arXiv:1509.07828v1 [] 25 Sep 2015

SUPPORT VARIETIES OVER COMPLETE INTERSECTIONS
MADE EASY
PETTER ANDREAS BERGH & DAVID A. JORGENSEN
Abstract. Using an alternate description of support varieties of pairs of modules over a complete intersection, we give several new applications of such
varieties, including results for support varieties of intermediate complete intersections. Simpler proofs of known results are also given.

1. Introduction
The point of this paper is to exploit an alternate description of support varieties
for cohomology of pairs of modules over a complete intersection. A coordinatized
version of this alternate description was first given in [AvB], and subsequently generalized to non-complete intersections in [Jo1]. However, this alternate description
has not really been explored; we find that it leads to conceptually clearer proofs
of known results, as well as new results for support varieties over related complete
intersections, and related support varieties.
The paper is organized as follows. In Section 2 we give the alternate definition
of support variety, and use it to reprove known results. In Section 3 we prove
new results for support varieties of related complete intersections, which we call
intermediate complete intersections. We give a new proof in Section 4 of realizability
of support varieties over complete intersections, namely, that every cone in affine
space occurs as the support variety of some finitely generated module, even one of
finite length. In Section 5 we give further applications of support varieties. Finally,
in the appendix we recount the usual definition of support varieties for cohomology
over complete intersections, as well as other facts used in the main part of the
paper.
2. Support varieties made easy
Throughout the paper we fix a regular local ring (Q, n, k) and an ideal I generated
by a regular sequence of length c contained in n2 . By 6.5 below we can assume that
the residue field k is algebraically closed. We denote by R the complete intersection
ring
R = Q/I,
and by V the k-vector space
V = I/ n I.
For an element f ∈ I, we let f denote its image in V .
2010 Mathematics Subject Classification. 13D02, 13D07.
Key words and phrases. Complete intersections, support varieties.
Part of this work was done while we were visiting the Mittag-Leffler Institute in February and
March 2015. We would like to thank the organizers of the Representation Theory program.
1

2

PETTER ANDREAS BERGH & DAVID A. JORGENSEN

We now give our working definition of support variety. A coordinatized version
of it was originally introduced in [AvB, Theorem 2.5] as an alternate description
of support variety (see also [Avr, Corollary (3.11)]. This alternate description was
subsequently generalized to non-complete intersections in [Jo1]. The fact that our
definition is coordinate-free leads to conceptually easier proofs of known results on
support varieties, as well as to several new applications.
Definition 2.1. Let M and N be finitely generated R-modules. We define the
support variety of the pair (M, N ) to be
VR (M, N ) = {f ∈ V | ExtiQ/(f ) (M, N ) 6= 0 for infinitely many i} ∪ {0}.
We set V(M ) = V(M, k).
Remark. There are two things that are admittedly unclear regarding this definition:
(1) That the definition makes sense, in that if f, g ∈ I are such that f = g in
V , then ExtiQ/(f ) (M, N ) = 0 for all i ≫ 0 if and only if ExtiQ/(g) (M, N ) = 0
for all i ≫ 0.
(2) That VR (M, N ) is actually a closed set in V ; there is no mention of coordinates in the definition.
In the appendix (see 6.4) we recall the usual definition of support variety of pairs of
modules over a complete intersection. Choosing a minimal generating set f1 , . . . , fc
of I, and thus coordinates of V , Avramov and Buchweitz prove in [AvB, Theorem
2.5] that these two notions of support variety agree under the identification V ∋
f i ↔ ei ∈ k c . This answers (2). That (1) holds is implicit in the course of the
proof of loc. cit. We mention also that, after coordinatizing V ∋ f i ↔ ei ∈ k c ,
both properties (1) and (2) are proved explicitly to hold more generally for finitely
generated modules over arbitrary quotients of integral domains in [Jo1, Theorem
2.1].
We first reiterate some basic properties of support varieties. Properties (1)-(7)
in the following result can be found in [AvB, Theorem 5.6], and (8) follows from
[AvI, Existence Theorem 7.4(4)]. Due to our definition, the proofs we give are
significantly simpler than those in the existing literature.
Theorem 2.2. The following hold for finitely generated R-modules M and N .
(1) VR (k) = V .
(2) If ExtiR (M, N ) = 0 for all i ≫ 0, then VR (M, N ) = {0}. In particular,
if M has finite projective dimension over R, then VR (M, N ) = {0} for all
R-modules N .
(3) VR (M, N ) = VR (M ) ∩ VR (N ).
(4) VR (M, M ) = VR (k, M ) = VR (M ).
(5) If M ′ is a syzygy of M and N ′ is a syzygy of N , then VR (M, N ) =
VR (M ′ , N ′ ).
(6) If 0 → M1 → M2 → M3 → 0 and 0 → N1 → N2 → N3 → 0 are short exact
sequence of finitely generated R-modules, then for {h, i, j} = {1, 2, 3} there
are inclusions
VR (Mh , N ) ⊆ VR (Mi , N ) ∪ VR (Mj , N );
VR (M, Nh ) ⊆ VR (M, Ni ) ∪ VR (M, Nj ).

