arXiv:1107.3102v2 [] 15 Dec 2013

VANISHING OF TATE HOMOLOGY
AND DEPTH FORMULAS OVER LOCAL RINGS
LARS WINTHER CHRISTENSEN AND DAVID A. JORGENSEN
To Hans-Bjørn Foxby on his 65th birthday
Abstract. Auslander’s depth formula for pairs of Tor-independent modules
over a regular local ring, depth(M ⊗R N ) = depth M + depth N − depth R,
has been generalized in several directions; most significantly it has been shown
to hold for pairs of Tor-independent modules over complete intersection rings.
In this paper we establish a depth formula that holds for every pair of Tate
Tor-independent modules over a Gorenstein local ring. It subsumes previous
generalizations of Auslander’s formula and yields new results on vanishing of
cohomology over certain Gorenstein rings.

Introduction
To infer properties, qualitative or quantitative, of a tensor product from properties
of its factors is a delicate task. For finitely generated modules M and N over a
commutative noetherian local ring R, Auslander [3] proved that the depth of the
tensor product is given by the formula
(A)

depthR (M ⊗R N ) = depthR M + depthR N − depth R,

provided that the projective dimension of M is finite and the two modules are Torindependent, that is, the homology modules TorR
i (M, N ) vanish for i > 1. In particular, the equality (A) holds for every pair of finitely generated Tor-independent
modules over a regular local ring.
The condition of finite projective dimension was first relaxed by Huneke and
Wiegand [21], who established the validity of (A) for pairs of finitely generated
Tor-independent modules over complete intersection local rings. Later, Araya and
Yoshino [1] and Iyengar [24] showed that (A) holds for Tor-independent modules
M and N , provided that M has finite complete intersection dimension.
In a different direction, Foxby [17] relaxed the condition of Tor-independence as
follows. Let M and N be modules over a commutative noetherian local ring R and
let P be a projective resolution of M . If M has finite projective dimension, then
there is an equality,
(B)

depthR (P ⊗R N ) = depthR M + depthR N − depth R.

Date: 13 December 2013.
2010 Mathematics Subject Classification. Primary 13D07. Secondary 13D02.
Key words and phrases. AB ring, complete intersection ring, depth formula, Gorenstein ring,
rigidity of Tor, Tate homology.
This research was partly supported by NSA grants H98230-11-0214 (L.W.C.) and H9823010-0197 (D.A.J.). Part of the work was done during the authors’ visits to the University of
Bielefeld (D.A.J) and the University of Paderborn (L.W.C.); the hospitality of both institutions
is acknowledged with gratitude.

2

L. W. CHRISTENSEN AND D. A. JORGENSEN

Implicit in this formula is an extension of the invariant “depth” to complexes of
modules; we recall it in (1.3). The homology of the complex P ⊗R N is TorR
∗ (M, N ),
and if M and N are Tor-independent, then (B) reduces to Auslander’s formula (A).
∗∗∗
The notion of Tate homology for modules over group algebras has a natural extension to modules over Gorenstein rings and, more generally, to modules of finite
Gorenstein projective dimension over any ring. This theory was recently treated by
Iacob [22]. We recall the basics in (2.2); a broader discussion is given in [14, sec. 2].
Let R be a commutative noetherian local ring. The central result in this paper,
d R (M, N ) as a sufficient
Theorem (2.3), establishes vanishing of Tate homology Tor
∗
condition for the equality (B) to hold for a pair of R-modules (M, N ), where M has
finite Gorenstein projective dimension. The Tate homology for such a pair vanishes
if M has finite projective dimension over R, and if R is complete intersection, then
Tor-independence implies vanishing of Tate homology. Thus the main theorem
subsumes all of the aforementioned generalizations of Auslander’s depth formula.
In fact, it goes further and subsumes several other generalizations obtained over
the half-century that has passed since [3] appeared.
What is more significant, though, is that Theorem (2.3) applies to modules over
all Gorenstein rings. The works of Iyengar and of Huneke and Wiegand established
the depth formula in the realm of complete intersection rings. The depth is a cohomological invariant, and the cohomological behavior of modules over Gorenstein
rings can stray dramatically from that of modules over complete intersection rings.
Therefore, it came as a surprise to us that vanishing of Tate homology is sufficient
to tame depth in this vastly wider context.
In Sections 4 and 5 we explore the consequences of the main theorem for modules
over different classes of Gorenstein rings. For so-called AB rings, a notion coined
by Huneke and Jorgensen [19], we obtain in Theorem (3.6) a precise bound on
vanishing of cohomology of finitely generated modules. For modules over complete
intersection rings, we obtain a derived depth formula for modules that satisfy an efd R (M, N )
fectively verifiable condition on vanishing of Tate homology. That is, if Tor
i
vanishes for a finite number of consecutive indices—a number that only depends
on R—then (B) holds; this is Theorem (4.6).
In the final section we prove a statement, dual to Theorem (2.3), for the width
invariant and use it to establish a bound on vanishing of cohomology of finitely
d ∗ (M, N ).
generated modules with vanishing Tate cohomology Ext
R
1. Depth of complexes
In this paper, R-complexes—that is, complexes of R-modules—are graded homologically. A complex
M:

M
∂i+1

∂M

· · · −→ Mi+1 −−−−→ Mi −−i−→ Mi−1 −→ · · ·

is called acyclic if the homology complex H(M ) is the zero-complex. We use the
M
. For n ∈ Z the n-fold shift
notation Ci (M ) for the cokernel of the differential ∂i+1
n
M
n
n
.
of M is the complex Σ M given by (Σ M )i = Mi−n and ∂iΣ M = (−1)n ∂i−n
The notation sup M and inf M is used for the supremum and infimum of the set
{ i ∈ Z | Mi 6= 0 }, with the conventions sup ∅ = −∞ and inf ∅ = ∞. A complex

VANISHING OF TATE HOMOLOGY AND DEPTH FORMULAS OVER LOCAL RINGS

3

M is called bounded above if sup M is finite, it is called bounded below if inf M is
finite, and it is called bounded if it is bounded above and below.
For some of our proofs, we shall need the following variation on [5, lem. 4.4.F].
(1.1) Lemma. Let F and M be R-complexes, and let P be a complex of finitely
generated R-modules. If one of the following conditions holds
(a) F and M are bounded above, and P is bounded below, or
(b) M and P are bounded
and P is a complex of projective R-modules or F is a complex of flat R-modules,
then there is a natural isomorphism of R-complexes
HomR (P, M ) ⊗R F ∼
= HomR (P, M ⊗R F ).
Proof. For R-modules P , M , and F the tensor evaluation map
ωP MF : HomR (P, M ) ⊗R F → HomR (P, M ⊗R F )
given by
ωP MF (ψ ⊗ f )(p) = ψ(p) ⊗ f,
is a homomorphism. It is an isomorphism if P is finitely generated and projective,
and also if P is finitely generated and F is flat; see [5, lem. 4.4.F].
Under either assumption, (a) or (b), the complex M is bounded above and P is
bounded below. Assume, therefore, without loss of generality that one has Mu = 0
for all u > 0 and Pu = 0 for all u < 0. For every n ∈ Z one then has
`
HomR (P, M )i ⊗R Fn−i
(HomR (P, M ) ⊗R F )n =
i∈Z
` Q
HomR (Pj , Mj+i )) ⊗R Fn−i
(
=
i∈Z j∈Z

