arXiv:1711.04052v1 [] 11 Nov 2017

BIG COHEN-MACAULAY MODULES,
MORPHISMS OF PERFECT COMPLEXES, AND
INTERSECTION THEOREMS IN LOCAL ALGEBRA
LUCHEZAR L. AVRAMOV, SRIKANTH B. IYENGAR, AND AMNON NEEMAN
Abstract. There is a well known link from the first topic in the title to the
third one. In this paper we thread that link through the second topic. The
central result is a criterion for the tensor nilpotence of morphisms of perfect
complexes over commutative noetherian rings, in terms of a numerical invariant of the complexes known as their level. Applications to local rings include a
strengthening of the Improved New Intersection Theorem, short direct proofs
of several results equivalent to it, and lower bounds on the ranks of the modules in every finite free complex that admits a structure of differential graded
module over the Koszul complex on some system of parameters.

1. Introduction
A big Cohen-Macaulay module over a commutative noetherian local ring R is a
(not necessarily finitely generated) R-module C such that some system of parameters of R forms a C-regular sequence. In [16] Hochster showed that the existence
of such modules implies several fundamental homological properties of finitely generated R-modules. In [17], published in [18], he proved that big Cohen-Macaulay
modules exist for algebras over fields, and conjectured their existence in the case
of mixed characteristic. This was recently proved by Y. André in [2]; as a major
consequence many “Homological Conjectures” in local algebra are now theorems.
A perfect R-complex is a bounded complex of finite projective R-modules. Its
level with respect to R, introduced in [6] and defined in 2.3, measures the minimal
number of mapping cones needed to assemble a quasi-isomorphic complex from
bounded complexes of finite projective modules with differentials equal to zero.
The main result of this paper, which appears as Theorem 3.3, is the following
Tensor Nilpotence Theorem. Let f : G → F be a morphism of perfect complexes
over a commutative noetherian ring R.
If f factors through a complex whose homology is I-torsion for some ideal I of
R with height I ≥ levelR HomR (G, F ), then the induced morphism
⊗nR f : ⊗nR G → ⊗nR F
is homotopic to zero for some non-negative integer n.
Date: November 15, 2017.
2010 Mathematics Subject Classification. 13D22 (primary); 13D02, 13D09 (secondary).
Key words and phrases. big Cohen-Macaulay module, homological conjectures, level, perfect
complex, rank, tensor nilpotent morphism.
Partly supported by NSF grants DMS-1103176 (LLA) and DMS-1700985 (SBI).
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L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN

Big Cohen-Macaulay modules play an essential, if discreet role in the proof, as a tool
for constructing special morphisms in the derived category of R; see Proposition 3.7.
In applications to commutative algebra it is convenient to use another property
of morphisms of perfect complexes: f is fiberwise zero if H(k(p) ⊗R f ) = 0 holds for
every p in Spec R. Hopkins [21] and Neeman [25] have shown that this is equivalent
to tensor nilpotence; this is a key tool for the classification of the thick subcategories
of perfect R-complexes.
It is easy to see that the level of a complex does not exceed its span, defined
in 2.1. Due to these remarks, the Tensor Nilpotence Theorem is equivalent to the
Morphic Intersection Theorem. If f is not fiberwise zero and factors through
a complex with I-torsion homology for an ideal I of R, then there are inequalities:
span F + span G − 1 ≥ levelR HomR (G, F ) ≥ height I + 1 .
In Section 4 we use this result to prove directly, and sometimes to generalize
and sharpen, several basic theorems in commutative algebra. These include the
Improved New Intersection Theorem, the Monomial Theorem and several versions
of the Canonical Element Theorem. All of them are equivalent, but we do not
know if they imply the Morphic Intersection Theorem; a potentially significant
obstruction to that is that the difference span F − levelR F can be arbitrarily big.
Another application, in Section 5, yields lower bounds for ranks of certain finite
free complexes, related to a conjecture of Buchsbaum and Eisenbud, and Horrocks.
In [7] a version of the Morphic Intersection Theorem for certain tensor triangulated categories is proved. This has implications for morphisms of perfect complexes
of sheaves and, more generally, of perfect differential sheaves, over schemes.
2. Perfect complexes
Throughout this paper R will be a commutative noetherian ring.
This section is a recap on the various notions and construction, mainly concerning
perfect complexes, needed in this work. Pertinent references include [6, 26].
2.1. Complexes. In this work, an R-complex (a shorthand for ‘a complex of Rmodules’) is a sequence of homomorphisms of R-modules
X :=

∂X

X
∂n−1

n
→ Xn−1 −−−→ Xn−2 −→ · · ·
· · · −→ Xn −−−

such that ∂ X ∂ X = 0. We write X ♮ for the graded R-module underlying X. The ith
i
X
suspension of X is the R-complex Σi X with (Σi X)n = Xn−i and ∂nΣ X = (−1)i ∂n−i
for each n. The span of X is the number
span X := sup{i | Xi 6= 0} − inf{i | Xi 6= 0} + 1
Thus span X = −∞ if and only if X = 0, and span X = ∞ if and only if Xi 6= 0
for infinitely many i ≥ 0. The span of X is finite if and only if span X is an natural
number. When span X is finite we say that X is bounded.
Complexes of R-modules are objects of two distinct categories.
In the category of complexes C(R) a morphism f : Y → X of R-complexes is a
family (fi : Yi → Xi )i∈Z or R-linear maps satisfying ∂iX fi = fi−1 ∂iY . It is a quasiisomorphism if H(f ), the induced map in homology, is bijective. Complexes that
can be linked by a string of quasi-isomorphisms are said to be quasi-isomorphic.