SUPPORT VARIETIES OVER COMPLETE INTERSECTIONS MADE EASY

3

(7) If M is Cohen-Macaulay of codimension m, then
VR (M ) = VR (Extm
R (M, R)).
In particular, if M is a maximal Cohen-Macaulay R-module, then VR (M ) =
VR (HomR (M, R)).
(8) If x1 , . . . , xd is an M -regular sequence, then
VR (M ) = VR (M/(x1 , . . . , xd )M ).
Proof. (1): This follows since k has infinite projective dimension over every singular
quotient ring of Q, in particular for every Q/(f ), when 0 6= f ∈ V .
(2): By the change of rings long exact sequence of Ext (see for example [Jo1,
2.1]) it is well-known that if A is a local ring and B = A/(x), where x is a nonzerodivisor of A, then vanishing of all higher Ext of a pair of B-modules over A
holds provided it does so over B. Inductively, the same is true for the quotient by a
regular sequence. Thus, if ExtiR (M, N ) = 0 for all i ≫ 0, then ExtiQ/(f ) (M, N ) = 0
for all i ≫ 0, and for every non-zero f in V .
(3): Suppose that f is not in V(M, N ). Then ExtiQ/(f ) (M, N ) = 0 for all
i ≫ 0. By taking syzygies of M and N over Q/(f ), we may assume that M and
N are maximal Cohen-Macaulay Q/(f )-modules. Then we have the conversion
Q/(f )
∗
∼
isomorphisms ExtiQ/(f ) (Ω−t
Q/(f ) (M ), N ) = Tort−i−1 (M , N ) for t ≥ 3 and 1 ≤ i ≤
−t
t−2, where ΩQ/(f ) (M ) denotes the tth cosyzygy of M [HuJ, Lemma 1.1]. Choosing
t large enough, and noting that ExtiQ/(f ) (Ω−t
Q/(f ) (M ), N ) = 0 for all i > 0, we realize
Q/(f )

(M ∗ , N ) vanishing for i > 0. Thus by [HuW, (1.9)], either
two consecutive Tori
M ∗ or N is free, and so either M or N is free. This means for the original M
and N that either f is not in V(M ) or f is not in V(N ), in other words, f is
not in V(M ) ∩ V(N ), and we have shown V(M ) ∩ V(N ) ⊆ V(M, N ). The reverse
containment is an easy exercise, using (2).
(4): This statement follows immediately from (3) and (1).
(5): Suppose that M ′ = ΩdR (M ) and N ′ = ΩeR (N ). Then we have
i−d+e
d
e
′
′
ExtiQ/(f ) (M, N ) ∼
= Exti−d+e
Q/(f ) (ΩQ/(f ) (M ), ΩQ/(f ) (N )) = ExtQ/(f ) (M , N )

for all i ≫ 0. Thus the first Ext vanishes for all i ≫ 0 if and only if so does the
third. The result follows.
(6): The long exact sequence of Ext over Q/(f ) corresponding to the first short
exact sequence is
· · · → ExtnQ/(f ) (M3 , N ) → ExtnQ/(f ) (M2 , N ) → ExtnQ/(f ) (M1 , N ) → · · ·
If f is not in V(Mi , N )∪V(Mj , N ) for {h, i, j} = {1, 2, 3}, then f is not in V(Mi , N )
and f is not in V(Mj , N ). Thus we have ExtnQ/(f ) (Mi , N ) = ExtnQ/(f ) (Mj , N ) = 0
for all n ≫ 0. Therefore ExtnQ/(f ) (Mh , N ) = 0 for all n ≫ 0, and so f is not
in V(Mh , N ). We have shown V(Mh , N ) ⊆ V(Mi , N ) ∪ V(Mj , N ). The other
containment is proved similarly.
(7) If M is Cohen-Macaulay of codimension m over R, then it is Cohen-Macaulay
of codimension m + c − 1 over Q/(f ). If moreover M has finite projective dimension
over Q/(f ), then it is a perfect Q/(f )-module of grade m + c − 1. It follows that