=

−i
` L
HomR (Pj , Mj+i )) ⊗R Fn−i
(

i60 j=0

and
HomR (P, M ⊗R F )n =

Q

HomR (Pj , (M ⊗R F )j+n )

Q

HomR (Pj ,

Q

HomR (Pj ,

j∈Z

=

j>0

=

`

Mh ⊗R Fj+n−h )

h∈Z

`

Mj+i ⊗R Fn−i ).

i6−j

j>0

If (a) holds, then one can assume that Fu is zero for all u > 0, whence
(HomR (P, M ) ⊗R F )n =

0 L
−i
L

HomR (Pj , Mj+i ) ⊗R Fn−i

i=n j=0

and
HomR (P, M ⊗R F )n =

−n
L

HomR (Pj ,

j=0

=

0 L
−i
L

−j
L

Mj+i ⊗R Fn−i )

i=n

HomR (Pj , Mj+i ⊗R Fn−i ).

i=n j=0

The map from HomR (P, M ⊗R F ) to HomR (P, M ) ⊗R F with degree n component
L0 L−i
j(n−i)
ωPj Mj+i Fn−i is a morphism of complexes; this is elementary
i=n
j=0 (−1)

4

L. W. CHRISTENSEN AND D. A. JORGENSEN

to verify. It follows that it is an isomorphism if the modules in P are projective or
the modules in F are flat.
If (b) holds, then a similar argument applies.

(1.2) Resolutions. A morphism of R-complexes that induces an isomorphism in
homology is called a quasi-isomorphism and indicated by the symbol ‘≃’.
An R-complex P is called semi-projective if each module Pi is projective, and the
functor HomR (P, −) preserves quasi-isomorphisms. Every bounded below complex
of projective R-modules is semi-projective. Similarly, an R-complex I is called
semi-injective if each module Ii is injective, and the functor HomR (−, I) preserves
quasi-isomorphisms. Every bounded above complex of injective R-modules is semiinjective. The following facts are proved in [6].
(P) Every R-complex M has a semi-projective resolution. That is, there is a
quasi-isomorphism π : P → M , where P is a semi-projective complex with
Pi = 0 for all i < inf M . Moreover, if H(M ) is bounded below, then M has a
≃
semi-projective resolution P ′ −−−→ M with Pi′ = 0 for all i < inf H(M ).
(I) Every R-complex M has a semi-injective resolution. That is, there is a quasiisomorphism ι : M → I, where I is semi-injective with Ii = 0 for all i > sup M .
Moreover, if H(M ) is bounded above, then M has a semi-injective resolution
≃
M −−−→ I ′ with Ii′ = 0 for all i > sup H(M ).
For an R-module M , a semi-projective (-injective) resolution is just a projective
(injective) resolution in the classic sense; see [9].
We use the standard notations − ⊗L
R − and RHomR (−, −) for the derived tensor
product and derived Hom of complexes; they are computed by way of the resolutions
described above. Extending the usual definitions of Tor and Ext for modules, set
L
TorR
i (M, N ) = Hi (M ⊗R N )

and

ExtiR (M, N ) = H−i (RHomR (M, N ))

for R-complexes M and N and i ∈ Z. In another extension of classic notions, define
the projective and injective dimension of an R-complex by
≃

pdR M = inf{ sup P | P −−−→ M is a semi-projective resolution}
and

≃

idR M = inf{ − inf I | M −−−→ I is a semi-injective resolution}.
Setup. From this point, R denotes a local ring with maximal ideal m and residue field k = R/m. The embedding dimension of R, written edim R, is the minimal
number of generators of m. The codepth and codimension of R are the differences
codepth R = edim R − depth R

and

codim R = edim R − dim R,

where dim R denotes the Krull dimension of R.
The depth of an R-complex is defined by extension of the homological characterization of depth of finitely generated modules.
(1.3) Depth. Let M be an R-complex. The depth of M is defined as
depthR M = − sup H(RHomR (k, M )).
≃

If H(M ) is bounded above, then M has a semi-injective resolution M −−−→ I with
Ii = 0 for i > sup H(M ); see (1.2)(I). Thus, for every R-complex M one has
(1.3.1)

depthR M > − sup H(M ).

VANISHING OF TATE HOMOLOGY AND DEPTH FORMULAS OVER LOCAL RINGS

5

(1.4) The derived depth formula. Let M and N be R-complexes. We say that
the derived depth formula holds for M and N if there is an equality
(1.4.1)

depthR (M ⊗L
R N ) = depthR M + depthR N − depth R.

Note that this is just a rewrite of the equality (B) in the introduction. By [17,
lem. (2.1)] the derived depth formula holds for complexes M and N if M has finite
projective dimension and H(N ) is bounded above.
The next result is due to Dwyer and Greenlees [16, 6.5] and to Foxby and Iyengar
[18, 2.3 and 4.1].
(1.5) Proposition. Let K be the Koszul complex on a set of generators for m. For
an R-complex M , the following conditions are equivalent.
(i)
(ii)
(iii)
(iv )

H(k ⊗L
R M ) = 0;
H(K ⊗R M ) = 0;
H(HomR (K, M )) = 0;
H(RHomR (k, M )) = 0.



2. Depth and vanishing of Tate homology—the main theorem
We start by recalling some facts from [14] and [22].
(2.1) Complete resolutions. An acyclic complex T of projective R-modules is
called totally acyclic, if the complex HomR (T, Q) is acyclic for every projective
R-module Q. An R-module G is called Gorenstein projective if there exists such a
totally acyclic complex T with C0 (T ) ∼
= G.
Let M be an R-complex. A complete (projective) resolution of M is a diagram
(2.1.1)

τ

π

T −→ P −−→ M,

where π is a semi-projective resolution, T is a totally acyclic complex of projective R-modules, and τi is an isomorphism for i ≫ 0. The Gorenstein projective
dimension of M , written GpdR M , is the least integer n such that there exists a
complete resolution (2.1.1) where τi is an isomorphism for all i > n. In particular, GpdR M is finite if and only if M has a complete resolution. Notice that the
homology H(M ) is bounded above if GpdR M is finite. Note also that a complex
of finite projective dimension has finite Gorenstein projective dimension; indeed,
0 → P → M is a complete resolution for every semi-projective resolution P → M
with P bounded above.
(2.2) Tate homology. Let M be an R-complex of finite Gorenstein projective
dimension, and let T → P → M be a complete resolution. For an R-complex N ,
the Tate homology of M with coefficients in N is defined as
dR
Tor
i (M, N ) = Hi (T ⊗R N ).