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3

The derived category D(R) is obtained from C(R) by inverting all quasi-isomorphisms. For constructions of the localization functor C(R) → D(R) and of the
derived functors ?⊗LR ? and RHomR (¿, ?), see e.g. [13, 31, 24]. When P is a complex
of projectives with Pi = 0 for i ≪ 0, the functors P ⊗LR ? and RHomR (P, ?) are
represented by P ⊗R ? and HomR (P, ?), respectively. In particular, the localization
functor induces for each n a natural isomorphism of abelian groups
∼ HomD(R) (P, Σn X) .
(2.1.1)
H−n (RHomR (P, X)) =
2.2. Perfect complexes. In C(R), a perfect R-complex is a bounded complex
of finitely generated projective R-modules. When P is perfect, the R-complex
P ∗ := HomR (P, R) is perfect and the natural biduality map
P −→ P ∗∗ = HomR (HomR (P, R), R)
is an isomorphism. Moreover for any R-complex X the natural map
P ∗ ⊗R X −→ HomR (P, X)
is an isomorphism. In the sequel these properties are used without comment.
2.3. Levels. A length l semiprojective filtration of an R-complex P is a sequence
of R-subcomplexes of finitely generated projective modules
0 = P (0) ⊆ P (1) ⊆ · · · ⊆ P (l) = P
♮

♮

such that P (i − 1) is a direct summand of P (i) and the differential of P (i)/P (i−1)
is equal to zero, for i = 1, . . . , l. For every R-complex F , we set


F is a retract of some R-complex P that
R
.
level F = inf l ∈ N
has a semiprojective filtration of length l
By [6, 2.4], this number is equal to the level F with respect to R, defined in [6, 2.1].
In particular, levelR F is finite if and only if F is quasi-isomorphic to some perfect
complex. When F is quasi-isomorphic to a perfect complex P , one has
levelR F ≤ span P .

(2.3.1)

Indeed, if P := 0 → Pb → · · · → Pa → 0, then consider the filtration by subcomplexes P (n) := P<n+a . The inequality can be strict; see 2.7 below.
When R is regular, any R-complex F with H(F ) finitely generated satisfies
levelR F ≤ dim R + 1 .

(2.3.2)

For R-complexes X and Y one has
(2.3.3)

levelR (Σi X) = levelR X

for every integer i, and

R

level (X ⊕ Y ) = max{levelR X, levelR Y }.

These equalities follow easily from the definitions.
Lemma 2.4. The following statements hold for every perfect R-complex P .
(1) levelR (P ∗ ) = levelR P .
(2) For each perfect R-complex Q there are inequalities
levelR (P ⊗R Q) ≤ levelR P + levelR Q − 1 .
levelR HomR (P, Q) ≤ levelR P + levelR Q − 1 .

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L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN

Proof. (1) If P is a retract of P ′ , then levelR P ≤ levelR P ′ and P ∗ is a retract of
(P ′ )∗ . Thus, we can assume P itself has a finite semiprojective filtration {P (n)}ln=0 .
The inclusions P (l − i) ⊆ P (l − i + 1) ⊆ P define subcomplexes
∗

P ∗ (i) := Ker(P ∗ −→ P (l − i) ) ⊆ Ker(P ∗ −→ P (l − i + 1)) =: P ∗ (i + 1)
of finitely generated projective modules. They form a length l semiprojective filtration of P ∗ , as P ∗ (i − 1)♮ is a direct summand of P ∗ (i)♮ and there are isomorphisms
∗

P ∗ (n) ∼ P (l − n + 1)
.
=
P ∗ (n − 1)
P (l − n)
This gives levelR P ∗ ≤ levelR P , and the reverse inequality follows from P ∼
= P ∗∗ .
(2) Assume first that P has a semiprojective filtration {P (n)}ln=0 and Q has a
semiprojective filtration {Q(n)}m
n=0 . For all h, i, we identity P (h) ⊗R Q(i) with a
subcomplex of P ⊗R Q. For each non-negative integer n ≥ 0 form the subcomplex
X
C(n) :=
P (j + 1) ⊗R Q(n − j)
j>0

of P ⊗R Q. A direct computation yields an isomorphism of R-complexes
Q(n − j)
C(n) ∼ X P (j + 1)
⊗R
.
=
C(n − 1)
P (j)
Q(n − j − 1)
j>0

l+m−1
{C(n)}n=0

Thus
is a semiprojective filtration of P ⊗R Q.
The second inequality in (2) follows from the first one, given (1) and the isomorphism HomR (P, Q) ∼
= P ∗ ⊗R Q. Next we verify the first inequality. There is nothing
to prove unless the levels of P and Q are finite. Thus we may assume that P is a
retract of a complex P ′ with a semiprojective filtration of length l = levelR P and Q
is a retract of a complex Q′ with a semiprojective filtration of length m = levelR Q.
Then P ⊗R Q is a retract of P ′ ⊗R Q′ , and—by what we have just seen—this
complex has a semiprojective filtration of length l + m − 1, as desired.

