# Maximal Cohen-Macaulay complexes and
their uses: A partial survey

Srikanth B. Iyengar  Department of Mathematics, University of Utah, Salt Lake
City, UT 84112, USA<EMAIL_ADDRESS>,  Linquan Ma  Department of
Mathematics, Purdue University, 150 N. University street, IN 47907
<EMAIL_ADDRESS>,  Karl Schwede  Department of Mathematics, University of
Utah, Salt Lake City, UT 84112, USA<EMAIL_ADDRESS>and  Mark E. Walker
Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.
<EMAIL_ADDRESS>
(Date: 15th June 2021)

###### Abstract.

This work introduces a notion of complexes of maximal depth, and maximal
Cohen-Macaulay complexes, over a commutative noetherian local ring. The
existence of such complexes is closely tied to the Hochster’s “homological
conjectures”, most of which were recently settled by André. Various
constructions of maximal Cohen-Macaulay complexes are described, and their
existence is applied to give new proofs of some of the homological
conjectures, and also of certain results in birational geometry.

###### Key words and phrases:

homological conjectures, maximal Cohen-Macaulay complex, multiplier ideal,
resolution of singularities

###### 2020 Mathematics Subject Classification:

13D02 (primary); 13D22, 13D45, 14E15, 14F18 (secondary)

Part of this work was partly supported by National Science Foundation DMS
grants 2001368 (SBI); 1901672 and FRG grant 1952366 (LM), 1801849 and FRG
grant 1952522 (KS), and 1901848 (MW). LM was supported by a fellowship from
the Sloan Foundation. KS was supported by a fellowship from the Simons
foundation.

## Introduction

Big Cohen-Macaulay modules over (commutative, noetherian) local rings were
introduced by Hochster around fifty years ago and their relevance to local
algebra is established beyond doubt. Indeed, they play a prominent role in
Hochster’s lecture notes [Hochster:1975], where he describes a number of
homological conjectures that can be proved using big Cohen-Macaulay modules,
and their finitely generated counterparts, the maximal Cohen-Macaulay modules;
the latter are sometimes called, as in _loc. cit._ , “small” Cohen-Macaulay
modules. Hochster [Hochster:1973, Hochster:1975] proved that big Cohen-
Macaulay modules exist when the local ring contains a field; and conjectured
that even rings of mixed-characteristic possess such modules. This conjecture
was proved by André [Andre:2018], thereby settling a number of the homological
conjectures. In fact, by results of Hochster and Huneke [Hochster/Huneke:1992,
Hochster/Huneke:1995], and André [Andre:2018] there exist even big Cohen-
Macaulay _algebras_ over any local ring. The reader will find a survey of
these developments in [Hochster:2017, Ma/Schwede:2019].

In this work we introduce three versions of the Cohen-Macaulay property that
apply also to complexes of modules, discuss various constructions that give
rise to them, and present some consequences that follow from their existence.
In fact such complexes have come up earlier, in the work of Roberts
[Roberts:1980, Roberts:1980a], recalled in §3.18, and in recent work of Bhatt
[Bhatt:2020], though only in passing. What we found is that results that were
proved using big Cohen-Macaulay modules can often be proved using one of their
complex variants. This assertion is backed up the material presented in
Sections 2 and 4. Moreover, as will be apparent in the discussion in Section
3, the complex versions are easier to construct, and with better finiteness
properties. It thus seems worthwhile to shine an independent light on them.
Let us begin by defining them.

We say that a complex $M$ over a local ring $R$ with maximal ideal
$\mathfrak{m}$ has _maximal depth_ if $0pt_{R}M=\operatorname{dim}R$, where
depth is as in §1.6; we ask also that $\operatorname{H}(M)$ be bounded and the
canonical map
$\operatorname{H}_{0}(M)\to\operatorname{H}_{0}(k\otimes_{R}^{\operatorname{L}}M)$
be non-zero. Any complex that satisfies the last condition has depth at most
$\operatorname{dim}R$, whence the name “maximal depth”. An $R$-module has
maximal depth precisely when it is big Cohen-Macaulay. The depth of a complex
can be computed in terms of its local cohomology modules,
$\operatorname{H}^{i}_{\mathfrak{m}}(M)$, with support on $\mathfrak{m}$. Thus
$0pt_{R}M=\operatorname{dim}R$ means that
$\operatorname{H}^{i}_{\mathfrak{m}}(M)$ is zero for $i<\operatorname{dim}R$,
and nonzero for $i=\operatorname{dim}R$. A complex of maximal depth is _big
Cohen-Macaulay_ if $\operatorname{H}^{i}_{\mathfrak{m}}(M)=0$ for
$i>\operatorname{dim}R$ as well. When in addition the $R$-module
$\operatorname{H}(M)$ is finitely generated, $M$ is _maximal Cohen-Macaulay_
(MCM). Thus an MCM module is what we know it to be. These notions are
discussed in detail in Section 2 and 3.