This definition is independent of the choice of complete resolution; see [22] or [14,
sec. 2] for details. In particular, one has
(2.2.1)

R
∼
dR
Tor
i (M, N ) = Tori (M, N ) for i > GpdR M + sup N.

If M has finite projective dimension or if N is bounded above and of finite projective
d R (M, N ) = 0 for all i ∈ Z; see [14, prop. (2.5) and lem. (2.7)].
dimension, then Tor
i

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L. W. CHRISTENSEN AND D. A. JORGENSEN

The next theorem is our central result. For an R-complex M of finite projective
dR
dimension one has Tor
∗ (M, −) = 0, so the theorem subsumes [17, lem. (2.1)]. The
boundedness condition on the complex N , as opposed to its homology, reflects the
fact that Tate homology is not a functor on the derived category over R; see the
remarks before [14, prop. (2.5)].
(2.3) Theorem. Let M be an R-complex of finite Gorenstein projective dimension
d R (M, N ) = 0 for all i ∈ Z,
and let N be a bounded above R-complex. If one has Tor
i
then the derived depth formula holds for M and N . That is, one has
depthR (M ⊗L
R N ) = depthR M + depthR N − depth R.
Note that the homology complex H(M ⊗L
R N ) is bounded above by (2.2.1).
τ

≃

Proof. Choose a complete resolution T −→ P → M and let π ′ : P ′ −−→ N be a
≃
semi-projective resolution. The quasi-isomorphism P ⊗R π ′ : P ⊗R P ′ −−→ P ⊗R N
is then a semi-projective resolution, and the Künneth formula yields
L
′
∼
H((M ⊗L
R N ) ⊗R k) = H((P ⊗R P ) ⊗R k)
∼
= H((P ⊗R k) ⊗k (P ′ ⊗R k))
∼ H(M ⊗L k) ⊗k H(N ⊗L k).
=
R

R

It now follows from (1.3.1) and Proposition (1.5) that depthR (M ⊗L
R N ) is finite
if and only if depthR M and depthR N are both finite. In particular, the left- and
right-hand sides of the equality we aim to prove are simultaneously finite.
Assume that both M and N have finite depth. Consider the degreewise split
exact sequence of R-complexes 0 → P → Cone τ → ΣT → 0 and apply the functor
− ⊗R N to it. By assumption, the complex (ΣT ) ⊗R N ∼
= Σ(T ⊗R N ) is acyclic,
≃
so there is a quasi-isomorphism P ⊗R N −−−→ (Cone τ ) ⊗R N . In high degrees
K = Cone τ is isomorphic to the mapping cone of an isomorphism, and the mapping cone of an isomorphism is contractible. Therefore there exist homomorphisms
K
σi for i ≫ 0. Since K is a
σi : Ki → Ki+1 , such that one has 1Ki = σi−1 ∂iK + ∂i+1
K
Im ∂i+1
K
complex of projective modules, and ∂i+1 σi = 1
holds for i ≫ 0, it follows that
K ∼
the modules Ker ∂iK = Im ∂i+1
= Ci+1 (K) are projective for i ≫ 0. Fix n ≫ 0 and
K
consider the contractible subcomplex J = · · · → Kn+2 → Kn+1 → Im ∂n+1
→ 0.
The sequence 0 → J → K → K/J → 0 is split exact, because the quotient complex L = K/J = 0 → Cn (K) → Kn−1 → · · · consists of projective modules. The
complex J ⊗R N is contractible, so there are quasi-isomorphisms,
(1)

≃

≃

P ⊗R N −−−→ K ⊗R N −−−→ L ⊗R N .
≃

Choose a semi-injective resolution ι : N −−−→ I, where ι is injective and I is bounded
ι
above; see (1.2)(I). Consider the exact sequence 0 → N −→ I → C → 0 of
R-complexes. The complex C = Coker ι is bounded above and acyclic, and hence
so is the complex L ⊗R C; cf. [12, lem. 2.13]. Thus, there is a quasi-isomorphism
(2)

≃

L ⊗R N −−−→ L ⊗R I.

The complex L ⊗R I is bounded above and consists of injective R-modules, so it
follows from (1), (2), and [5, 1.4.I] that it is a semi-injective resolution of the

VANISHING OF TATE HOMOLOGY AND DEPTH FORMULAS OVER LOCAL RINGS

7

complex P ⊗R N ≃ M ⊗L
R N . The third equality in the next computation follows
from Lemma (1.1).
L
depthR (M ⊗L
R N ) = − sup H(RHomR (k, M ⊗R N ))

= − sup H(HomR (k, I ⊗R L))
= − sup H(HomR (k, I) ⊗R L)
= − sup H(HomR (k, I) ⊗k k ⊗R L)
= − sup H(HomR (k, I)) − sup H(k ⊗R L)
= depthR N − sup H(k ⊗R L)
For N = R this equality reads
depthR M = depth R − sup H(k ⊗R L).
The desired equality follows by elimination of the quantity sup H(k ⊗R L).



(2.4) Example. Let M be an R-module of finite Gorenstein projective dimension.
d R (M, N ) = 0 holds
(1) If N is an R-module of finite injective dimension, then Tor
∗
by [12, lem. 2.3 and prop. 3.9], so the derived depth formula holds for M and N .
(2) If N is an R-module of finite projective dimension, and ι : N → I is an
injective preenvelope, then the derived depth formula holds for M and N ′ = Coker ι,
′
dR
dR
dR
as Tor
∗ (M, N ) = 0 and Tor∗ (M, I) = 0 force Tor∗ (M, N ) = 0.
(3) If N is an R-module of finite injective dimension, and π : P → N is a projective precover, then the derived depth formula holds for M and Ker π.
Part (1) is known from [13, thm. 6.3], while (2) and (3) appear to be new.
(2.5) Gorenstein rings. Let R be Gorenstein. Every R-complex with bounded
above homology has finite Gorenstein projective dimension; see [31, thm. 3.11].
Thus by Theorem (2.3) the derived depth formula holds for R-complexes M and
d R (M, N ) = 0.
N with H(M ) and N bounded above and Tor
∗
For finitely generated modules, the Gorenstein projective dimension coincides
with Auslander and Bridger’s notion of G-dimension; see [12, prop. 3.8]. The
following equality is known as the Auslander–Bridger Formula; it holds for every
finitely generated module M of finite G-dimension,
(2.5.1)

G-dimR M = depth R − depthR M.

dR
For finitely generated R-modules with Tor
∗ (M, N ) = 0 there is hence an equality

(2.5.2)

G-dimR (M ⊗L
R N ) = G-dimR M + G-dimR N,

which represents a natural generalization of [3, cor. 1.3] to G-dimension.
(2.6) Remark. In [27] Jorgensen and Şega give an example of an artinian Gorenstein ring R and a finitely generated R-module M , such that for every s > 0 there
exists a finitely generated R-module Ns with
L
s = sup H(M ⊗L
R Ns ) = − depthR (M ⊗R Ns ).

