arXiv:1801.02491v1 [] 8 Jan 2018

ON CARLSON’S DEPTH CONJECTURE
JAMES A. SCHAFER

1. Introduction
Let A = ⊕i≥0 Ai be a finitely generated graded commutative k-algebra where k
is a field and M = ⊕i≥0 Mi a finitely generated graded k-module.
Definition 1.1. ωA (M ) = min{dim P | P ∈ Ass A M }. For M = A denote this
simply by ω A.
It is well known (shown originally by Serre) that depth A M ≤ ωA M . For a proof
of this as well as other standard results of commutative algebra adapted to the
graded commutative k-algebra setting a good reference is [16]. Clearly ωA M ≤
dimA M and so depth A M ≤ ωA M ≤ dimA M for all finitely generated M .
It is quit easy to produce examples where depth A M < ωA M . However based on
the work of Benson and Carlson in [5], Carlson in [7] asked whether it might not be
the case that ω A = depth A if A = H ∗ (G, k) where k is a field whose characteristic
divides the order of the finite group G. This is obviously true if H ∗ (G, k) is CohenMacaulay since depth A = dim A in that case. Carlson showed in the same paper
the conjecture is true if dim H ∗ (G, k) = 2. Except for innumerable calculations
showing the conjecture was true the only other general result is one of D.J Green in
[13]. A basic result of Duflot [8] states that if k is a field of characteristic p > 0 then
the p-rank of the center of G is a lower bound for the depth of H ∗ (G, k). Green
shows that if G is a p-group and the depth of H ∗ (G, k) equals Duflot lower bound
then depth H ∗ (G, k) = ω H ∗ (G, k).
Note that because of the result of Duflot if dim H ∗ (G, k) = 2 the gap between
the dimension and the depth of H ∗ (G, k) is at most one. Carlson’s result could be
interpreted to say that if the gap is less than or equal to one (and the dimension
is two) then the conjecture is true. It is the aim of this paper to demonstrate the
following
Theorem. Let G be a finite group and k a field whose characteristic divides the order of G. If dim H ∗ (G, k)−depth H ∗ (G, k) = 1, then depth H ∗ (G, k) = ω H ∗ (G, k).
2. Reduction to the case A = k[x1 , . . . , xd ]
Proposition 2.1. 1) Let f : S → A be a homomorphism of Noetherian rings and
suppose A is finitely generated as an S-module. Let M be a finitely generated Amodule. Then Ass S M = f −1 (Ass A M ). If P ∈ Ass A M then dim P = dim f −1 P
and hence ωA M = ωS M .
2) Let f : S → A be an epimorphism of Noetherian rings and M a finitely
generated A-module. Suppose either S and A are local and f (mS ) ⊆ mA or S and
A are graded commutative finitely generated k algebras where k is a field, f is a
Date: January 9, 2018.
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JAMES A. SCHAFER

graded homomorphism and M a finitely generated graded module. Then depth S M =
depth A M .
3) In the graded commutative case f can also be an inclusion with A integral
over S.
Proof. 1) If Q is an A-primary in M with associated prime P = r(ann A M/Q) then
it is easily seenTthat Q is S-primary with associated prime f −1 P = r(ann S M/Q).
Hence if 0 = Q is a minimal primary decomposition of 0 as an A-module we
obtain a minimal decomposition of 0 as an S-module by intesecting all those Q
whose associated primes have the same inverse image under f . If Pi is the prime
associated to the A-primaray module Qi then this gives an S-primary decompostion
(0) = ∩Q̄rj where Q̄rj = ∩{Qi | f −1 Pri = f −1 Prj }. This is a primary decomposition of (0) with distinct primes and it must be minimal for if ∩j ′ 6=j Q̄rj′ ⊆ Q̄rj
then ∩s6=rj Qs ⊆ Qrj which is impossible since (0) = ∩Qi is a minimal primary
decomposition. The last statement follows since S/f −1 P ⊆ A/P and the latter is
finitely generated over the former.
2) If f is an epimorphism and {x1 , . . . , xk } is a maximal M -sequence in A+ , we
may choose yi ∈ S + (of the same degree) with f (yi ) = xi . Since yi m = xi m for all
m ∈ M , {y1 , . . . , yk } is an M -sequence in S + . Hence depth S M ≥ depth A M . If
{y1 , . . . yt } is an M -sequence in S + then since yi m = f (yi )m, {f (y1 ), . . . , f (yt )} is
an M -sequence in A+ and so depth A M ≥ depth S M .
3) In [12] Evens shows this is true in the strictly commutative case if A is integral
extension of S. An M -sequence for M consists of algebraically independent elements
since any M -sequence can be exended to a system of parameters for M and these are
algebraically independent, [6], theorem 2.2.7. Hence any M -sequence is contained in
S ev or Aev . Hence depth S ev M = depth S M and therefore depth A M = depth S M .