2.5. Ghost maps. A ghost is a morphism g : X → Y in D(R) such that H(g) = 0;
see [11, §8]. Evidently a composition of morphisms one of which is ghost is a ghost.
The next result is a version of the “Ghost Lemma”; cf. [11, Theorem 8.3], [29,
Lemma 4.11], and [5, Proposition 2.9].
Lemma 2.6. Let F be an R-complex and c an integer with c ≥ levelR F .
When g : X → Y is a composition of c ghosts the following morphisms are ghosts
F ⊗LR g : F ⊗LR X −→ F ⊗LR Y

and

RHomR (F, g) : RHomR (F, X) −→ RHomR (F, Y )
Proof. For every R-complex W there is a canonical isomorphism
≃

RHomR (F, R) ⊗LR W −−→ RHomR (F, W ) ,
so it suffices to prove the first assertion. For that, we may assume that F has a
semiprojective filtration {F (n)}ln=0 , where l = levelR F . By hypothesis, g = h ◦ f
where f : X → W is a (c − 1)fold composition of ghosts and h : W → Y is a ghost.
Tensoring these maps with the exact sequence of R-complexes
ι

π

0 −→ F (1) −−→ F −−→ G −→ 0

TENSOR NILPOTENT MORPHISMS

5

where G := F/F (1), yields a commutative diagram of graded R-modules
H(F (1) ⊗R X)

// H(F ⊗R X)

H(F (1)⊗f )

// H(G ⊗R X)

H(F ⊗f )


H(F (1) ⊗R W )

H(ι⊗W )


// H(F ⊗R W )

H(F (1)⊗h)

H(G⊗f )

H(π⊗W )


// H(G ⊗R W )

H(F ⊗h)


H(F (1) ⊗R Y )

H(ι⊗Y )


// H(F ⊗R Y )


// H(G ⊗R Y )

where the rows are exact. Since levelR G ≤ l − 1 ≤ c − 1, the induction hypothesis
implies G ⊗ f is a ghost; that is to say, H(G ⊗ f ) = 0. The commutativity of the
diagram above and the exactness of the middle row implies that
Im H(F ⊗ f ) ⊆ Im H(ι ⊗ W )
This entails the inclusion below.
Im H(F ⊗ g) = H(F ⊗ h)(Im H(F ⊗ f ))
⊆ H(F ⊗ h)(Im H(ι ⊗ W ))
⊆ Im H(ι ⊗ Y ) H(F (1) ⊗ h))
=0
The second equality comes from the commutativity of the diagram. The last one
holds because F (1) is graded-projective and H(h) = 0 imply H(F (1) ⊗ h) = 0. 
2.7. Koszul complexes. Let x := x1 , . . . , xn be elements in R.
♮
We write K(x) for the Koszul complex on x. Thus K(x) is the exterior algebra
K
on a free R-module K(x)1 with basis {e
x1 , . . . , x
en }, and ∂ is the unique R-linear
map that satisfies the Leibniz rule and has ∂(e
xi ) = xi for i = 1, . . . , n. In particular,
K(x) is a DG (differential graded) algebra, and so its homology H(K(x)) is a graded
algebra with H0 (K(x)) = R/(x). This implies (x) H(K(x)) = 0.
Evidently K(x) is a perfect R-complex; it is indecomposable when R is local;
see [1, 4.7]. As K(x)i is non-zero precisely for 0, . . . , n, from (2.3.1) one gets
levelR K(x) ≤ span K(x) = n + 1 .
Equality holds if R is local and x is a system of parameters; see Theorem 4.2 below.
However, span K(x) − levelR K(x) can be arbitrarily large; see [1, Section 3].
For any Koszul complex K on n elements, there are isomorphisms of R-complexes
n
M
n
Σi K ( i ) .
K∗ ∼
=
= Σ−n K and K ⊗R K ∼
i=0

See [8, Propositions 1.6.10 and 1.6.21]. It thus follows from (2.3.3) that
(2.7.1)

levelR HomR (K, K) = levelR (K ⊗R K) = levelR K .

In particular, the inequalities in Lemma 2.4(2) can be strict.
3. Tensor nilpotent morphisms
In this section we prove the Tensor Nilpotence Theorem announced in the introduction. We start by reviewing the properties of interest.

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L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN

3.1. Tensor nilpotence. Let f : Y → X be a morphism in D(R).
The morphism f is said to be tensor nilpotent if for some n ∈ N the morphism
f ⊗L · · · ⊗L f : Y ⊗LR · · · ⊗LR Y −→ X ⊗LR · · · ⊗LR X
| R {z R }
n

is equal to zero in D(R); when the R-complexes X, Y are perfect this means that
the morphism ⊗n f : ⊗nR Y → ⊗nR X is homotopic to zero. When X is perfect and
f : X → Σl X is a morphism with ⊗n f homotopic to zero the n-fold composition
Σl f

Σ2l f

Σnl

X −→ Σl X −−−→ Σ2n X −−−−→ · · · −−−→ Σnl X
is also homotopic to zero. The converse does not hold, even when R is a field for
in that case tensor nilpotent morphisms are zero.
3.2. Fiberwise zero morphisms. A morphism f : Y → X that satisfies
k(p) ⊗LR f = 0

in D(k(p))