4

PETTER ANDREAS BERGH & DAVID A. JORGENSEN

Extm+c−1
Q/(f ) (M, Q/(f )) is also a perfect Q/(f )-module of grade m + c − 1 [BrH, Exerm+c−1
∼
cise 1.4.26]. By Rees’ Theorem we have Extm
R (M, R) = ExtQ/(f ) (M, Q/(f )). Thus
m
M and ExtR (M, R) have finite projective dimension over Q/(f ) simultaneously.
(8) By induction it suffices to consider a single non-zerodivisor x. We have
x
the short exact sequence of R-modules 0 → M −
→ M → M/xM → 0. The
corresponding long exact sequence of Ext over Q/(f ) is
x

· · · → ExtiQ/(f ) (M, k) −
→ ExtiQ/(f ) (M, k) → Exti+1
Q/(f ) (M/xM, k) → · · ·
We have f is not in VR (M ) if and only if ExtiQ/(f ) (M, k) = 0 for all i ≫ 0. By
Nakayama’s Lemma this is equivalent to ExtiQ/(f ) (M/xM, k) = 0 for all i ≫ 0, that

is, f is not in VR (M/xM ).
3. Intermediate complete intersections and support varieties
Our definition of support variety makes it possible to relate varieties over related
complete intersections.
Support varieties of intermediate complete intersections. Any complete
intersection R = Q/I of codimension at least 2 has a myriad of related complete
intersections. Namely, if W is a subspace of V , then choosing preimages in I of a
basis of W we obtain another regular sequence [BrH, Theorem 2.1.2(c,d)], and the
ideal J ⊆ I it generates. We thus get natural projections of complete intersections
Q → Q/J → Q/I = R. We call Q/J a complete intersection intermediate to Q and
R, or when the context is clear, simply an intermediate complete intersection.
Consider the natural map of k-vector spaces
ϕJ : J/ n J → W ⊆ V
defined by f + n J 7→ f + n I. This is an isomorphism: it is onto by construction,
and one-to-one since J ∩ n I = n J. A basic question is how the support variety
of a pair of R-modules (M, N ) over R relates to that of the pair (M, N ) over the
intermediate complete intersection R′ = Q/J. Using our definition of support
variety, together with the map just defined, we get a very simple and appealing
answer to this question.
Theorem 3.1. Let W be a subspace of V . Consider the intermediate complete intersection R′ = Q/J, where J is an ideal generated by a regular sequence consisting
of preimages in I of a basis of W . Then for all finitely generated R-modules M
and N we have
ϕJ (VR′ (M, N )) = VR (M, N ) ∩ W
Proof. Let f + n I ∈ ϕJ (VR′ (M, N )), where f ∈ J. Then f + n J is in VR′ (M, N ),
which means that ExtiQ/(f ) (M, N ) is nonzero for infinitely many i. But then f + n I
is in VR (M, N ) ∩ W . The reverse inclusion is equally tautological.

Equivalent intermediate complete intersections. Let Q/J and Q/J ′ be two
complete intersections intermediate to Q and R. The condition that
ϕJ (J/ n J) = ϕJ ′ (J ′ / n J ′ )
in V defines an equivalence relation on the set of such intermediate complete intersections. The next result states that equivalent intermediate complete intersections
have the same support varieties.

SUPPORT VARIETIES OVER COMPLETE INTERSECTIONS MADE EASY

5

Proposition 3.2. Suppose that R′ = Q/J and R′′ = Q/J ′ are equivalent complete
intersections intermediate to Q and R, that is, ϕJ (J/ n J) = ϕJ ′ (J ′ / n J ′ ). Then
for all finitely generated R-modules M and N we have
ϕJ (VR′ (M, N )) = ϕJ ′ (VR′′ (M, N ))
Proof. This just follows from Theorem 3.1:
ϕJ (VR′ (M, N )) = VR (M, N ) ∩ ϕJ (J/ n J)
= VR (M, N ) ∩ ϕJ ′ (J ′ / n J ′ )
= ϕJ ′ (VR′′ (M, N )).

4. realizability
Recall that V = I/ n I. For the following results, we associate to V the coordinate
ring k[x1 , . . . , xc ], coordinatized by xi ↔ f i ∈ V , where fi ∈ Q, 1 ≤ i ≤ c, form a
minimal generating set for the ideal I. For a polynomial p ∈ k[x1 , . . . , xc ], we let
Z(p) denote the zero set of p in V .
Lemma 4.1. Let p be a homogeneous polynomial in k[x1 , . . . , xc ], the coordinate
ring of V , and M a finitely generated R-module. Then there exists a finitely generated R-module Kp such that
VR (Kp ) = VR (M ) ∩ Z(p)
Proof. Let f ∈ I be an element such that f is nonzero in V . Then we have the
natural projection Q/(f ) → R, which induces natural maps ϕ : ExtR (M, k) →
ExtQ/(f ) (M, k). Consider the map ξk : k[χ1 , . . . , χc ] → ExtR (k, k) from the polynomial ring of cohomology operators associated to f1 , . . . , fc (see 6.4), and the map
′
ξM
: k[χ] → ExtQ/(f ) (k, k), from the polynomial ring of the cohomology operator
associated to f . Write f = a1 f1 + · · · + ac fc . Then by [AvB, 1.1.1] (which follows
from [Eis, Proposition 1.7]) we have commutativity of the squares
ExtiR (M, k)