Thus, over a Gorenstein ring, boundedness of H(M ⊗L
R N )—that is, vanishing of
TorR
(M,
N
)—does
not
per
se
guarantee
that
the
derived
depth formula holds.
≫0
This phenomenon disappears over so-called AB rings, where vanishing of homology
is easier to control.

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L. W. CHRISTENSEN AND D. A. JORGENSEN

3. AB rings
Recall from Huneke and Jorgensen [19] that a local ring R is called AB if it is
Gorenstein, and the following holds for all finitely generated R-modules M and N ,
ExtiR (M, N ) = 0 for i ≫ 0 implies ExtiR (M, N ) = 0 for i > dim R.
At the end of this section we apply our main theorem to provide a precise bound
for the vanishing of cohomology ExtiR (M, N ) for modules over AB rings; it turns
out to depend only on M .
(3.1) Tate cohomology. Let M be an R-complex of finite Gorenstein projective
dimension, and let T → P → M be a complete resolution. For an R-complex N ,
the Tate cohomology of M with coefficients in N is defined as
d iR (M, N ) = H−i (HomR (T, N )).
Ext

This definition is independent of the choice of complete resolution; see [31] for
details. In particular, one has
(3.1.1)

d iR (M, N ) ∼
Ext
= ExtiR (M, N ) for i > GpdR M − inf N.

If M has finite projective dimension or if N is bounded below and of finite injective
d i (M, N ) = 0 for all i ∈ Z; see [31, thm. 4.5] and [14,
dimension, then one has Ext
R
lem. (4.2)].
In some sense, the conditions in the main theorem (2.3) are easier to verify for
moduless over AB rings. Not only is finiteness of Gorenstein projective dimenion
automatic, per the next lemma one only needs vanishing of homology in high degrees. For modules over a familiar class of AB rings, namely complete intersections,
it becomes truly easier, as one only needs vanishing of a finite number of homology
modules; see Theorem (4.6).
(3.2) Proposition. Let R be AB and let M and N be finitely generated R-modules.
The following assertions hold.
dR
(a) TorR
i (M, N ) = 0 for i ≫ 0 implies Tor (M, N ) = 0 for all i ∈ Z.
i

d i (M, N ) = 0 for all i ∈ Z.
(b) ExtiR (M, N ) = 0 for i ≫ 0 implies Ext
R

Proof. As R is Gorenstein, M has finite G-dimension. Let T → P → M be a
complete resolution of M . For all integers i and n with i > n one has
(1)

d R (M, N ) = Hi (T ⊗R N ) = TorR (Cn (T ), N ).
Tor
i
i−n

The Krull dimension d = dim R is an upper bound for the G-dimension of a finitely
generated R-module, cf. (2.5.1). It follows from (2.2.1) that there are isomorphisms
R
d R (M, N ) ∼
Tor
= TorR
i (M, N ) for i > d. Thus, if the homology modules Tori (M, N )
i
R
d (M, N ). For every n ∈ Z it follows
vanish for i ≫ 0, then so do the modules Tor
i
that the modules TorR
(C
(T
),
N
)
vanish
for
j ≫ 0, and since R is AB they vanish
n
j
for j > 0; see [19, thm. 3.4]. Now it follows from (1) that all the Tate homology
d R (M, N ) vanish. The proves part (a); the proof of (b) is similar.
modules Tor

i
(3.3) Remark. If R is AB and M and N are finitely generated R-modules with
TorR
i (M, N ) = 0 for i ≫ 0, then it follows from (2.5) and Proposition (3.2)(a) that
the derived depth formula holds for M and N , and hence that (2.5.2) holds.

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9

(3.4) Remark. Let R be AB and let M and N be finitely generated Gorenstein
projective R-modules with (minimal) complete resolutions T and A. If one has
TorR
i (M, N ) = 0 for i ≫ 0, then it follows from Proposition (3.2)(b) and [14,
cor. (6.2)] that the module M ⊗R N is Gorenstein projective with (minimal) com✶
plete resolution T ⊗✶
R A. Here T ⊗R A denotes the pinched tensor product defined
in [14].
Recall that R is complete intersection if there exists a surjective ring homomorb where Q is a complete regular local ring, and Ker π is generated
phism π : Q ։ R,
by a Q-regular sequence. The least length of such a sequence equals codim R. Complete intersection rings are perhaps the best known examples of AB rings, and they
are studied further in the next section. Here we mention, in passing, a result of
Celikbas and Dao [10, cor. 1.3] that constitutes a partial converse to Remark (3.3).
(3.5) Let R be complete intersection of codimension c, and assume that Rp is regular
for every prime ideal p in R with htR p 6 c. If the tensor product M ⊗R N of two
finitely generated Gorenstein projective R-modules is Gorenstein projective, then
one has TorR
i (M, N ) = 0 for all i > 1.
The next result on vanishing of cohomology was established for complete intersection rings by Araya and Yoshino [1, thm. 4.2].
(3.6) Theorem. Assume that R is AB and let M and N be finitely generated
R-modules. If one has ExtiR (M, N ) = 0 for i ≫ 0, then the next equality holds,
sup{ i ∈ Z | ExtiR (M, N ) 6= 0 } = depth R − depthR M.
Proof. As R is Gorenstein, the module M has finite G-dimension. Choose a comτ
plete resolution T −→ P → M . By Proposition (3.2)(b) the Tate cohomology
d i (M, N ) vanish for all i, so the complex HomR (T, N ) is acyclic. Let
modules Ext
R
E be the injective hull of the residue field. The complex
∼ T ⊗R HomR (N, E),
HomR (HomR (T, N ), E) =
is then acyclic as well; the isomorphism is homomorphism evaluation [5, 4.4.I] in
d R (M, HomR (N, E))
each degree. It follows that the Tate homology modules Tor
i
vanish for all i, so Theorem (2.3) applies to the modules M and HomR (N, E). The
latter module has depth 0, so one has
(1)

depthR (M ⊗L
R HomR (N, E)) = depthR M − depth R.

The module HomR (N, E) is only supported on the maximal ideal of R, and so are
the homology modules of the complex M ⊗L
R HomR (N, E) ≃ P ⊗R HomR (N, E).
This explains the first equality in the next chain; the second equality is homomorphism evaluation.
− depthR (P ⊗R HomR (N, E)) = sup H(P ⊗R HomR (N, E))
= sup H(HomR (HomR (P, N ), E))
(2)

= − inf H(HomR (P, N ))
= − inf H(RHomR (M, N ))
= sup{ i ∈ Z | ExtiR (M, N ) 6= 0 }

The desired equality now follows from (1) and (2).