for every p ∈ Spec R

is said to be fiberwise zero. This is equivalent to requiring k ⊗LR f = 0 in D(k) for
every homomorphism R → k with k a field. In D(k) a morphism is zero if (and
only if) it is a ghost, so the latter condition is equivalent to H(k ⊗LR f ) = 0.
In D(k), a morphism is tensor nilpotent exactly when it is zero. Thus if f is
tensor nilpotent, it is fiberwise zero. There is a partial converse: If a morphism
f : G → F of perfect R-complexes is fiberwise zero, then it is tensor nilpotent. This
was proved by Hopkins [21, Theorem 10] and Neeman [25, Theorem 1.1].
The next result is the Tensor Nilpotence Theorem from the Introduction. Recall
that an R-module is said to be I-torsion if each one of its elements is annihilated
by some power of I.
Theorem 3.3. Let R be a commutative noetherian ring and f : G → F a morphism
of perfect R-complexes. If for some ideal I of R the following conditions hold
(1) f factors through some complex with I-torsion homology, and
(2) levelR HomR (G, F ) ≤ height I ,
then f is fiberwise zero. In particular, f is tensor nilpotent.
The proof of the theorem is given after Proposition 3.7.
Remark 3.4. Lemma 2.4 shows that the inequality (2) is implied by
levelR F + levelR G ≤ height I + 1 ;
the converse does not hold; see (2.7.1).
On the other hand, condition (2) cannot be weakened: Let (R, m, k) be a local
ring and G the Koszul complex on some system of parameters of R and let
f : G −→ (G/G6d−1 ) ∼
= Σd R
be the canonical surjection with d = dim R. Then G is an m-torsion complex and
levelR G = d + 1; see 2.7. Evidently H(k ⊗R f ) 6= 0, so f is not fiberwise zero.
In the proof of Theorem 3.3 we exploit the functorial nature of I-torsion.

TENSOR NILPOTENT MORPHISMS

7

3.5. Torsion complexes. The derived I-torsion functor assigns to every X in
D(R) an R-complex RΓI X; when X is a module it computes its local cohomology:
HnI (X) = H−n (RΓI X) holds for each integer n. There is a natural morphism
t : RΓI X −→ X in D(R) that has the following universal property: Every morphism
Y → X such that H(Y ) is I-torsion factors uniquely through t; see Lipman [24,
Section 1]. It is easy to verify that the following conditions are equivalent.
(1) H(X) is I-torsion.
(2) H(X)p = 0 for each prime ideal p 6⊇ I.
(3) The natural morphism t : RΓI X → X is a quasi-isomorphism.
When they hold, we say that X is I-torsion. Note a couple of properties:
(3.5.1)
(3.5.2)

If X is I-torsion, then X ⊗LR Y is I-torsion for any R-complex Y .
There is a natural isomorphism RΓI (X ⊗L Y ) ∼
= (RΓI X) ⊗L Y .
R

R

Indeed, H(Xp ) ∼
= H(X)p = 0 holds for each p 6⊇ I, giving Xp = 0 in D(R). Thus
(X ⊗LR Y )p ∼
= Xp ⊗LR Y ∼
=0
holds in D(R). It yields H(X ⊗LR Y )p ∼
= H((X ⊗LR Y )p ) = 0, as desired.
A proof of the isomorphism in (3.5.2) can be found in [24, 3.3.1].
3.6. Big Cohen-Macaulay modules. Let (R, m, k) be a local ring.
A (not necessarily finitely generated) R-module C is big Cohen-Macaulay if every
system of parameters for R is a C-regular sequence, in the sense of [8, Definition
1.1.1]. In the literature the name is sometimes used for R-modules C that satisfy
the property for some system of parameters for R; however, the m-adic completion
of C is then big Cohen-Macaulay in the sense above; see [8, Corollary 8.5.3].
The existence of big Cohen-Macaulay was proved by Hochster [16, 17] in case
when R contains a field as a subring, and by André [2] when it does not; for the
latter case, see also Heitmann and Ma [15].
In this paper, big Cohen-Macaulay modules are visible only in the next result.
Proposition 3.7. Let I be an ideal in R and set c := height I.
When C is a big Cohen-Macaulay R-module the following assertions hold.
(1) The canonical morphism t : RΓI C → C from the I-torsion complex RΓI C
(see 3.5) is a composition of c ghosts.
(2) If a morphism g : G → C of R-complexes with levelR G ≤ c factors through
some I-torsion complex, then g = 0.
Proof. (1) We may assume I = (x), where x = {x1 , . . . , xc } is part of a system of
parameters for R; see [8, Theorem A.2]. The morphism t factors as
RΓ(x1 ,...,xc ) (C) −→ RΓ(x1 ,...,xc−1 ) (C) −→ · · · −→ RΓ(x1 ) (C) −→ C .
Since the sequence x1 , . . . , xc is C-regular, we have Hi (RΓ(x1 ,...,xj ) (C)) = 0 for
i 6= −j; see [8, (3.5.6) and (1.6.16)]. Thus every one of the arrows above is a ghost,
so that t is a composition of c ghosts, as desired.
(2) Suppose g factors as G → X → C with X an I-torsion R-complex. As noted
in 3.5, the morphism X → C factors through t, so g factors as
g′

g′′

t

G −→ X −→ RΓI C −
→C.

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L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN

In view of the hypothesis levelR G ≤ c and part (1), Lemma 2.6 shows that
RHomR (G, t) : RHomR (G, RΓI C) −→ RHomR (G, C)
is a ghost. Using brackets to denote cohomology classes, we get
[g] = [tg ′′ g ′ ] = H0 (RHomR (G, t))([g ′′ g ′ ]) = 0 .
Due to the isomorphism (2.1.1), this means that g is zero in D(R).