ξk (χj )·

/ Exti+2 (M, k)
R
ϕi+2

ϕi


ExtiQ/(f ) (M, k)

′
aj ξk
(χ)·


/ Exti+2 (M, k)
Q/(f )

for all i ≥ 0 and all 1 ≤ j ≤ c. In other words, the equality
ϕ ◦ ξk (χj ) = aj ξk′ (χ) ◦ ϕ
holds in each degree and for all 1 ≤ j ≤ c. A straightforward computation yields
ϕ ◦ ξk (p(χ1 , . . . , χc )) = p(a1 , . . . , ac )ξk′ (χ)d ◦ ϕ
in each degree, where d is the degree of p. We note that since Q/(f ) is a hypersurface, if ExtiQ/(f ) (M, k) 6= 0 for all i ≫ 0, then ξk′ (χ) is an isomorphism in all
sufficiently high degrees (see 6.6 in the appendix). Consequently, so is ξk′ (χ)d under
the same condition.
Consider the short exact sequence of R-modules associated to p(χ1 , . . . , χc ) (see
6.1 in the appendix):
0 → M → Kp → Ωd−1
R M →0

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PETTER ANDREAS BERGH & DAVID A. JORGENSEN

This gives rise to the commutative diagram for i ≫ 0
Exti+d−1
(M, k)
R

/ ExtiR (Kp , k)

/ ExtiR (M, k)






Exti+d−1
Q/(f ) (M, k)

/ Exti
Q/(f ) (Kp , k)

/ Exti
Q/(f ) (M, k)

ϕi+d−1

αi

/ Exti+d (M, k)
R
ϕi+d

ϕi
βi


/ Exti+d (M, k)
Q/(f )

where αi = ξk (p(χ1 , . . . , χc ))·, βi = p(a1 , . . . , ac )ξk′ (χ)d ·, and we have used the fact
i
d−1
∼
that ExtiQ/(f ) (Ωd−1
R (M ), k) = ExtQ/(f ) (ΩQ/(f ) (M ), k) for all i ≫ 0 (see 6.2 in the
appendix).
Now we are ready to prove the lemma. Suppose that f ∈ V is not in VR (Kp ).
Then ExtQ/(f ) (Kp , k) = 0 for all i ≫ 0. In this case, either ExtQ/(f ) (M, k) = 0
for all i ≫ 0, in which case f is not in VR (M ), or the maps βi in the diagram
above are isomorphisms for all i ≫ 0. The latter happens only if p(a1 , . . . , ac ) 6= 0,
in other words, f is not in Z(p). Thus f is not in VR (M ) ∩ Z(p) and we have
VR (M ) ∩ Z(p) ⊆ VR (Kp ).
Now suppose that f is not in VR (M ) ∩ Z(p). If f is not in VR (M ), then
ExtiQ/(f ) (M, k) = 0 for all i ≫ 0, and this forces ExtiQ/(f ) (Kp , k) = 0 for all
i ≫ 0, so that f is not in VR (Kp ). So assume f is in VR (M ) and not in Z(p).
Then p(a1 , . . . , ac ) 6= 0 and the maps βi are isomorphisms for all i ≫ 0. Thus
ExtiQ/(f ) (Kp , k) = 0 for all i ≫ 0 and so f is not in VR (Kp ). This finishes the
proof.

Theorem 4.2. Let C be any cone in V . Then there exists a finitely generated
R-module M such that
VR (M ) = C
Proof. Since C is a cone in V , there exist homogeneous polynomials p1 , . . . , pn in
k[x1 , . . . , xc ] such that C = Z(p1 , . . . , pn ). We induct on n.
For the base case n = 1, we apply Lemma 4.1 to the R-module M = k. Thus
there exists a finitely generated R-module K1 such that
VR (K1 ) = VR (k) ∩ Z(p1 ) = V ∩ Z(p1 ) = Z(p1 )
Now suppose that we have constructed a finitely generated R-module Kn−1 such
that VR (Kn−1 ) = Z(p1 , . . . , pn−1 ). Applying Lemma 4.1 to Kn−1 now, we obtain
a finitely generated R-module Kn such that
VR (Kn ) = VR (Kn−1 ) ∩ Z(pn ) = Z(p1 , . . . , pn−1 ) ∩ Z(pn ) = Z(p1 , . . . , pn )