10

L. W. CHRISTENSEN AND D. A. JORGENSEN

4. Complete intersections
Homology of finitely generated modules over complete intersection rings is rigid in
the following sense; see [20, remarks before thm. 1.9].
(4.1) Fact. Let R be complete intersection of codimension c and let M and N be
finitely generated R-modules. Let n > 0 be an integer; if one has TorR
i (M, N ) = 0
for all i with n + c > i > n, then one has TorR
i (M, N ) = 0 for all i > n.
Combined with Theorem (2.3) and [4, thm. 4.9] this fact has the following consequence.
(4.2) Corollary. Let R be complete intersection of codimension c, and let M and
N be finitely generated R-modules. If one has TorR
i (M, N ) = 0 for c+1 consecutive
d R (M, N ) vanish for all i ∈ Z,
values of i > 0, then the Tate homology modules Tor
i
and the derived depth formula holds for M and N .

One goal of this section is to obtain a similar result for modules that are not
finitely generated; see Theorem (4.6).
Recall from [7] that a (codimension c) quasi-deformation of R is a diagram of
ρ
π
local homomorphism R −→ R′ ←−− Q, where ρ is flat, and π is surjective with
kernel generated by a Q-regular sequence (of length c).
(4.3) Lemma. Let M and N be R-complexes with N bounded above. If there
exists a codimension c quasi-deformation R → R′ ← Q such that pdQ (R′ ⊗R M )
is finite, then GpdR M is finite, and the following conditions are equivalent.
d R (M, N ) = 0 for all i ∈ Z.
(i) Tor
i

(ii) TorR
i (M, N ) = 0 for all i ≫ 0.
d R (M, N ) = 0 for all i ≪ 0.
(ii’ ) Tor
i
(iii) TorR
i (M, N ) = 0 for c + 1 consecutive values of i > GpdR M + sup N .
d
(iv ) TorR (M, N ) = 0 for c + 1 consecutive values of i.
i

To not interrupt the flow, we defer the proof of this lemma to the end of the section.
We say that an R-complex M has finite CI-dimension, if there is a quasideformation of R such that pdQ (R′ ⊗R M ) is finite. For modules M and N with
TorR
>1 (M, N ) = 0 the next theorem recovers Iyengar’s [24, thm. 4.3]; it also subsumes a recent generalization of this result due to Sahandi, Sharif, and Yassemi
[28, thm. 3.3].
(4.4) Theorem. Let M and N be R-complexes with H(N ) bounded above. If M
has finite CI-dimension, and one of the following conditions holds.
(a) one has TorR
i (M, N ) = 0 for i ≫ 0; or
d R (M, N ) = 0 for i ≪ 0;
(b) the complex N is bounded above, and one has Tor
i
R
d
then the Tate homology modules Tori (M, N ) vanish for all i ∈ Z, and the derived
depth formula holds for M and N .
We precede the proof with a technical observation.
(4.5) Remark. Let N be an R-complex with H(N ) bounded above. Set s =
sup H(N ) and let N ′ be the soft truncation of N at s, i.e. the bounded above

VANISHING OF TATE HOMOLOGY AND DEPTH FORMULAS OVER LOCAL RINGS

11

complex 0 → Cs (N ) → Ns−1 → · · · . The natural morphism N ։ N ′ is a quasiisomorphism; in particular, one has depthR N ′ = depthR N . For every R-complex
M there are isomorphisms
R
′ ∼
TorR
i (M, N ) = Tori (M, N ) for all i ∈ Z.
′
L
Moreover, one has depthR (M ⊗L
R N ) = depthR (M ⊗R N ), so the depth formula
holds for M and N if and only if it holds for M and N ′ .

Proof of Theorem (4.4). Under the assumption that N is bounded above and
d R (M, N ) = 0 holds for i ≪ 0, the assertions follow immediately from Lemma (4.3)
Tor
i
and Theorem (2.3).
Assume now that TorR
i (M, N ) = 0 holds for i ≫ 0. By Remark (4.5) we can
replace N by its soft truncation at sup H(N ); that is, we can assume that N is
bounded above. Now the assertions follow as above.

d R (M, N ) for
Over a complete intersection ring, vanishing of Tate homology Tor
i
all i ∈ Z can be inferred from a finite gap in (Tate) homology, and the length of
that gap is independent of M and N .
(4.6) Theorem. Let R be complete intersection of codimension c and dimension d.
Let M and N be R-complexes with H(M ) and H(N ) bounded above. If one of the
following conditions holds,
(a) TorR
i (M, N ) = 0 for c+1 consecutive values of i > d+sup H(M )+sup H(N ); or
d R (M, N ) = 0 for c + 1
(b) the complex N is bounded above, and one has Tor
i
consecutive values of i;
d R (M, N ) vanish for all i ∈ Z, and the derived
then the Tate homology modules Tor
i
depth formula holds for M and N .
Proof. A complete intersection ring is Gorenstein, so by [31, thm. 3.10] one has
GpdR M 6 d + sup H(M ) < ∞.
b where Q is a complete regular
By assumption there is a homomorphism π : Q ։ R,
local ring, and Ker π is generated by a Q-regular sequence x1 , . . . , xc . Thus, the
b և Q is a codimension c quasi-deformation of R, and because
diagram R → R
b
b ⊗R M ) < ∞.
H(R ⊗R M ) is bounded above, one has pdQ (R
d R (M, N ) = 0 for c + 1 consecutive values
If N is bounded above and one has Tor
i
R
d (M, N ) = 0 for all i ∈ Z. As GpdR M is finite,
of i, then Lemma (4.3) yields Tor
i
the derived depth formula holds for M and N by Theorem (2.3).
Now, set s = sup H(N ) and assume that one has TorR
i (M, N ) = 0 for c + 1
consecutive values of i > d + sup H(M ) + s. By Remark (4.5) we can replace N
by its soft truncation at s; that is, we can assume that N is bounded above with
sup N = s. Now the assertions follow as above.

(4.7) Remark. Over a Gorenstein ring R of dimension d, the number d+sup H(M )
is an upper bound for the Gorenstein projective dimension of every R-complex M .
The proof above and the fact that Tate homology is balanced, see [14, sec. 5], shows
that one can replace the quantity d + sup H(M ) + sup H(N ) in the theorem with
min{GpdR M + sup H(N ), GpdR N + sup H(M )}.