Lemma 3.8. Let f : G → F be a morphism of perfect R-complexes, where G is
finite free with Gi = 0 for i ≪ 0 and F is perfect. Let f ′ : F ∗ ⊗R G → R denote
the composed morphism in the next display, where e is the evaluation map:
F ∗ ⊗R f

e

F ∗ ⊗R G −−−−−−→ F ∗ ⊗R F −−→ R .
If f factors through some I-torsion complex, then so does f ′ .
The morphism f ′ is fiberwise zero if and only if so is f .
Proof. For the first assertion, note that if f factors through an I-torsion complex
X, then F ∗ ⊗R f factors through F ∗ ⊗R X, and the latter is I-torsion.
For the second assertion, let k be field and R → k be a homomorphism of rings.
Let (−) and (−)∨ stand for k ⊗R (−) and Homk (−, k), respectively. The goal is to
prove that f = 0 is equivalent to f ′ = 0.
Since F is perfect, there are canonical isomorphisms
∼
∨
=
F ∗ ⊗ k −−→ HomR (F, k) ∼
= Homk (k ⊗R F, k) = (F ) .

Given this, it follows that f ′ can be realized as the composition of morphisms
∨

(F ) ⊗k f

∨

∨

e

(F ) ⊗k G −−−−−−→ (F ) ⊗k F −−→ k .
If F is zero, then f = 0 and f ′ = 0 hold. When F is nonzero, it is easy to verify
6 0 is equivalent to f ′ 6= 0, as desired.

that f =
Proof of Theorem 3.3. Given morphisms of R-complexes G → X → F such that F
and G are perfect and X is I-torsion for an ideal I with
levelR HomR (G, F ) ≤ height I ,
we need to prove that f is fiberwise zero. This implies the tensor nilpotence of f ,
as recalled in 3.1.
By Lemma 3.8, the morphism f ′ : F ∗ ⊗R G → R factors through an I-torsion
complex, and if f ′ is fiberwise zero, so if f . The isomorphisms of R-complexes
(F ∗ ⊗R G)∗ ∼
= G∗ ⊗R F ∼
= HomR (G, F )
and Lemma 2.4 yield levelR (F ∗ ⊗R G) = levelR HomR (G, F ). Thus, replacing f
by f ′ , it suffices to prove that if f : G → R is a morphism that factors through an
I-torsion complex and satisfies levelR G ≤ height I, then f is fiberwise zero.
Fix p in Spec R. When p 6⊇ I we have Xp = 0, by 3.5(2). For p ⊇ I we have
levelRp Gp ≤ levelR G ≤ height I ≤ height Ip ,
where the first inequality follows directly from the definitions; see [6, Proposition
3.7]. It is easy to verify that Xp is Ip -torsion. Thus, localizing at p, we may further
assume (R, m, k) is a local ring, and we have to prove that H(k ⊗R f ) = 0 holds.
Let C be a big Cohen-Macaulay R-module. It satisfies mC 6= C, so the canonical
γ

ε

f

γ

map π : R → k factors as R −
→C −
→ k. The composition G −
→R−
→ C is zero in

TENSOR NILPOTENT MORPHISMS

9

D(R), by Proposition 3.7. We get πf = εγf = 0, whence H(k ⊗R π) H(k ⊗R f ) = 0.
Since H(k ⊗R π) is bijective, this implies H(k ⊗R f ) = 0, as desired.

The following consequence of Theorem 3.3 is often helpful.
Corollary 3.9. Let (R, m, k) be a local ring, F a perfect R-complex, and G an
R-complex of finitely generated free modules.
If a morphism of R-complexes f : G → F satisfies the conditions
(1) f factors through some m-torsion complex,
(2) sup F ♮ − inf G♮ ≤ dim R − 1, and
then H(k ⊗R f ) = 0.
Proof. An m-torsion complex X satisfies k(p) ⊗LR X = 0 for any p in Spec R \ {m}.
Thus a morphism, g, of R-complexes that factors through X is fiberwise zero if and
only if k ⊗LR g = 0. This remark will be used in what follows.
Condition (2) implies Gn = 0 for n ≪ 0. Let f ′ denote the composition
F ∗ ⊗R f

e

F ∗ ⊗R G −−−−−−→ F ∗ ⊗R F −−→ R ,
where e is the evaluation map. Since inf (F ∗ ⊗R G)♮ = − sup F ♮ +inf G♮ , Lemma 3.8
shows that it suffices to prove the corollary for morphisms f : G → R.
As f factors through some m-torsion complex, so does the composite morphism
f

G60 ⊆ G −−→ R
It is easy to check that if the induced map H(k ⊗R G60 ) → H(k ⊗R R) = k is zero,
then so is H(k ⊗R f ). Thus we may assume Gn = 0 for n 6∈ [−d + 1, 0], where
d = dim R. This implies levelR G ≤ d, so Theorem 3.3 yields the desideratum. 
For some applications the next statement, with weaker hypothesis but also
weaker conclusion, suffices. The example in Remark 3.4 shows that the result
cannot be strengthened to conclude that f is fiberwise zero.
Theorem 3.10. Let R be a local ring and f : G → F a morphism of R-complexes.
If there exists an ideal I of R such that
(1) f factors through an I-torsion complex, and
(2) levelR F ≤ height I,
then H(C ⊗LR f ) = 0 for every big Cohen-Macaulay module C.
Proof. Set c := height I and let t : RΓI C → C be the canonical morphism. It
follows from (3.5.1) that C ⊗LR f also factors through an I-torsion R-complex. Then
the quasi-isomorphism (3.5.2) and the universal property of the derived I-torsion
functor, see 3.5, implies that C ⊗LR f factors as a composition of the morphisms:
t⊗L F

R
C ⊗LR G −→ (RΓI C) ⊗LR F −−−−
−→ C ⊗LR F

By Proposition 3.7(1) the morphism t is a composition of c ghosts. Thus condition
(2) and Lemma 2.6 imply t ⊗LR F is a ghost, and hence so is C ⊗LR f .