Remark. In fact every cone in V is realized by an R-module of finite length. Indeed,
suppose that C is a cone in V . Then from Theorem 4.2 there exists an R-module
M such that VR (M ) = C. From (5) of Theorem 2.2 we can assume that M is
maximal Cohen-Macaulay. Letting x denote a maximal regular sequence on R, we
have that it is also a regular sequence on M . Thus by (8) of Theorem 2.2 we have
that C = VR (M ) = VR (M/(x)M ), and M/(x)M has finite length.

SUPPORT VARIETIES OVER COMPLETE INTERSECTIONS MADE EASY

7

5. Applications
The following theorem shows that support varieties for complementary complete
intersections can be realized by a single module.
Theorem 5.1. Suppose that V = W1 ⊕ · · · ⊕ Wr , for subspaces Wi , 1 ≤ i ≤ r, of
V . Choose complete intersections Ri = Q/Ji intermediate to Q and R such that
ϕJi (Ji / n Ji ) = Wi , i ≤ i ≤ r. Let Mi be a maximal Cohen-Macaulay Ri -module
for 1 ≤ i ≤ r. Then for all 1 ≤ i ≤ r we have
ϕJi (VRi (Mi )) = VR (M1 ⊗Q · · · ⊗Q Mr ) ∩ Wi
Proof. By induction it suffices to prove the theorem when r = 2. By symmetry it
suffices to prove that ϕJ1 (VR1 (M1 )) = VR (M1 ⊗Q M2 ) ∩ W1 , and by Theorem 3.1,
for this it suffices to prove that VR1 (M1 ) = VR1 (M1 ⊗Q M2 ). Choose f in W1 .
Then f is regular on R2 . Since M2 is a maximal Cohen-Macaulay R2 -module, we
have that f is also regular on M2 . Thus a (necessarily) finite free resolution F of
M2 over Q yields the finite free resolution Q/(f ) ⊗Q F of Q/(f ) ⊗Q M2 over Q/(f ).
Since f annihilates M1 , we have that M1 is a Q/(f )-module, and we have the
Q/(f )
Q
(M1 , Q/(f ) ⊗Q M2 ) ∼
standard isomorphism Tori
= Tori (M1 , M2 ) for all i > 0.
Now by [Jo2, 2.2] we have TorQ
i (M1 , M2 ) = 0 for all i > 0. Thus a free resolution
of M1 ⊗Q M2 ∼
= M1 ⊗Q (Q/(f ) ⊗Q M2 ) over Q/(f ) is gotten by tensorring a free
resolution of M1 over Q/(f ) with a free resolution of Q/(f ) ⊗Q M2 over Q/(f ).
Since the latter resolution is finite, M1 ⊗Q M2 will have a finite free resolution over
Q/(f ) if and only if M1 has one, and this is what we needed to show.

One may ask how support varieties change upon moving to higher codimension.
As a corollary of Theorem 5.1, we give one basic case where it does not change.
Corollary 5.2. Suppose that W and W ′ are subspaces of V with V = W ⊕ W ′ . Let
R′ = Q/J and R′′ = Q/J ′ be intermediate complete intersections corresponding to
W and W ′ , so that ϕJ (J/ n J) = W and ϕJ ′ (J ′ / n J ′ ) = W ′ . Let M be a maximal
Cohen-Macaulay R′ -module. Then VR (M/J ′ M ) ⊆ W , and we have
ϕJ (VR′ (M )) = VR (M/J ′ M )
Proof. We first establish the inclusion. Since VR′′ (R′′ ) = 0, by Theorem 5.1 we
have 0 = ϕJ ′ (VR′′ (R′′ )) = VR (M ⊗Q R′′ ) ∩ W ′ . Thus VR (M ⊗Q R′′ ) ⊆ W ,
and since M ⊗Q R′′ ∼
= M/J ′ M , we have the desired inclusion. Noting that J ′ is
generated by an M -regular sequence, and by using (8) of Theorem 2.2 together with
Theorem 5.1, we now have ϕJ (VR′ (M )) = ϕJ (VR′ (M/J ′ M )) = VR (M/J ′ M ) ∩
W = VR (M/J ′ M ).