12

L. W. CHRISTENSEN AND D. A. JORGENSEN

For complexes with bounded and degreewise finitely generated homology, for
finitely generated modules in particular, the CI-dimension agrees with Avramov,
Gasharov, and Peeva’s notion of CI-dimension; see [7, 29].
(4.8) Corollary. Let R be complete intersection of codimension c and let M and N
d R (M, N ) = 0 for c + 1 consecutive
be finitely generated R-modules. If one has Tor
i
R
values of i or Tori (M, N ) = 0 for c + 1 consecutive values of i > 0, then the
following equality holds,
CI-dimR (M ⊗L
R N ) = CI-dimR M + CI-dimR N.
Proof. It follows from the theorem and Corollary (4.2) that the derived depth
formula holds for M and N . For every R-complex X with H(X) finitely generated,
there is an Auslander–Buchsbaum-type formula CI-dimR X = depth R − depthR X;
see [29, prop. 3.3]. Now the desired equality follows from the depth formula.

(4.9) Theorem. Let R be of codepth c, let M be a finitely generated R-module
d R (M, N ) = 0
of finite CI-dimension, and let N be an R-module. If one has Tor
i
R
for c + 1 consecutive values of i or Tori (M, N ) = 0 for c + 1 consecutive values
d R (M, N ) vanish for all i ∈ Z,
of i > depth R, then the Tate homology modules Tor
i
and the derived depth formula holds for M and N .
Proof. By [7, thm. (1.4) and (5.6)] the number d = depth R is an upper bound
for the CI-dimension of M , and the complexity of M is at most c. If one has
TorR
i (M, N ) = 0 for c + 1 consecutive values of i > d, then [26, cor. 2.3] yields
TorR
i (M, N ) = 0 for all i > d, and then it follows from Lemma (4.3) and Theorem (2.3) that the derived depth formula holds for M and N .
Let T → P → M be a complete resolution. By [7, lem. (1.5)] every syzygy of M
has finite CI-dimension. In particular, Cd (T ) has finite CI-dimension, and it follows
d R (M, N ) = 0 for
that Ci (T ) has finite CI-dimension for every i ∈ Z. If one has Tor
i
c + 1 consecutive values of i, then there exists an ı ∈ Z such that one has
R
R
0 = TorR
d+1 (Cı (T ), N ) = Tord+2 (Cı (T ), N ) = · · · = Tord+c+1 (Cı (T ), N ).

Now [26, cor. 2.3] yields TorR
i (Cı (T ), N ) = 0 for all i > d, and then it follows from
Lemma (4.3) and Theorem (2.3) that the desired formula holds for M and N . 
Proof of Lemma (4.3). The complex M has finite CI-dimension. If the homology
complex H(M ) is bounded, then it follows from [30, thm. 5.1.(b) and rmk. 2.5] that
GpdR M is finite. Here we give a direct but similar argument that does not use
boundedness of H(M ).
ρ
π
Let R −→ R′ ←−− Q be a quasi-deformation such that pdQ (R′ ⊗R M ) is finite.
Assume, without loss of generality, that the rings Q and, therefore, R′ are complete.
Then Q has a dualizing complex D, and the complex D′ = RHomQ (R′ , D) is dua′
lizing for R′ ; see [25, sec. 3]. The complex H(D ⊗L
Q (R ⊗R M )) is bounded above
by [5, thm. 2.4.F and 2.3.F], and it follows from [25, prop. 7.3] that the complex M
belongs to the Auslander category Â(R). Let x1 , . . . , xc be a Q-regular sequence
that generates Ker π. The Koszul complex on x1 , . . . , xc is a projective resolution
′
of R′ over Q, so one has D′ ≃ Σ−c (D ⊗L
Q R ). Now it is straightforward to verify
′
′
′
that H(D′ ⊗L
R′ (R ⊗R M )) is bounded above and that R ⊗R M belongs to Â(R ).
≃
′
Therefore, g = GpdR′ (R ⊗R M ) is finite by [25, thm. 8.1]. Let P −−−→ M be a

VANISHING OF TATE HOMOLOGY AND DEPTH FORMULAS OVER LOCAL RINGS

13

semi-projective resolution over R. To prove that GpdR M is finite, it suffices to
show that the module Cg (P ) has finite Gorenstein projective dimension or, equivalently, finite Gorenstein flat dimension; cf. [12, thm. 4.1]. The quasi-isomorphism
R′ ⊗R P → R′ ⊗R M is a semi-projective resolution over R′ . By [31, thm. 3.4] the
module Cg (R′ ⊗R P ) ∼
= R′ ⊗R Cg (P ) is Gorenstein projective, in particular it is
Gorenstein flat, and then so is Cg (P ) by [15, thm. A]. Thus GpdR M is finite.
The implications
(iv) ⇐= (ii′ ) ⇐= (i) =⇒ (ii) =⇒ (iii) =⇒ (iv)
are trivial or follow from (2.2.1). It remains to prove that (iv) implies (i).
Let T → P → M be a complete resolution. The morphism R′ ⊗R P → R′ ⊗R M
is then a semi-projective resolution over R′ , and it is straightforward to verify that
R′ ⊗R T is a totally acyclic complex of projective R′ -modules. Thus, by flatness of
R′ one has
′
∼ d R′ ′
dR
R′ ⊗R Tor
i (M, N ) = Tori (R ⊗R M , R ⊗R N )
for all i ∈ Z. Without loss of generality, assume that ρ is the identity map. The
assumptions are now that R is isomorphic to Q/(x1 , . . . , xc ), that pdQ M is finite,
and that there is an integer h such that
d R (M, N ) = Tor
d R (M, N ) = · · · = Tor
d R (M, N ).
0 = Tor
h+1
h+c
h

(1)

For j ∈ Z set Mj = Cj (T ). Notice that because R and M have finite projective
dimension over Q, each module Mj has finite projective dimension over Q as well.
Set d = depth Q; fix a j and let F be a projective resolution of Mj over Q of length
pdQ Mj 6 d. Let L be a semi-projective resolution of N over R. Without loss of
generality, assume that one has sup N = 0. The filtrations F and L defined by
a
a
(F p (F ⊗Q L))n =
Fi ⊗Q Ln−i and (Lp (L ⊗Q F ))n =
Li ⊗Q Fn−i
i6p

i6p

are bounded; they give rise to spectral sequences
F

2

Ep,q =⇒ Hn (F ⊗Q L)
p

and

L

2

Ep,q =⇒ Hn (L ⊗Q F ).
p

2

The E -terms are the iterated homologies of the underlying double complexes, obtained by first taking homology along columns and then along rows. The terms
F

2

Ep,q = TorQ
p (Mj , Hq (N ))

vanish for q > 0, and they vanish for p not in {0, . . . , d}. In particular, for n > d one
2
has F Ei,n−i = 0 for all i ∈ Z. It follows that the homology modules Hn (F ⊗Q L)
vanish for n > d; see [9, prop. 5.5]. From the isomorphism H(F ⊗R L) ∼
= H(L ⊗R F )
and [9, prop. 5.3a] one now gets
L

(2)

∞

Ei,n−i = 0 for all n > d and all i ∈ Z.