10

L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN

4. Applications to local algebra
In this section we record applications the Tensor Nilpotence Theorem to local
algebra. To that end it is expedient to reformulate it as the Morphic Intersection
Theorem from the Introduction, restated below.
Theorem 4.1. Let R be a commutative noetherian ring and f : G → F a morphism
of perfect R-complexes.
If f is not fiberwise zero and factors through a complex with I-torsion homology
for an ideal I of R, then there are inequalities:
span F + span G − 1 ≥ levelR HomR (G, F ) ≥ height I + 1 .
Proof. The inequality on the left comes from Lemma 2.4 and (2.3.1). The one on
the right is the contrapositive of Theorem 3.3.

Here is one consequence.
Theorem 4.2. Let R be a local ring and F a complex of finite free R-modules:
F :=

0 → Fd → Fd−1 → · · · → F0 → 0

For each ideal I such that I · Hi (F ) = 0 for i ≥ 1 and I · z = 0 for some element z
in H0 (F ) \ m H0 (F ), where m is the maximal ideal of R, one has
d + 1 ≥ span F ≥ levelR F ≥ height I + 1 .
Proof. Indeed, the first two inequalities are clear from definitions. As to third
one, pick ze ∈ F0 representing z in H0 (F ) and consider the morphism of complexes
f : R → F given by r 7→ re
z . Since z is not in m H0 (F ), one has
H0 (k ⊗R f ) = k ⊗R H0 (f ) 6= 0

for k = R/m. In particular, k⊗R f is nonzero. On the other hand, f factors through
the inclusion X ⊆ F , where X is the subcomplex defined by
(
Fi
i≥1
Xi =
Re
z + d(F1 ) i = 0
By construction, we have Hi (X) = Hi (F ) for i ≥ 1 and I H0 (X) = 0, so H(X) is
I-torsion. The desired inequality now follows from Theorem 4.1 applied to f . 
The preceding result is a stronger form of the Improved New Intersection Theorem1 of Evans and Griffith [12]; see also [19, §2]. First, the latter is in terms of
spans of perfect complexes whereas the one above is in terms of levels with respect
to R; second, the hypothesis on the homology of F is weaker. Theorem 4.2 also
subsumes prior extensions of the New Intersection Theorem to statements involving
levels, namely [6, Theorem 5.1], where it was assumed that I · H0 (F ) = 0 holds,
and [1, Theorem 3.1] which requires Hi (F ) to have finite length for i ≥ 1.
In the influential paper [18], Hochster identified certain canonical elements in the
local cohomology of local rings, conjectured that they are never zero, and proved
that statement in the equal characteristic case. He also gave several reformulations
that do not involve local cohomology. Detailed discussions of the relations between
these statements and the histories of their proofs are presented in [28] and [20].
1This and the other statements in this section were conjectures prior to the appearance of [2].

TENSOR NILPOTENT MORPHISMS

11

Some of those statements concern properties of morphisms from the Koszul complex on some system of parameters to resolutions of various R-modules. This makes
them particularly amenable to approaches from the Morphic Intersection Theorem.
In the rest of this section we uncover direct paths to various forms of the Canonical
Element Theorem and related results.
We first prove a version of [18, 2.3]. The conclusion there is that fd is not zero,
but the remarks in [18, 2.2(6)] show that it is equivalent to the following statement.
Theorem 4.3. Let (R, m, k) be a local ring, x a system of parameters for R, and
K the Koszul complex on x, and F a complex of free R-modules.
If f : K → F is a morphism of R-complexes with H0 (k ⊗R f ) 6= 0, then one has
Hd (S ⊗R f ) 6= 0

for

S = R/(x)

and

d = dim R .

Proof. Recall from 2.7 that ∂(K) lies in (x)K, that K1 has a basis x
e1 , . . . , x
ed and K ♮
is the exterior algebra on K1 . Thus Kd is a free R-module with basis x = x
e1 · · · x
ed
and Hd (S ⊗R K) = S(1 ⊗ x), so we need to prove f (Kd ) 6⊆ (x)Fd + ∂(Fd+1 ).
Arguing by contradiction, we suppose the contrary. This means
f (x) = x1 y1 + · · · + xd yd + ∂ F (y)
with y1 , . . . , yd ∈ Fd and y ∈ Fd+1 . For i = 1, . . . , d set x∗i := x
e1 · · · x
ei−1 x
ei+1 · · · x
ed ;
thus {x∗1 , . . . , x∗d } is a basis of the R-module Kd−1 . Define R-linear maps
hd−1 : Kd−1 → Fd

by

hd : Kd → Fd+1

by

hd−1 (x∗i ) = (−1)i−1 yi

for i = 1, . . . , d .

hd (x) = y .