The following theorem gives a useful criterion for determining when syzygies
of modules over complete intersections split (see, for example [OPo]). It uses a
result of the first author, [Ber, Theorem 3.1], which says that if M is a maximal
Cohen-Macaulay R-module, then VR (M ) is irreducible if M is indecomposable.
Theorem 5.3. Let M be a maximal Cohen-Macaulay R-module, and W a subspace of V . Let R′ be an intermediate complete intersection corresponding to W .
Suppose that VR (M ) is irreducible and VR (M ) ∩ W is reducible. Then M is an
indecomposable R-module, but the R′ -syzygies of M split.

8

PETTER ANDREAS BERGH & DAVID A. JORGENSEN

Proof. By Theorem 3.1 we have that VR′ (M ) = VR (M ) ∩ W decomposes as a
union of non-zero closed sets with zero intersection. Thus the same is true for any
syzygy of M over R′ . Choose a syzygy M ′ of M over R′ such that M ′ is a maximal
Cohen-Macaulay R′ -module. Then by [Ber, Theorem 3.1], the module M ′ splits as
a direct sum of nonzero submodules.

Example. Consider Q = k[[x, y, z]], and R = Q/(x2 , y 2 , z 2 ). Then R is a codimension 3 complete intersection. We know there exists an R-module M such that
VR (M ) = Z(x1 x2 − x23 ). We actually construct said module. One can show that k
has a minimal free resolution F over R of the form
∂

∂

1
4
R→k→0
R10 → R6 → R3 −→
· · · → R21 → R15 −→

where ξk (χ1 χ2 − χ23 ) ∈ Ext4R (k, k) corresponds to the map R15 → R that sends
e1 to −1, e13 to 1, and ei to 0 for i 6= 1, 13, where the ei denote a standard basis
of R15 . With respect to the standard bases, the differentials ∂1 and ∂4 of F are
represented by the matrices

x y z
and


















z
0
0
0
0
0
0
0
0
0

y
z
0
0
0
0
0
0
0
0

x
0
z
0
0
0
0
0
0
0

0
−y
0
z
0
0
0
0
0
0

0
0
0
−x 0
0
y −x 0
0
0 −y
0
0
z
z
0
0
0
0
0
0
z
0
0
0
0
0
0
0

0
0
0
x
0
−y
z
0
0
0

0
0 0
0
0 0
0
0 0
0
0 0
0
0 y
−x 0 0
0
0 0
−y −x 0
z
0 0
0
z 0

0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
x 0
0 0
0 0
0 0
y −x 0 0
0 0
0 0
0 y −x 0
0 0
y x


















respectively. Thus, according to Theorem 4.2, the module M having support variety
Z(x1 x2 − x23 ) is the cokernel of the map represented by the following 11 × 18 matrix


z y x 0
0
0
0
0
0
0 0 0 0
0 0 0 0 0
 0 z 0 −y −x 0
0
0
0
0 0 0 0
0 0 0 0 0 


 0 0 z 0
y
−x
0
0
0
0
0
0
0
0 0 0 0 0 


 0 0 0 z
0
0 −y x
0
0 0 0 0
0 0 0 0 0 


 0 0 0 0
0
0
z
0
0
0 y x 0
0 0 0 0 0 


 0 0 0 0
z
0
0 −y −x 0 0 0 0
0 0 0 0 0 


 0 0 0 0
0
0
0
z
0
0 0 y −x 0 0 0 0 0 


 0 0 0 0
0
z
0
0 −y −x 0 0 0
0 0 0 0 0 


 0 0 0 0
0
0
0
0
z
0 0 0 y −x 0 0 0 0 


 0 0 0 0
0
0
0
0
0
z 0 0 0
y x 0 0 0 
−1 0 0 0
0
0
0
0
0
0 0 0 1
0 0 x y z
6. Appendix
We establish core facts that were used in the main body of the paper.

SUPPORT VARIETIES OVER COMPLETE INTERSECTIONS MADE EASY

9

6.1. Chain maps and short exact sequences. Let A be a ring and M an Amodule. Suppose that F is a projective resolution of M over A. Choose [p] ∈
ExtdA (M, M ). Then [p] is a homotopy equivalence class of a chain map p : Σ−d F →
F . Consider the augmented mapping cone given by:








−∂d 0 
−∂d+1 0 

p
∂1
p
∂2
augcone(p) : · · · → Fd+1 ⊕ F2 −−−−−−−−−−−−−→ Fd ⊕ F1 −−−−−−−−−−−→ Fd−1 ⊕ F0


We have the obvious short exact sequence of complexes
(1)

0 → F → augcone(p) → Σ−d+1 F → 0

which gives rise to a long exact sequence of homology, which, by taking into account
exactness, is
0 → M → Kp → Ωd−1
A (M ) → 0


−∂d 0
.
where Kp is the cokernel of the map
p
∂1
Now applying HomA (−, N ) to (1) we obtain the long exact sequence
ξN (p)·