The homology in degree q within the pth column in the double complex L ⊗R F
Q
is isomorphic to Lp ⊗R Hq (R ⊗L
Q Mj ). The Koszul complex K (x1 , . . . , xc ) is a free
resolution of R over Q, and the elements x1 , . . . , xc act trivially on Mj , so one has
c
L ⊗ H (M ⊗L R) ∼
= L ⊗ M (q) for all q. Thus the E2 -terms in the second
p

R

q

j

p

Q

R

j

sequence are
L

2
(qc) ∼
R
(qc) .
Ep,q = TorR
(N,
M
p
j ) = Torp (Mj , N )

14

L. W. CHRISTENSEN AND D. A. JORGENSEN
2

2

Clearly the terms L Ep,q vanish for q not in {0, . . . , c}, and L Ep,q vanishes for all q
in {0, . . . , c} if and only if it vanishes for one of them.
Assume now that j < h − d + c − 1 holds. One then has h − j > 0, and the
2
hypothesis (1) yields L Ep,q = 0 for p in {h − j, . . . , h − j + c}. This yields the limit
terms
(3)

L

2

∞

Eh−j+c+1,0 = L Eh−j+c+1,0

and

L

2

∞

Eh−j−1,c = L Eh−j−1,c .

Moreover, the inequalities h − j + c + 1 > d and c + (h − j − 1) > d hold, so
2
2
(2) and (3) combine to yield L Eh−j+c+1,0 = 0 and L Eh−j−1,c = 0 and, therefore,
L

2

2

Eh−j+c+1,q = 0 = L Eh−j−1,q for all q.
2

Iterating this argument, one gets L Ep,q = 0 for all p > 0 and all q. Thus, for
dR
every j < h − d + c − 1 one has 0 = TorR
p (Mj , N ) = Torp+j (M, N ) for all p > 0;
d R (M, N ) = 0 holds for all i ∈ Z.
hence Tor

i
5. Auslander’s depth formula
The purpose of this section is to connect the derived depth formula (1.4.1) with
another generalization, due to Auslander [3], of the Auslander–Buchsbaum Formula.
In view of Lemma (4.3) the next result generalizes [11, thm. 3] by replacing
finitely generated modules with complexes of modules with mild boundedness conditions and no assumptions of finite generation.
(5.1) Theorem. Let M be an R-complex of finite Gorenstein projective dimension
d R (M, N ) = 0 for all i ∈ Z,
and let N be a bounded above R-complex. If one has Tor
i
L
then s = sup H(M ⊗R N ) is finite, there is an inequality,
−s 6 depthR M + depthR N − depth R,
and equality holds if and only if depthR Hs (M ⊗L
R N ) is zero. Moreover, if one has
s = 0 or depthR Hs (M ⊗L
R N ) 6 1, then there is an equality
depthR Hs (M ⊗L
R N ) − s = depthR M + depthR N − depth R.
Proof. By Theorem (2.3) the complex H(M ⊗L
R N ) is bounded above, and one has
depthR (M ⊗L
N
)
=
depth
M
+
depth
N
−
depth R; the inequality now follows
R
R
R
from (1.3.1). By [18, 1.5.(3)] there is an isomorphism
L
∼
H−s (RHomR (k, M ⊗L
R N )) = HomR (k, Hs (M ⊗R N )),

and it follows that the equality depthR (M ⊗L
R N ) = −s holds if an only if the
N
)
has
depth
zero.
Finally,
by [24, thm. 2.3] the desired equalmodule Hs (M ⊗L
R
L
N
)
− s holds provided that one has
N
)
=
depth
H
(M
⊗
ity depthR (M ⊗L
s
R
R
R
L
depthR Hs (M ⊗L
N
)
−
s
6
depth
H
(M
⊗
N
)
− i for all i > s.

i
R
R
R
(5.2) The depth formula. Let M and N be finitely generated R-modules. Following Choi and Iyengar [11] we say that the depth formula holds for M and N if
s = sup{ i | TorR
i (M, N ) 6= 0 } is finite and one has
depthR TorR
s (M, N ) − s = depthR M + depthR N − depth R.
Auslander [3, thm. 1.2] proved that the formula holds, if M has finite projective
dimension, and one has s = 0 or depthR TorR
s (M, N ) 6 1.

VANISHING OF TATE HOMOLOGY AND DEPTH FORMULAS OVER LOCAL RINGS

15

(5.3) Corollary. Let M and N be finitely generated R-modules, and assume that
s = 0 or depthR TorR
s (M, N ) 6 1. The depth formula holds for M and N if one of
the following conditions is satisfied.
d R (M, N ) = 0 for all i ∈ Z.
(a) M has finite G-dimension, and one has Tor
i

(b) R is AB, and one has TorR
i (M, N ) = 0 for i ≫ 0.

d R (M, N ) = 0 for
(c) R is complete intersection of codimension c, and one has Tor
i
c + 1 consecutive values of i.
(d) R is complete intersection of codimension c, and one has TorR
i (M, N ) = 0 for
c + 1 consecutive values of i > 0.
d R (M, N ) = 0 for codepth R + 1
(e) M has finite CI-dimension, and one has Tor
i
consecutive values of i > 0.
(f) M has finite CI-dimension, and one has TorR
i (M, N ) = 0 for codepth R + 1
consecutive values of i > depth R.
Proof. Part (a) is immediate from Theorem (5.1). Part (b) follows from (a) in
view of Proposition (3.2)(a). Similarly, part (c) follows in view of Theorem (4.6),
part (d) follows in view of Corollary (4.2), and parts (e) and (f) follow in view of
Theorem (4.9).

6. Vanishing of cohomology
Let M and N be finitely generated R-modules. If M has finite projective dimension
or N has finite injective dimension, then the largest index for which ExtiR (M, N )
does not vanish is i = depth R − depthR M . That is, the vanishing of cohomology
ExtiR (M, N ) only depends on M ; see Ischebeck [23, 2.6]. The next result is more
d ∗ (M, N ) vanishes if M has finite projective dimengeneral, as Tate cohomology Ext
R
sion or N has finite injective dimension, see (3.1), and finite projective/injective
dimension implies finite G-dimension/Gorenstein injective dimension.
(6.1) Theorem. Let M and N be finitely generated R-modules such that M has
finite G-dimension or N has finite Gorenstein injective dimension. If one has
d i (M, N ) = 0 for all i ∈ Z, then the next equality holds
Ext
R

sup{ i ∈ Z | ExtiR (M, N ) 6= 0 } = depth R − depthR M.

