Lemma
i
Ext (T, D) = 0 for all i ≫ 0.
x2
D1 → D2 → 0. A similar argument
We also have an exact sequence 0 → D1 −→
to the above yields that Exti (T, D1 ) = 0 for all i ≫ 0 if and only if Exti (T, D2 ) = 0
for all i ≫ 0. Combining this with the previous result we get Exti (T, D) = 0 for
all i ≫ 0 if and only if Exti (T, D2 ) = 0 for all i ≫ 0.
Iterating this argument yields the result.
4(b): This is similar to 4(a).

We now give
Proof of Theorem 1.2(2). Let x = x1 , . . . , xd−r be a maximal D-regular sequence.
Then by 5.3, x is also a D† regular sequence. Also by 5.3, we get D† /xD† =
(D/xD)† . Let E be the injective hull of the residue field of A.

10

TONY J. PUTHENPURAKAL

We have
ExtiA (M, D) = 0 for all i ≫ 0
⇐⇒ ExtiA (M, D/xD) = 0 for all i ≫ 0; see 5.3,
∨
⇐⇒ TorA
i (M, (D/xD) ) = 0 for all i ≫ 0; see 5.2 and [4, 3.2.12],
†
†
⇐⇒ TorA
i (M, D /xD ) = 0 for all i ≫ 0; see 5.3,
†
†
⇐⇒ ExtA
i (D /xD , N ) = 0 for all i ≫ 0; see Theorem 1.2(1),
†
⇐⇒ ExtA
i (D , N ) = 0 for all i ≫ 0; see 5.3.


We now give
Proof of Corollary 1.3. By Theorem 1.2(2) we get that
ExtiA (M, C) = 0 for all i ≫ 0 ⇐⇒ ExtiA (C † , N ) = 0 for all i ≫ 0.
We note that C † = ExtrA (C, A) ∼
= HomA/n (C, A/n), see 3.2. As C ∼n D we have
†
an exact sequence 0 → C → G → D → 0, where G is a finitely generated free
A/n-module. As n is a Gorenstein ideal in A we get projdimA G is finite. The result
follows.

6. Some preliminaries to prove Theorem 1.1
In this section we discuss a few preliminaries which will enable us to prove
Theorem 1.1. More precisely we need the notion of cohomological operators over a
complete intersection ring; see [7] and [6].
6.1. Let f = f1 , . . . , fc be a regular sequence in a local Noetherian ring (Q, n). We
assume f ⊆ n2 . Set I = (f ) and A = Q/I.
6.2. The Eisenbud operators, [6] are constructed as follows:
∂

∂

→ Fi → · · · be a complex of free A-modules.
→ Fi+1 −
Let F : · · · → Fi+2 −
Step 1: Choose a sequence of free Q-modules Fei and maps ∂e between them:
e
e
∂
∂
e : · · · → Fei+2 −
→ Fei → · · ·
→ Fei+1 −
F

e
so that F = A ⊗ F
Pc
Step 2: Since ∂e2 ≡ 0 modulo (f ), we may write ∂e2 = j=1 fj e
tj where tej : Fei →
e
Fi−2 are linear maps for every i.
Step 3: Define, for j = 1, . . . , c the map tj = tj (Q, f , F) : F → F(−2) by tj =
A⊗e
tj .
6.3. The operators t1 , . . . , tc are called Eisenbud’s operator’s (associated to f ) . It
can be shown that
(1) ti are uniquely determined up to homotopy.
(2) ti , tj commute up to homotopy.

6.4. Let R = A[t1 , . . . , tc ] be a polynomial ring over A with variables t1 , . . . , tc
of degree 2. Let M, N be finitely generated A-modules. By considering a free
resolution F of M we get well defined maps
tj : ExtnA (M, N ) → Extn+2
R (M, N )

for 1 ≤ j ≤ c and all n,

INVARIANTS OF LINKAGE OF MODULES

11

L
which turn Ext∗A (M, N ) = i≥0 ExtiA (M, N ) into a module over R. Furthermore
these structure depend on f , are natural in both module arguments and commute
with the connecting maps induced by short exact sequences.
6.5. Gulliksen, [7, 3.1], proved that if projdimQ M is finite then Ext∗A (M, N ) is a
finitely generated R-module. For N = k, the residue field of A, Avramov in [2,
3.10] proved a converse; i.e., if Ext∗A (M, k) is a finitely generated R-module then
projdimQ M is finite.
6.6. We need to recall the notion of complexity of a module. This notion was
th
introduced by Avramov in [2]. Let βiA (M ) = ℓ(TorA
Betti number
i (M, k)) be the i
of M over A. The complexity of M over A is defined by


β A (M )
cxA M = inf b ∈ N | lim n b−1 < ∞ .
n→∞ n

If A is a local complete intersection of codim c then cxA M ≤ c. Furthermore all
values between 0 and c occur.