· · · → Exti−d
−−−→ ExtiA (M, N ) → ExtiA (Kp , N ) → Exti−d+1
(M, N ) → · · ·
A (M, N ) −
A
where ξN (p)· is left multiplication by the element ξN (p) ∈ ExtdA (N, N ).
6.2. Syzygies. In this section we show that in some cases Ext is independent of
which ring syzygies of a module are taken over.
Proposition 6.1. Suppose that A is a local ring, I a proper ideal of A of finite
projective dimension and B = A/I. Let M and N be B-modules. Then
ExtiA (ΩnB (M ), N ) ∼
= ExtiA (ΩnA (M ), N )
for all n ≥ 0 and all i > pdA B.
Proof. We induct on n. The case n = 0 is trivial. Suppose the result is true for
n−1
(M ) → 0, where F
n − 1 ≥ 0. The short exact sequence 0 → ΩnB (M ) → F → ΩB
is a free B-module, gives rise to the long exact sequence
n−1
· · · → ExtiA (F, N ) → ExtiA (ΩnB (M ), N ) → Exti+1
A (ΩB (M ), N )

→ Exti+1
A (F, N ) → · · ·
Since ExtiA (F, N ) = 0 for all i > pdA B, we see that
n−1
ExtiA (ΩnB (M ), N ) ∼
= Exti+1
A (ΩB (M ), N )

for all i > pdA B. This fact, induction and dimension shifting thus give
n−1
ExtiA (ΩnB (M ), N ) ∼
= Exti+1
A (ΩB (M ), N )
∼ Exti+1 (Ωn−1 (M ), N )
=
A

A

∼
= ExtiA (ΩnA (M ), N )
for all i > pdA B.



10

PETTER ANDREAS BERGH & DAVID A. JORGENSEN

6.3. Regular Sequences. We recall some basic facts regarding regular sequences
in Cohen-Macaulay local rings. Let (A, n, k) be a Cohen-Macaulay local ring,
f1 , . . . , fc an A-regular sequence contained in n, and I = (f1 , . . . , fc ) the ideal
they generate.
(1) If g1 , . . . , gr is a part of a minimal generating set for I, then g1 , . . . , gr form
an A-regular sequence [BrH, Theorem 2.1.2(c)].
(2) A consequence of Nakayama’s Lemma is the following: g1 , . . . , gr are part
of a minimal generating set for I if and only if g1 + n I, . . . , gr + n I are
linearly independent in I/ n I.
6.4. Cohomology operators. Let R be a complete intersection of codimension
c, that is,
R = Q/(f1 , . . . , fc )
where (Q, n, k) is a regular local ring, and f1 , . . . , fc is a regular sequence contained
in n2 . If M is any R-module, then the above presentation of R induces, for each
i ∈ {1, . . . , c}, a degree −2 endomorphism of any free resolution of M , see [Eis] for
details. As shown in [Avr, Section 1] and [AvB, Section 1], these endomorphisms are
images of generators of a certain polynomial ring over R. Moreover, this polynomial
ring is “cohomologically central”, and all cohomology modules
def

n
Ext∗R (M, N ) = ⊕∞
n=0 ExtR (M, N )

over R are finitely generated over it. To be precise, there is a polynomial ring
R[χ] = R[χ1 , . . . , χc ], in c commuting cohomology operators χ1 , . . . , χc , satisfying
the following:
(1) The degree of each χi is 2.
(2) For every R-module M , there is a homomorphism
ϕM

R[χ] −−→ Ext∗R (M, M )
of graded R-algebras.
(3) For any pair (M, N ) of R-modules, the R[χ]-module structures on the cohomology Ext∗R (M, N ) via ϕN (left module structure) and ϕM (right module
structure) coincide. That is, if η ∈ R[χ] and θ ∈ Ext∗R (M, N ) are homogeneous elements, then ϕN (η) · θ = θ · ϕM (η).
(4) For any pair (M, N ) of finitely generated R-modules, the R[χ]-module
Ext∗R (M, N ) is finitely generated.
Motivated by the classical theory of cohomological support varieties over group
algebras, Avramov and Buchweitz introduced support varieties for complete intersections in [AvB]. Namely, the support variety of a pair of modules is the variety defined by the annihilator of the cohomology in k[χ]. By the last property
above, for all finitely generated R-modules M and N the graded k-vector space
k ⊗R Ext∗R (M, N ) is a finitely generated module over k ⊗R R[χ]. The latter is
naturally identified with the polynomial ring k[χ].
Definition 6.2 (Avramov-Buchweitz). Let (Q, n, k) be a regular local ring and
f1 , . . . , fc a regular sequence contained in n2 . Denote by R the complete intersection
ring R = Q/(f1 , . . . , fc ), and for R-modules M and N , denote the k[χ]-module
k ⊗R Ext∗R (M, N ) by E(M, N ).