3. Modules over regular local or ∗-local domains
In this section, S will always be a Noetherian, regular ring. By (S, m) we will
mean a regular finitely generated graded commutative k-algebra over a field k with
S0 = k and where m is the maximal homogeneous ideal of elements of positive
degree or a regular local ring and m is the maximal ideal. We will just say (S, m)
is a local ring in both cases. If M is a finitely generated S-module, graded in the
graded case, pd M denotes the projective dimension of M over S and HJi (M ) is
the local cohomology of the finitely generated module M with respect to the ideal
J ⊆ S.
Theorem 3.1. i) For finitely generated M , min{j | ExtjS (M, S)) 6= 0} = codim M .
ii) If (S, m) is local then max{j | ExtjS (M, S) 6= 0} = pd M .
Proof. i) codim M = codim (ann M ), codim I = depth I = depth (I, S) for all
ideals I ⊆ S since S is Cohen-Macauley. But depth (ann M, S) ≥ n if and only
if ExtjS (M, S) = 0 for all j < n by [9], Proposition 18.4.
The proof of ii) uses the following duality theorem for Gorenstein local rings.
Proofs for the local case may be found in [15], 11.8 and for the graded k-algebra case
in [1] 11.2.5 and [2], 3.58. N ∨ is the Matlis dual HomS (N, E) where E = E(S/m)
is the injective envelope of S/m = k. In the graded case N ∨ may be identified, [2],
3.6.16, with graded Hom functor
∗ Homk (N, k) where ∗ Homk (N, k)s = Homk (N−s , k).

ON CARLSON’S DEPTH CONJECTURE

3

Remark 3.2. In the graded case, define graded Hom, ∗ HomS (M, N )t = {f : M →
N | f (Mu ) ⊆ Nu+t for all u}. If M is finitely generated then ∗ HomS (M, N ) =
HomS (M, N ) and therefore ∗ Ext∗S (M, N ) = Ext∗S (M, N ) for all N .
Theorem 3.3. Let (S, m) be a Gorenstein, local ring of dimension d and M a
finitely generated S-module. Then there exists a natural isomorphism
Hi M ∼
= Extd−i (M, S)∨ .
m

S

Remark 3.4. In the graded case [2] we have an isomorphism of graded modules
j
i
(Hm
M )∨ ≃ Extd−i
S (M, S[a]) for some a. Since Hm M is Artinian and duality is an
anti equivalence of the full subcategories of the category of graded (S, m)-modules
consisting of the finitely generated modules and the Artinian
P modules we obtain
the dedsired isomorphism. If S = k[x1 , . . . , xn ] then a = − degree(xi ).[2] p.140.
Proof of ii) By Auslander-Buchsbaum pd M + depth M = d. But
j
depth M = min{j | Hm
M 6= 0} = d − max{e | ExteS (M, S)∨ 6= 0}

and so
pdM = max{e | ExteS (M, S)∨ 6= 0}.
But the Matlis duality functor is faithful (obvious in the graded case) and hence
the conclusion.

The following result by Eisenbud, Huenke and Vasconselos will be essential.
Theorem 3.5 ([11], Theorem 1.1). Let M be a finitely generated module over a
regular domain S and set Ie = ann ExteS (M, S):
(1) codim Ie ≥ e and M/(0 :M Ie ) has no associated primes of codim e. In
particular, if P ∈ Spec S and codim P = e then P ∈ Ass (M ) iff P ∈ V (Ie ).
(2) If c = codim M then hull (0, M ) is the kernel of the natural map
ϕ : M → ExtcS (ExtcS (M, S), S)).
If M is a finitely generated S-module, let (Ass M )e = {P ∈ Ass M | codim P = e}.
Proposition 3.6. Let (S, m) be a regular domain and M a finitely generated Smodule. Then
ExtjS (M, S) = 0 implies (Ass M )j = ∅ hence
max{j | (Ass M )j 6= ∅} ≤ max{j | ExtjS (M, S) 6= 0}.
Proof. Let P ∈ Ass M be of codimension j. Then PP is an SP -associated prime
of MP which is non-zero since P ∈ V (ann M ) = Supp M . Hence SP /PP ⊆ MP
and HP0 (MP ) 6= 0 since HP0 (MP ) is the maximal submodule M ′ of MP of finite
type. HP0 (MP /M ′ ) = 0 and therefore depth MP /M ′ > 0 and pd MP /M ′ < j by
Auslander-Buchsbaum. Hence
ExtjSP (MP , SP ) → ExtjSP (M ′ , SP )
is an isomorphism. Since dim SP = j, ExtjSP (M ′ , SP ) 6= 0 and therefore
ExtjSP (MP , SP ) = ExtjS (M, S)P 6= 0.