When $R$ is an excellent local domain with residue field of positive
characteristic, $R^{+}$, its integral closure in an algebraic closure of its
field of fractions, is big Cohen-Macaulay. This was proved by Hochster and
Huneke [Hochster/Huneke:1992], see also Huneke and Lyubeznik
[Huneke/Lyubeznik:2007], when $R$ itself contains a field of positive
characteristic. When $R$ has mixed characteristic this is a recent result of
Bhatt [Bhatt:2020]. Thus for such rings there is a canonical construction of a
big Cohen-Macaulay module, even an _algebra_. See also the work of André
[Andre:2020] and Gabber [Gabber:2018] concerning functorial construction of
big Cohen-Macaulay algebras; see also
[Ma/Schwede/Tucker/Waldron/Witaszek:2019, Appendix A]). On the other hand,
$R^{+}$ is never big Cohen-Macaulay when $R$ contains the rationals and is a
normal domain of Krull dimension at least $3$, by a stadard trace argument. As
far as we know, in this context there are no such “simple” models of big
Cohen-Macaulay modules, let alone algebras. See however Schoutens’ work
[Schoutens:2004].

When $R$ is essentially of finite type over a field of characteristic zero,
the derived push-forward of the structure sheaf of a resolution of
singularities of $\operatorname{Spec}R$ is an MCM complex [Roberts:1980]. What
is more, this complex is equivalent to a graded-commutative differential
graded algebra, as we explain in 3.18. This is noteworthy because when such a
ring $R$ is also a normal domain of dimension $\geq 3$ it cannot have any MCM
algebras, by the same trace argument as for $R^{+}$. For any local ring $R$
with a dualizing complex, there are simple constructions of MCM complexes—see
Corollaries 3.8 and 3.11—though we do not know any that are also differential
graded algebras. In [Bhatt:2014] Bhatt gives examples of complete local rings,
containing a field of positive characteristic, that do not have any MCM
algebras.

As to applications, in Section 2 we prove the New Intersection Theorem and its
improved version using complexes of maximal depth, extending the ideas from
[Iyengar:1999] where they are proved using big Cohen-Macaulay modules.
Hochster proved in [Hochster:1983] that the Improved New Intersection Theorem
is equivalent to the Canonical Element Theorem. In Section 3 we use results
from _loc. cit._ to prove that for local rings with dualizing complexes the
Canonical Element Theorem implies the existence of MCM complexes. An
interesting point emerges: replacing “module” with “complex” puts the
existence of big Cohen-Macaulay modules on par with the rest of the
homological conjectures.

In Section 4 we paraphrase Boutot’s proof of his theorem on rational
singularities to highlight the role of MCM complexes. We also give a new proof
of a subadditivity property for multiplier ideals. On the other hand, there
are applications of MCM modules that do require working with modules; see
3.17. Nevertheless, it is clear to us that big Cohen-Macaulay complexes and
MCM complexes have their uses, hence this survey.

##  1\. Local cohomology and derived completions

In this section, we recall basic definitions and results on local cohomology
and derived completions. Throughout $R$ will be a commutative noetherian ring.
By an _$R$ -complex_ we mean a complex of $R$-modules; the grading will be
upper or lower, depending on the context. In case of ambiguity, we indicate
the grading; for example, given an $R$-complex $M$, the supremum of
$\operatorname{H}(M)$ depends on whether the grading is upper or lower. So we
write $\sup\operatorname{H}_{*}(M)$ for the largest integer $i$ such that
$\operatorname{H}_{i}(M)\neq 0$, and $\sup\operatorname{H}^{*}(M)$ for the
corresponding integer for the upper grading.