SUPPORT VARIETIES OVER COMPLETE INTERSECTIONS MADE EASY

11

(1) For a pair (M, N ) of finitely generated R-modules, the support variety
VR (M, N ) is defined as
def

c

VR (M, N ) = {α ∈ k | φ(α) = 0 for all φ ∈ Annk[χ] E(M, N )},
where k is the algebraic closure of k.
(2) For a single finitely generated R-module M , the support variety VR (M ) is
defined as
def

VR (M ) = VR (M, M ).
We mention some properties of these cohomological support varieties. Firstly,
since Ext∗R (M, N ) is a graded module over the ring R[χ] of cohomology operators,
the k[χ]-module E(M, N ) is also graded. Its annihilator Annk[χ] E(M, N ) is therefore a homogeneous ideal, hence the support variety VR (M, N ) contains the origin
and is a cone in affine c-space Ac (k). Secondly, by [AvB, Theorem 5.6(9)], for every
finitely generated R-module M there are equalities
VR (M, M ) = VR (M, k) = VR (M, k).
Consequently, the support variety VR (M ) can be defined as any of these three
varieties. Finally, the main reason why this theory is so powerful, as illustrated
by the numerous results in [AvB], is the fact that cohomological finiteness holds.
In other words, the fact that for all finitely generated R-modules M and N , the
R[χ]-module Ext∗R (M, N ) is finitely generated. This is precisely the reason why
the support varieties encode many of the homological properties of the modules
involved. For example, the dimension of the variety of a module equals the complexity of the module. In particular, a module has trivial variety if and only if it
has finite projective dimension (or, equivalently, finite injective dimension, since
complete intersections are Gorenstein).
6.5. Algebraic closure. Let R be a complete intersection, that is, R = Q/I for
(Q, n, k) a regular local ring, and I an ideal generated by a Q-regular sequence contained in n2 . Let M and N be finitely generated R-modules. If k is not algebraically
closed, then there exists a faithfully flat extension Q → Q′ such that the residue
field k ′ of Q′ is algebraically closed (see [Bou, App., Théorèm 1, Corollarie]). In
this case R′ = Q′ /IQ′ is a complete intersection with algebraically closed residue
field, and for any ideal J of Q, the induced map Q/J → Q′ /JQ′ is a faithfully flat.
We define the support variety VR (M, N ) of the pair (M, N ) over R as the support
variety of the pair (M ⊗R R′ , N ⊗R R′ ) over R′ . Since ExtnQ/J (M, N ) vanishes if
and only if so does ExtnQ′ /JQ′ (M ⊗R R′ , N ⊗R R′ ) for every n, and for any ideal J
of Q, the results of the paper go through without assuming that k is algebraically
closed.
6.6. Hypersurfaces. In this subsection we prove the following.
Proposition 6.3. Suppose that R is a hypersurface ring, that is, a quotient of
a regular local ring (Q, n, k) by a principal ideal generated by a nonzero element
f ∈ n. Let M be a finitely generated R-module. If ExtiR (M, k) 6= 0 for infinitely
many i, then right multiplication on ExtR (M, k) by ξk (χ) is an isomorphism in all
sufficiently high degrees.

12

PETTER ANDREAS BERGH & DAVID A. JORGENSEN

Proof. The hypothesis that ExtiR (M, k) 6= 0 for infinitely many i implies that M has
infinite projective dimension over R. In this case R must be a singular hypersurface
(i.e. f ∈ n2 ) and it is well-known that k has an infinite minimal free resolution
over R, which is eventually 2-periodic [Eis]. Since R is a singular hypersurface, the
ring of cohomology operators is a polynomial ring in a single cohomology operator
variable χ. The map
ϕk : R[χ] → Ext∗R (k, k)
sends χ to the homotopy equivalence class of the periodicity endomorphism of
the minimal free resolution of k. This shows that left multiplication by ϕk (χ) on
Ext∗R (M, k) is an isomorphism in all sufficiently high degrees.

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[AvI]
[Ber]
[Bou]
[BrH]
[Eis]
[HuJ]
[HJK]
[HuW]
[Jo1]
[Jo2]
[OPo]

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Petter Andreas Bergh, Institutt for matematiske fag, NTNU, N-7491 Trondheim,
Norway
E-mail address: bergh@math.ntnu.no
David A. Jorgensen, Department of mathematics, University of Texas at Arlington,
Arlington, TX 76019, USA
E-mail address: djorgens@uta.edu