Theorem 3.7. Let e = pd M and I = Ie = ann ExteS (M, S). Then (Ass M )e 6= ∅
if and only if HI0 (M ) 6= 0.

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JAMES A. SCHAFER

Proof. Since e = pd M , (Ass M )f = ∅ for f > e. From the above result, codim I ≥ e
and (Spec S)e ∩ V (I) = (Ass M )e . Therefore (Ass M )e = ∅ implies (is equivalent
to) all associated primes P of M have I 6⊆ P which in turn implies IP = SP and
hence (HI0 (M ))P = HI0P (MP ) = (0) but this implies HI0 (M ) = (0) by [E], Cor. 3.5.
Conversely if (Ass M )e 6= ∅ then there exists P ∈ Ass M with I ⊆ P . But PP ∈
Ass SP MP and so depth (SP , MP ) = 0. But this is equivalent to HP0P MP 6= (0) and
since HP0P MP ⊆ HI0P MP we have (HI0 M )P 6= (0) and hence HI0 M 6= (0).

Remark 3.8. An even easier proof is to note that by [9] corollary 3.13 and [11],
Ass HI0 M = (Ass M )e .
Theorem 3.9. Let (S, m) be a local, regular domain and M a finitely generated
S-module with pd M = e. Let I = Ie = ann ExteS (M, S). Then the following are
equivalent.
(1) depth M = ωM .
(2) HI0 (M ) 6= 0.
(3) HomS (ExteS (M, S), M ) 6= 0.
(4) TorSe (M, M ) 6= 0.
Proof. Since pd M = e, ExtfS (M, S) = (0) if f > e, hence (Ass M )f = ∅ for f > e.
(1) ⇐⇒ (2). If dim S = s then depth M = s − pd M = s − e by AuslanderBuchsbaum. On the other hand ωM = s − max{j | (Ass M )j 6= ∅}. Therefore
depth M = ωM if and only if (Ass M )e 6= ∅ if and only if HI0 M 6= (0).
(2) ⇐⇒ (3) Let M = ExteS (M, S). From [E], Proposition 18.4, for all N with
ann N + ann M 6= S
depth (I, N ) = min{r | ExtrS (M , N ) 6= 0}.
But ann M ⊆ ann M and so
depth (I, M ) = min{r | ExtrS (M , M ) 6= 0}.
But from [15], 6.9, depth (I, M ) = min{t | HIt M 6= 0}. Therefore
HI0 M 6= 0 ⇐⇒ depth (I, M ) = 0 ⇐⇒ Hom(ExteS (M, S), M ) 6= 0.
(3) ⇐⇒ (4). If F∗ → M → 0 is a minimal free S-resolution of M then
HomS (Fe−1 , S) → HomS (Fe , S) → ExteS (M, S) → 0
is exact and hence
0 → HomS (ExteS (M, S), M ) → HomS (HomS (Fe , S), M )) → Hom(HomS (Fe−1 , S), M ))
is exact. But for any finitely generated projective module P , there is a natural
isomorphism P ⊗ M → HomS (HomS (P, S), M ) given by x ⊗ m 7→ g where g(f ) =
f (x)m. Hence there exists an exact sequence
0 → HomS (ExteS (M, S), M ) → Fe ⊗ M → Fe−1 ⊗ M
and therefore HomS (ExteS (M, S), M ) ≃ TorSe (M, M ). Hence HI0 M 6= 0 if and only
if TorSe (M, M ) 6= 0.

Remark 3.10. In the graded case since ∗ HomS (A, B)[a] = ∗ HomS (A[−a], B) =
∗ HomS (A, B[a]) for any a ∈ Z, we have for any a, b ∈ Z, ∗ ExtjS (A, B) 6= 0 if and
only if ∗ ExtjS (A[a], B[b]) 6= 0 .