6.7. Since m ⊆ ann ExtiA (M, k) for all i ≥ 0 we get that Ext∗A (M, k) is a module
over S = R/mR = k[t1 , . . . , tc ]. If projdimQ M is finite then Ext∗A (M, k) is a finitely
generated S-module of Krull dimension cx M .
6.8. If (Q, n) is regular then by [3, Theorem I(3)] we get that dimS Ext∗A (M, k) =
dimS Ext∗A (k, M ). In particular if M is maximal Cohen-Macaulay A-module then
cx M = cx M ∗ . Using this fact it is not difficult to show that if M is a CohenMacaulay A-module then cx M = cx M † .
6.9. Let Q is regular local with infinite residue field and let M be a finitely generated A-module with cx(M ) = r. The surjection Q → A factors as Q → R → A,
with the kernels of both maps generated by regular sequences, projdimR M < ∞
and cxA M = projdimR A (see [2, 3.9]).
We need the following:
Proposition 6.10. Let Q = k[x, y, z](x,y,z) where k is an infinite field. Let m be
the maximal ideal of Q. Let q be an m-primary Gorenstein ideal such that q is not a
complete intersection. Suppose q ⊇ (u, v) where u, v ∈ m2 is an Q-regular sequence.
Set A = Q/(u, v) and q = q/(u, v). Then cx A/q = 2.
Proof. As codim A = 2 we have that cx A/q = 0, 1 or 2. We prove cx A/q 6= 0, 1.
If cx A/q = 0 then A/q has finite projective dimension over A. So by AuslanderBuchsbaum formula projdimA A/q = 1. Therefore q is a principal ideal. It follows
that q is a complete intersection, a contradiction.
If cx A/q = 1 then by 6.9, the surjection Q → A factors as Q → R → A, with
the kernels of both maps generated by regular sequences, projdimR A/q < ∞ and
cxA A/q = projdimR A = 1. Thus dim R = 2. So R = Q/(h) for some h.
As A/q has finite length, by Auslander-Buchsbaum formula projdimR A/q = 2.
Consider the minimal resolution of A/q over R:
0 → Rb → Ra → R → A/q → 0.
As A/q is a Gorenstein ring we have b = 1 and so a = 2. Thus there exists α, β ∈ R
with A/q = R/(α, β) = Q/(h, α, β). It follows that q is a complete intersection
ideal, a contradiction.


12

TONY J. PUTHENPURAKAL

7. Proof of Theorem 1.1
In this section we prove Theorem 1.1. We first make the following:
7.1. Construction: Let Q = Q[x, y, z](x,y,z) and let m be its maximal ideal. We
construct a m-primary Gorenstein ideal q in Q such that
(1) q is not a complete intersection.
(2) There exists a Q-regular sequence f, u, v ∈ m2 such that
(f a , v b , u) ⊆ q ⊆ (f, u, v) for some a, b ≥ 2.
Set u = x2 + y 2 + z 2 , R = Q/(u), n = m/(u). Then note (x, y) is a reduction
of n and n2 = (x, y)n. It follows that x7 , y 7 is a regular sequence in R. So I =
(x7 , y 7 ) : (xy + yz + xz) is a Gorenstein ideal. Using Singular, [5], it can be shown
that I has 12 minimal generators and I ⊆ n6 . In particular I is not a complete
intersection in R. Also note that n6 ⊆ (x, y)5 ⊆ (x2 , y 2 ). Let q be an ideal in Q
containing u such that q/(u) = I. Then q has 12 or 13 minimal generators. So q is
not a complete intersection. Also clearly q is m-primary. Note
((x2 )4 , (y 2 )4 , u) ⊆ (x7 , y 7 , u) ⊆ q ⊆ (x2 , y 2 , u).
Set f = x2 , v = y 2 and a = b = 4.
We now give
Proof of Theorem 1.1. Let Q = Q[x, y, z](x,y,z) and let m be its maximal ideal.
We make the construction as in 7.1. Let E be a non-free stable maximal CohenMacaulay Q/(f )-module. Let
0 → Qr → Qr → E → 0
be a minimal resolution of E as a Q-module. Note u is Q/(f )-regular and so
E-regular. Thus we have an exact sequence
0 → (Q/(u))r → (Q/(u))r → E/uE → 0.
Set A = Q/(u, f a ) and q = q/(u, f a ). Then note that q is a quasi-Gorenstein
ideal in A which is not a complete intersection. Furthermore notice that E/uE is
a maximal Cohen-Macaulay Q/(f, u)-module and so a maximal Cohen-Macaulay
A-module. Furthermore notice that cx E/uE = 1 as an A-module.
We now note that v is A-regular and so E/uE-regular. Set M = E/(u, v)E.
Thus M is an A-module of finite length. Also cxA M = 1. Furthermore
q ⊆ (f, u, v) ⊆ annQ M.
So we have q ⊆ annA M . Thus we have finite length module M of complexity one
and a quasi-Gorenstein ideal q of complexity two. Set B = A/q. Clearly M does
not have B as a direct summand. Set N = ΩB (TrB (M )). Then M is horizontally
linked to N as B-modules, see [10, Proposition 8, p. 596]. So M ∼q N as Amodules. If t ∈ annA M is a regular element then it is not difficult to show that
there exists i ≥ 1 such that the C = A/(ti )-module M has no free summands as a
C-module. Let M ∼ti L as A-modules.
By Lemma 7.2 we have: cxA N = 2 and cxA L = 1.
This finishes the proof as for any module P the complexity of a maximal CohenMacaulay approximation of P is equal to complexity of P . We note that N is
evenly linked to L and cx X(N ) = cx N = 2 while cx X(L) = cx L = 1. It follows
that X(N ) is not stably isomorphic to X(L).