Extend them to a degree one map h : K → F with hi = 0 for i 6= d − 1, d. The map
g := f − ∂ F h − h∂ K : K → F
is easily seen to be a morphism of complexes that is homotopic to f and satisfies
gd = 0. This last condition implies that g factors as a composition of morphisms
g′

K −−→ F<d ⊆ F .
The complex K is m-torsion; see 2.7. Thus Corollary 3.9, applied to g ′ , yields
H(k ⊗R g ′ ) = 0. This gives the second equality below:
H0 (k ⊗R f ) = H0 (k ⊗R g) = H0 (k ⊗R g ′ ) = 0 .
The first one holds because f and g are homotopic, and the last one because g0 = g0′ .
The result of the last computation contradicts the hypotheses on H0 (k ⊗R f ). 
A first specialization is the Canonical Element Theorem.
Corollary 4.4. Let I be an ideal in R containing a system of parameters x1 , . . . , xd .
With K the Koszul complex on x and F a free resolution of R/I, any morphism
f : K → F of R-complexes lifting the surjection R/(x) → R/I has fd (K) 6= 0. 
As usual, when A is a matrix, Id (A) denotes the ideal of its minors of size d.
Corollary 4.5. Let R be a local ring, x a system of parameters for R, and y a
finite subset of R with (y) ⊇ (x).
If A is a matrix such that Ay = x, then Id (A) 6⊆ (x) for d = dim R.

12

L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN

Proof. Let K and F be the Koszul complexes on x and y, respectively. The matrix
A defines a unique morphism of DG R-algebras f : K → F . Evidently, H0 (k ⊗R f )
is the identity map on k, and hence is not zero. Since fd can be represented by a
column matrix whose entries are the various d×d minors of A, the desired statement
is a direct consequence of Theorem 4.3.

A special case of the preceding result yields the Monomial Theorem.
Corollary 4.6. When y1 , . . . , yd is a system of parameters for local ring, one has
(y1 · · · yd )n 6∈ (y1n+1 , . . . , ydn+1 )
Proof. Apply Corollary 4.5 to the inclusion
 n
y1 0
 0 y2n

A:= .
..
 ..
.
0

0

for every integer

n ≥ 1.

(y1n+1 , . . . , ydn+1 ) ⊆ (y1 , . . . , yd ) and

··· 0
··· 0 


..  .
..
.
.
· · · ydn

We also deduce from Theorem 4.3 another form of the Canonical Element Theorem. Roberts [27] proposed the statement and proved that it is equivalent to the
Canonical Element Theorem; a different proof appears in Huneke and Koh [22].
Recall that for any pair (S, T ) of R-algebras the graded module TorR (S, T ) carries a natural structure of graded-commutative R-algebra, given by the ⋔-product
of Cartan and Eilenberg [10, Chapter XI.4].

Lemma 4.7. Let R be a commutative ring, I an ideal of R, and set S := R/I. Let
G → S be some R-free resolution, K be the Koszul complex on some generating set
of I, and g : K → G a morphisms of R-complexes lifting the identity of S.
For every surjective homomorphism ψ : S → T of of commutative rings there is
a commutative diagram of strictly graded-commutative S-algebras
S ⊗R K

V

S H1 (S ⊗R K)

α
V

ψ

// // V TorR
1 (S, S)
S

µS

TorR
1 (S,ψ)

V

T



TorR
1 (S, T )

// TorR (S, S)
TorR (S,ψ)

µT


// TorR (S, T )

where α1 = H1 (S⊗R g), the map α is defined by the functoriality of exterior algebras,
and the maps µ? are defined by the universal property of exterior algebras.
V
Proof. The equality follows from ∂ K (K) ⊆ IK and K ♮ = R K1 . The resolution
G can be chosen to have G61 = K61 ; this makes α1 surjective, and the surjectivity of α follows. The map TorR
1 (S, ψ) is surjective because it can be identified
with the natural map I/I 2 → I/IJ, where J = Ker(R → T ); the surjectivity of
V
R
ψ Tor1 (S, ψ) follows. The square commutes by the naturality of ⋔-products. 

Theorem 4.8. Let (R, m, k) be a local ring, I a parameter ideal, and S := R/I.
For each surjective homomorphism S → T the morphism of graded T -algebras
V
R
µT : T TorR
1 (S, T ) −→ Tor (S, T )
has the property that µT ⊗T k is injective. In particular, µk is injective.

TENSOR NILPOTENT MORPHISMS

13

Proof. The functoriality of the construction of µ implies that the canonical surjection π : T → k induces a commutative diagram of graded-commutative algebras
V

T

V

π

TorR
1 (S, T )

µT

// TorR (S, T )

TorR
1 (S,π)

V

k

TorR (S,π)



TorR
1 (S, k)

µk


// TorR (S, k)

It is easy to verify that π induces a bijective map
R
R
TorR
1 (S, π) ⊗T k : Tor1 (S, T ) ⊗T k → Tor1 (S, k) ,

so (∧π TorR (S, π)) ⊗T k is an isomorphism. Thus it suffices to show µk is injective.
≃
Let K be the Koszul complex on a minimal generating set of I. Let G −
→ S and
≃
F −
→ k be R-free resolutions of S and k, respectively. Lift the identity map of S
and the canonical surjection ψ : S → k to morphisms g : K → G and h : G → F ,
respectively. We have µS α = H(S ⊗R g) and TorR (S, ψ) = H(S ⊗R h). This implies
the second equality in the string
R
S
µkd TorR
d (S, π)αd = Tord (S, ψ)µd αd = Hd (S ⊗R hg) 6= 0 .