ON CARLSON’S DEPTH CONJECTURE

5

Theorem 3.11. Let (S, m) = k[x1 , . . . , xn ] be a graded polynomial ring over the
field k and M a finitely generated graded S-module of depth d. Then
d
depth S M = ωS M if and only if ∗ HomS (M ∨ , Hm
M ) 6= (0)

where M ∨ is the Matlis dual of M ,
M ∨ = ∗ Homk (M, k) ≃ ∗ HomS (M, ES (k)).
Proof. By Auslander-Buchsbaum pd M = e = n−d andPwe know depth S M = ωS M
if and only if HomS (ExteS (M, S), M ) 6= 0. If σ =
degree xi then the above
remarks say depth S M = ωS M if and only if HomS (ExteS (M, S[−σ]), M ) 6= 0.
Since ExteS (M, S[−σ]) is finitely generated ,
HomS (ExteS (M, S[−σ]), M ) = ∗ HomS (ExteS (M, S[−σ]), M ).
Since Matlis duality is an anti-equivalence of the full subcategories of graded Smodules consisting of finitely generated graded modules and Artinian modules
depth S M = ωS M if and only if ∗ HomS (M ∨ , (ExteS (M, S[−σ]))∨ ) 6= 0. By lod
cal duality, 3.3, we have (ExteS (M, S[−σ]), M )∨ ≃ Hm
M.

4. Finitely generated graded k-algebras and H ∗ (G, k)
Theorem 4.1. Let (A, m) be a finitely generated graded commutative k-algebra
and M a finitely generated graded module of depth d. Let M ∗ be the k-dual of M
with A-action given by (af )m) = f (am). Then depth A M = ωA M if and only if
d
∗ HomA (M ∗ , Hm
M ) 6= (0).
Proof. Let π : (S, mS ) = (k[x1 , . . . , xn ], (x1 , . . . , xn )) → (A, m) be an epimorphism.
By 2.1 depth A M = ωA M if and only if depth S M = ωS M and this occurs if
d
M ) 6= (0). Now by change of rings for local cohoand only if HomS (M ∨ , Hm
S
i
i
M since π(mS ) = m. If
M ≃ Hm
molgy there is a canonical isomorphism Hm
S
i
the A-module Hm M is given an S-module structure by means of π then this
isomorphism is an isomorphism of S-modules. Since M ∨ ≃ M ∗ and the Amodule and S-module structures on M ∗ correspond under π we can conclude
d
that depth A M = ωA M if and only if HomS (M ∗ , Hm
M ) 6= (0) where both M ∗
d
and Hm M are A-modules which are S-modules via π. Since π is an epimorphism
d
d
HomS (M ∗ , Hm
M ) = HomA (M ∗ , Hm
M ).

Corollary 4.2. Suppose A = H ∗ (G, k) then depth A = ωA if and only if there
depth A
exists a non-zero graded H ∗ (G, k)-map, λ : H∗ (G, k) → Hm
(H ∗ (G, k)) where
∗
H∗ (G, k) is an H (G, k)-module via cap product.
Proof. (A∗ )s = Homk (A−s , k) ≃ H−s (G, k) and it is well known that the action of
A on A∗ is the cap product.

Theorem 4.3. Let G be a finite group and k a field whose characteristic divides the
order of G. If dim H ∗ (G, k)−depth H ∗ (G, k) = 1 then depth H ∗ (G, k) = ωH ∗ (G, k).
Proof. In [14] and again from a more algebraic aspect in [3] and [5] a convergent
spectral sequence {Erp,q | p ≥ 0, r ≥ 2} of H ∗ (G, k)-modules is developed with
p,q ∗
p,q ∗
p
E2p,q = Hm
H (G, k) where Hm
H (G, k)) is the degree q part of Hm
(H ∗ (G, k))
and converging to Hq−p (G, k). That is there is an decreasing filtration F p H∗ (G, k)
p,−q
with E∞
≃ F p (Hq−p (G, k)/F p+1 Hq−p (G, k). Grothendieck’s Vanishing Theorem
p,q
says E2 = (0) if p < depth H ∗ (G, k) or p > dim H ∗ (G, k) and that both E2depth ,∗

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JAMES A. SCHAFER

and E2dim,∗ are non-zero. There is a graded H ∗ (G, k) edge homomophism ν :
depth ,∗
depth ,∗
H∗ (G, k) → E2depth ,∗ = Hm
(H ∗ (G, k)) whose image is E∞
. Hence if
depth ,∗
∗
E∞
6= (0) we have a non-zero graded H (G, k)-homomorphism
∗
ν : H∗ (G, k) → Hm
H ∗ (G, k)

and hence depth H∗ (G, k) = ωH∗ (G, k). If dim H∗ (G, k) − depth H∗ (G, k) = 1 this
spectral sequence has only two non-zero columns and so E2 = E∞ . Grothendieck’s
Theorem gives the result.

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Department of Mathematics
University of Maryland
College Park, Maryland 20742
E-mail address: jas@math.umd.edu