INVARIANTS OF LINKAGE OF MODULES

13


We now state and prove the Lemma we need to finish the proof of Theorem 1.1.
Lemma 7.2. Let (Q, n) be a regular local ring and let f = f1 , . . . , fc ∈ n2 be a
regular sequence. Set A = Q/(f ). Let M ∈ CMg (A). Also let M ∼q N where q is
a quasi-Gorenstein ideal in A. Then
(1) If q is a Gorenstein ideal then cx M = cx N .
(2) If projdim A/q = ∞ and cx A/q > cx M then cx N = cx A/q.
Proof. Set B = A/q. Let M † = ExtgA (M, A) ∼
= HomB (M, B), by Lemma 3.2. As
M ∼q N we have an exact sequence
(7.2.3)

0 → M † → G → N → 0;

where G is free B-module.

By [3, 3.3] we get cx M † = cx M . The result now follows from the exact sequence
7.2.3 and 6.7.

8. Proof of Theorem 1.4
In this section we prove Theorem 1.4. We need to prove several preliminary
results first.
8.1. Let M, N be finitely generated A-modules. By β(M, N ) we mean the subset
of HomA (M, N ) which factor through a finitely generated free A-module. We first
prove the following
Proposition 8.2. Let (A, m) be a Gorenstein local ring. Let M be a maximal
Cohen-Macaulay A-module with no free summands. Then β(M, M ) ⊆ rad End(M ).
Proof. Let f ∈ β(M, M ). Say f = v ◦ u where u : M → F , v : F → M and
F = An . Let u = (u1 , . . . , un ) where ui : M → A. As M does not have a free
summand we get that ui (M ) ⊆ m for each i. Thus u(M ) ⊆ mF . It follows that
f (M ) ⊆ mM . Thus f ∈ HomA (M, mM ). However it is well-known and easy to
prove that HomA (M, mM ) ⊆ rad End(M ).

8.3. It can be easily seen that β(M, M ) is a two-sided ideal in End(M ). Set
End(M ) = End(M )/β(M, M ). Assume A is Gorenstein and M is maximal CohenMacaulay with no free summands. Let
u

π

0→N −
→F −
→ M → 0,
be a minimal presentation. Note that N is also maximal Cohen-Macaulay with no
free-summnads. We construct a ring homomorphism
σ : End(M ) → End(N )
as follows: Let θ ∈ End(M ). Let δ ∈ End(N ) be a lift of θ. We first note that if
δ ′ is another lift of θ then it can be easily shown that there exists ξ : F → N such
that ξ ◦ u = δ − δ ′ . Thus we have a well defined element σ(θ) ∈ End(N ). Thus we
have a map
σ
e : End(M ) → End(N )
It is easy to see that σ
e is a ring homomorphism. We prove

Proposition 8.4. (with hypotheses as above) If f ∈ β(M, M ) then σ
e (f ) = 0.

14

TONY J. PUTHENPURAKAL

Proof. Let F be a minimal resolution of M with F0 = F . Suppose f = ψ ◦ φ
where ψ : G → M , φ : M → G and G = Am for some m. We may take G be a
minimal resolution of G with G0 = G and Gn = 0 for n > 0. We can construct a
lift fe: F → F of f by composing a lift of v with that of u. It follows that fen = 0
for n ≥ 1. An easy computation shows that for this lift fe the corresponding map
δ : N → N is infact zero. Thus σ
e(f ) = 0.