The first equality comes from Lemma 4.7, with T = k, and the non-equality from
Theorem 4.3, with f = hg. In particular, we get µkd 6= 0. We have an isomorphism
∼ V k d of graded k-algebras, so µk is injective by the next remark. 
TorR
1 (S, k) =
k
V
Remark 4.9. If Q is a field, d is a non-negative integer, and λ : Q Qd → B is a
homomorphism of graded Q-algebras with λd 6= 0, then λ is injective.
V
Vd
Indeed, the graded subspace Q Qd of exterior algebra Q Qd is contained in
every every non-zero ideal and has rank one, so λd 6= 0 implies Ker(λ) = 0.
5. Ranks in finite free complexes
This section is concerned with DG modules over Koszul complexes on sequences
of parameters. Under the additional assumptions that R is a domain and F is a
resolution of some R-module, the theorem below was proved in [3, 6.4.1], and earlier
for cyclic modules in [9, 1.4]; background is reviewed after the proof. The Canonical
Element Theorem, in the form of Theorem 4.3 above, is used in the proof.
Theorem 5.1. Let (R, m, k) be a local ring, set d = dim R, and let F be a complex
of finite free R-modules with H0 (F ) 6= 0 and Fi = 0 for i < 0.
If F admits a structure of DG module over the Koszul complex on some system
of parameters of R, then there is an inequality
 
d
(5.1.1)
rankR (Fn ) ≥
for each n ∈ Z .
n
Proof. The desired inequality is vacuous when d = 0, so suppose d ≥ 1. Let x be
the said system of parameters of R and K the Koszul complex on x. Since F is a
DG K-module, each Hi (F ) is an R/(x)-module, and hence of finite length.
First we reduce to the case when R is a domain. To that end, let p be a prime
ideal of R such that dim(R/p) = d. Evidently, the image of x in R is a system of

14

L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN

parameters for R/p. By base change, (R/p) ⊗R F is a DG module over (R/p) ⊗R K,
the Koszul complex on x with coefficients in R/p, with
∼ R/p ⊗ H0 (F ) 6= 0 .
H0 ((R/p) ⊗R F ) =
♮

Moreover, the rank of F ♮ as an R-module equals the rank of (R/p) ⊗R F as an
R/p-module. Thus, after base change to R/p we can assume R is a domain.
Choose a cycle z ∈ F0 that maps to a minimal generator of the R-module H0 (F ).
Since F is a DG K-module, this yields a morphism of DG K-modules
f: K →F

with

f (a) = az .

This is, in particular, a morphism of complexes. Since k ⊗R H0 (F ) 6= 0, by the
choice of z, Theorem 4.3 applies, and yields that f (Kd ) 6= 0. Since R is a domain,
this implies f (Q ⊗R Kd ) is non-zero, where Q is the field of fractions of R.
♮
Set Λ := (Q ⊗R K) and consider the homomorphism of graded Λ-modules
V

λ := Q ⊗R f ♮ : Λ → Q ⊗R F ♮ .

Qd , Remark 4.9 gives the inequality in the display
 
d
rankR (Fn ) = rankQ (Q ⊗R Fn ) ≥ rankQ (Λn ) =
.
n

As Λ is isomorphic to

Q

Both equalities are clear.



The inequalities (5.1.1) are related to a major topic of research in commutative
algebra. We discuss it for a local ring (R, m, k) and a bounded R-complex F of
finite free modules with F<0 = 0, homology of finite length, and H0 (F ) 6= 0.
5.2. Ranks of syzygies. The celebrated and still open Rank Conjecture of Buchsbaum and Eisenbud [9, Proposition 1.4], and Horrocks [14, Problem 24] predicts
that (5.1.1) holds whenever F is a resolution of some module ofPfinite length.
That conjecture is known for d ≤ 4. Its validity would imply n rankR Fn ≥ 2d .
For d = 5 and equicharacteristic R, this was proved in [4, Proposition 1] by using
Evans and Griffth’s Syzygy Theorem [12]; in view of [2], it holds for all R.
M. Walker [30] used methods from K-theory to prove that
P In a breakthrough,
d
rank
F
≥
2
holds
when R contains 12 and is complete intersection (in particR
n
n
ular, regular), and when R is an algebra over some field of positive characteristic.
5.3. Obstructions for DG module structures. Theorem 5.1 provides a series
of obstruction for the existence of any DG module structure on F . In particular, it
implies that if rankR F < 2d holds with d = dim R, then F supports no DG module
structure over K(x) for any system of parameters x. Complexes satisfying the
restriction on ranks were recently constructed in [23, 4.1]. These complexes have
nonzero homology in degrees 0 and 1, so they are not resolutions of R-modules.
5.4. DG module structures on resolutions. Let F be a minimal resolution of
an R-module M of nonzero finite length and x a parameter set for R with xM = 0.
When F admits a DG module structure over K(x) the Rank Conjecture holds,
by Theorem 5.1. It was conjectured in [9, 1.2′ ] that such a structure exists for all
F and x. An obstruction to its existence was found in [3, 1.2], and examples when
that obstruction is not zero were produced in [3, 2.4.2]. On the other hand, by [3,
1.8] the obstruction vanishes when x lies in m annR (M ).

TENSOR NILPOTENT MORPHISMS

15

It is not known if F supports some DG K(x)-module structure for special choices
of x; in particular, for high powers of systems of parameters contained in annR (M ).
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L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN

[29] R. Rouquier, Dimensions of triangulated categories, J. K-Theory 1 (2008), 193–256, 257–258.
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Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.
E-mail address: avramov@math.unl.edu
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, U.S.A.
E-mail address: iyengar@math.utah.edu
Centre for Mathematics and its Applications, Mathematical Sciences Institute Australian National University, Canberra, ACT 0200, Australia.
E-mail address: Amnon.Neeman@anu.edu.au