The proof is given at the end of the section; note that under the assumption that M
has finite G-dimension, the desired equality follows from the proof of Theorem (3.6).
The notion of Gorenstein injective dimension is dual to that of Gorenstein projecd ∗ (M, N ) can be extended to the situation
tive dimension, and Tate cohomology Ext
R
where the second variable N has finite Gorenstein injective dimension; see [2, 14].
Before we start the proof of Theorem (6.1) we record an easy consequence.
(6.2) Remark. Let M and N be finitely generated R-modules. Under each of the
following conditions,
(a) M has finite G-dimension and pdR N is finite
(b) N has finite Gorenstein injective dimension and idR M is finite
d ∗ (M, N ) vanishes by [31, thm. 4.5] and [2, thm. 3.9], whence
Tate cohomology Ext
R
one has sup{ i ∈ Z | ExtiR (M, N ) 6= 0 } = depth R − depthR M .

16

L. W. CHRISTENSEN AND D. A. JORGENSEN

As a first step towards a proof of (6.1) we recall the notion of width.
(6.3) Width. The width of an R-complex M is defined as
widthR M = inf H(k ⊗L
R M ).
There is an obvious inequality
(6.3.1)

widthR M > inf H(M ),

and equality holds if H(M ) is bounded below and degreewise finitely generated.
(6.4) Proposition. Let N be an R-complex of finite Gorenstein injective dimension
d i (M, N ) = 0 for all i ∈ Z,
and let M be a bounded above R-complex. If one has Ext
R
then the next equality holds,
widthR RHomR (M, N ) = depthR M + widthR N − depth R.
By [14, (5.6.1)] the homology complex H(RHomR (M, N )) is bounded below.
υ

≃

Proof. Choose a complete injective resolution N → I −−→ U and let π : P −−→ M
be a semi-projective resolution. The induced quasi-isomorphism
≃

HomR (π, I) : HomR (M, I) −−→ HomR (P, I)
is a semi-injective resolution, and the Künneth formula yields
H(RHomR (k, RHomR (M, N ))) ∼
= H(HomR (k, HomR (P, I)))
∼
= H(HomR (k ⊗R P , I))
∼
= H(Homk (k ⊗R P , HomR (k, I)))
∼ Homk (H(k ⊗L M ), H(RHomR (k, N ))).
=
R

It follows from (6.3.1) and Proposition (1.5) that widthR RHomR (M, N ) is finite
if and only if widthR M and depthR N are both finite. In particular, the left- and
right-hand sides of the desired equality are simultaneously finite.
Assume that widthR M and depthR N are finite. Set K = Σ−1 (Cone ι); consider
the degreewise split exact sequence 0 → Σ−1 U → K → I → 0 and apply the
functor HomR (M, −). By assumption, the complex HomR (M, U ) is acyclic, cf. [14,
≃
def. (5.5)], so there is a quasi isomorphism HomR (M, K) −−−→ HomR (M, I). In low
degrees, K is isomorphic to the mapping cone of an isomorphism. Therefore, there
K
σi holds for
exist homomorphisms σi : Ki → Ki+1 such that 1Ki = σi−1 ∂iK + ∂i+1
K
K
i ≪ 0. Since K is a complex of injective modules, and one has ∂i+1 σi = 1Im ∂i+1
K ∼
for i ≪ 0, it follows that the modules Ker ∂iK = Im ∂i+1
= Ci+1 (K) are injective
K
for i ≪ 0. Fix n ≪ 0; the subcomplex J = · · · → Kn+2 → Kn+1 → Im ∂n+1
→0
consists of injective modules, the sequence 0 → J → K → K/J → 0 is split exact,
and the quotient complex K/J = 0 → Cn (K) → Kn−1 → · · · is contractible. It
follows that there are quasi-isomorphisms,
(1)

≃

≃

HomR (M, J) −−−→ HomR (M, K) −−−→ HomR (M, I).
≃

Choose a semi-injective resolution ι′ : M −−−→ I ′ , where ι′ is injective and I ′ is
bounded above; see (1.2)(I). Then the complex C = Coker ι′ is bounded
above and
ι′
acyclic. Apply HomR (−, J) to the exact sequence 0 → M −−→ I ′ → C → 0 of

VANISHING OF TATE HOMOLOGY AND DEPTH FORMULAS OVER LOCAL RINGS

17

R-complexes. The complex HomR (C, J) is acyclic by [12, lem. 2.5], so there is a
quasi-isomorphism
(2)

≃

HomR (I ′ , J) −−−→ HomR (M, J).

The complex HomR (I ′ , J) is bounded below and consists of flat R-modules, so it
is semi-flat in the sense of [5, 6], and it follows from (1) and (2) that there is a
≃
quasi-isomorphism HomR (I ′ , J) −−−→ HomR (M, I). The third equality in the next
chain uses homomorphism evaluation, a variation on [5, lem. 4.4.I] which is proved
similarly to Lemma (1.1).
widthR RHomR (M, N ) = inf H(k ⊗L
R RHomR (M, N ))
= inf H(k ⊗R HomR (I ′ , J))
= inf H(HomR (HomR (k, I ′ ), J))
= inf H(Homk (HomR (k, I ′ ), HomR (k, J)))
= inf H(HomR (k, J)) − sup H(HomR (k, I ′ ))
= inf H(HomR (k, J)) + depthR M.
For M = R this equality reads widthR N = inf H(HomR (k, J)) + depthR R, and
the desired equality follows.

A dual argument yields the next result, which is also invoked in the proof of
Theorem (6.1). The special case of Proposition (6.5) where M and N are finitely
generated, which is the context of (6.1), follows from the proof of Theorem (3.6).
(6.5) Proposition. Let M be an R-complex of finite Gorenstein projective dimend i (M, N ) = 0 for all
sion and let N be a bounded below R-complex. If one has Ext
R
i ∈ Z, then the next equality holds,
widthR RHomR (M, N ) = depthR M + widthR N − depth R.



Note that by (3.1.1) the homology complex H(RHomR (M, N )) is bounded below.
Proof of Theorem (6.1). It follows from [14, (5.6.1)] and (3.1.1) that the complex
RHomR (M, N ) has bounded below homology. Moreover, each homology module
H−i (RHomR (M, N )) = ExtiR (M, N ) is finitely generated, as M and N are finitely
generated. Now (6.3.1) and Propositions (6.4) and (6.5) yield
sup{ i ∈ Z | ExtiR (M, N ) 6= 0 } = − inf RHomR (M, N )
= − widthR RHomR (M, N )
= − depthR M − widthR N + depth R
= depth R − depthR M.



Acknowledgments
We thank Olgur Celikbas and Srikanth Iyengar for their comments on an earlier
version of this paper. We also acknowledge an anonymous referee whose pertinent
suggestions helped us tighten the presentaion.

18

L. W. CHRISTENSEN AND D. A. JORGENSEN

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Department of Math. and Stat., Texas Tech University, Lubbock, TX 79409, U.S.A.
E-mail address: lars.w.christensen@ttu.edu
URL: http://www.math.ttu.edu/~ lchriste
Department of Mathematics, University of Texas, Arlington, TX 76019, U.S.A.
E-mail address: djorgens@uta.edu
URL: http://dreadnought.uta.edu/~ dave