Thus we have a ring homomorphism σ : End(M ) → End(N ). Our next result is

Proposition 8.5. (with hypotheses as above) σ is an isomorphism.
Proof. We construct a ring homomorphism τ : End(N ) → End(M ) which we show
is the inverse of σ.
Let θ : N → N be A-linear. Then θ∗ : N ∗ → N ∗ is also A-linear. We dualize the
exact sequence 0 → N → F → M → 0 to get an exact sequence
π∗

u∗

0 → M ∗ −→ F ∗ −→ N ∗ → 0.
We can lift θ∗ to an A-linear map δ : M ∗ → M ∗ . Also if δ ′ is another lift then as
before it is easy to see δ − δ ′ ∈ β(M ∗ , M ∗ ). We define τe : End(N ) → End(M ) by
τe(θ) = δ ∗ . It is clear that τe is a ring homomorphism. Finally as in Proposition 8.4,
it can be easily proved that if θ ∈ β(N, N ) then τe(θ) = 0. Thus we have a ring
homomorphism τ : End(M ) → End(N ). Finally it is tautalogical that
τ ◦ σ = 1End(M)

and σ ◦ τ = 1End(N ) .


8.6. If φ : R → S is an isomophism of rings then it is easy to see that φ(rad R) =
rad S and thus we have an isomorphism of rings R/ rad R ∼
= S/ rad S. As a corollary
we obtain
Corollary 8.7. (with hypotheses as above)
End(M )/ rad End(M ) ∼
= End(N )/ rad End(N ).
Proof. By Proposition 8.5 we have an isomorphism σ : End(M ) → End(N ). By 8.2
we have that β(M, M ) ⊆ rad End(M ). It follows that
rad End(M ) = rad End(M )/β(M, M ).
Similarly rad End(N ) = rad End(N )/β(N, N ). The result follows from 8.6.



We now give
Proof of Theorem 1.4. It suffices to consider to prove that if
M0 ∼q M1
∼
then EndA (M0 )/ rad EndA (M0 ) = (EndA (M0 )/ rad EndA (M0 ))op . By assumption
M0 , M1 ∈ CMg (A). It follows that q is a codimension g quasi-Gorenstein ideal [10,
Lemma 14, p. 616]. Set B = A/q. Then B is a Gorenstein ring. Notice M0 , M1 are
maximal Cohen-Macaulay B-modules. Furthermore they are stable B-modules, see
[10, Proposition 3, p. 593]. Notice HomA (Mi , Mi ) = HomB (Mi , Mi ) for i = 0, 1.
As M0 is horizontally linked to M1 we get that M0∗ ∼
= Ω(M1 ). By 8.7 we get
that
End(M1 )/ rad End(M1 ) ∼
= End(M0∗ )/ rad End(M0∗ ).

INVARIANTS OF LINKAGE OF MODULES

15

Furthermore it is easy to see that EndB (M0 ) ∼
= EndB (M0∗ )op and this is preserved
when we go mod radicals. Thus
op
End(M1 )/ rad End(M1 ) ∼
= (End(M0 )/ rad End(M0 )) .

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approximations, Colloque en l’honneur de Pierre Samuel (Orsay, 1987). Mm. Soc. Math.
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101.
[3] L. L. Avramov and R-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), no. 2, 285-318.
[4] W. Bruns and J. Herzog, Cohen-Macaulay Rings, revised edition, Cambridge Stud. Adv.
Math., vol. 39, Cambridge University Press, Cambridge, (1998).
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algebra system for polynomial computations. http://www.singular.uni-kl.de (2015).
[6] D. Eisenbud, Homological algebra on a complete intersection, with an application to group
representations, Trans. Amer. Math. Soc. 260 (1980), 35–64.
[7] T. H. Gulliksen, A change of ring theorem with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974), 167–183.
[8] C. Huneke, Linkage and the Koszul homology of ideals, Amer. J. Math. 104 (1982), no. 5,
1043-1062.
[9] C. Huneke, Numerical invariants of liaison classes, Invent. Math. 75 (1984), no. 2, 301-325.
[10] A. Martsinkovsky and J. R. Strooker, Linkage of modules, J. Algebra 271 (2004), no. 2,
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[11] H. Matsumura, Commutative ring theory, Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press,
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Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai
400 076, India
E-mail address: tputhen@math.iitb.ac.in