We write $\operatorname{D}(R)$ for the (full) derived category of $R$ viewed
as a triangulated category with translation $\mathsf{\Sigma}$, the usual
suspension functor on complexes. We take [Lipman:1999, Dwyer/Greenlees:2002]
as basic references, augmented by [Avramov/Foxby:1991, Roberts:1980], except
that we use the term “semi-injective” in place of “q-injective” as in
[Lipman:1999], and “DG-injective”, as in [Avramov/Foxby:1991]. Similarly for
the projective and flat analogs.

###  1.1. Derived $I$-torsion

Let $I$ an ideal in $R$. The _$I$ -power torsion_ subcomplex of an $R$-complex
$M$ is

$\varGamma_{I}{M}\colonequals\\{m\in M\mid I^{n}m=0\text{ for some $n\geq
0$}\\}.$

By $m\in M$ we mean that $m$ is in $M_{i}$ for some $i$. The corresponding
derived functor is denoted $\operatorname{R}\\!\varGamma_{\\!I}(M)$; thus
$\operatorname{R}\\!\varGamma_{\\!I}(M)=\varGamma_{I}{J}$ where
$M\xrightarrow{\sim}J$ is any semi-injective resolution of $M$. In fact, one
can compute these derived functors from any complex of injective $R$-modules
quasi-isomorphic to $M$; see [Lipman:1999, §3.5]. By construction there is a
natural morphism $\operatorname{R}\\!\varGamma_{\\!I}(M)\to M$ in the
$\operatorname{D}(R)$. The $R$-modules

$\operatorname{H}^{i}_{I}(M)\colonequals\operatorname{H}^{i}(\operatorname{R}\\!\varGamma_{\\!I}(M))\quad\text{for
$i\in\mathbb{Z}$}$

are the local cohomology modules of $M$, supported on $I$. Evidently, these
modules are $I$-power torsion. Conversely, when the $R$-module
$\operatorname{H}(M)$ is $I$-power torsion, the natural map
$\operatorname{R}\\!\varGamma_{\\!I}(M)\to M$ is an isomorphism in
$\operatorname{D}(R)$; see [Dwyer/Greenlees:2002, Proposition 6.12], or
[Lipman:1999, Corollary 3.2.1].

In what follows we will use the fact that the class of $I$-power torsion
complexes form a localizing subcategory of $\operatorname{D}(R)$; see
[Dwyer/Greenlees:2002, §6], or [Lipman:1999, §3.5]. This has the consequence
that these complexes are stable under various constructions. For example, this
class of complexes is closed under $L\otimes_{R}^{\operatorname{L}}(-)$ for
any $L$ in $\operatorname{D}(R)$. Thus, for any $R$-complexes $L$ and $M$ the
natural map

(1.1.1)
$\operatorname{R}\\!\varGamma_{\\!I}(L\otimes_{R}^{\operatorname{L}}M)\longrightarrow
L\otimes_{R}^{\operatorname{L}}\operatorname{R}\\!\varGamma_{\\!I}(M)$

is a quasi-isomorphism.

###  1.2. Derived $I$-completion

The $I$-adic completion of an $R$-complex $M$ with respect to the ideal $I$,
denoted $\varLambda^{I}{M}$, is

$\varLambda^{I}{M}\colonequals\lim_{n\geqslant 0}M/I^{n}M\,.$

This complex is thus the limit of the system

$\cdots\longrightarrow M/I^{n+1}M\longrightarrow
M/I^{n}M\longrightarrow\cdots\longrightarrow M/IM\,.$

The canonical surjections $M\to M/I^{n}M$ induce an $R$-linear map
$M\to\varLambda^{I}{M}$. If this is an isomorphism we say that $M$ is _$I$
-adically complete_, or just _$I$ -complete_, though we reserve this name
mainly for modules. The _left-derived completion_ with respect to $I$ of an
$R$-complex $M$ is the $R$-complex

$\operatorname{L}\\!\varLambda^{I}(M)\colonequals\varLambda^{I}{P}\quad\text{where
$P\simeq M$ is a semi-projective resolution.}$

This complex is well-defined in $\operatorname{D}(R)$, and there is a natural
morphism

$M\longrightarrow\operatorname{L}\\!\varLambda^{I}(M)\,.$

We say $M$ is _derived $I$-complete_ if this map is a quasi-isomorphism;
equivalently if each $\operatorname{H}_{i}(X)$ is derived $I$-complete; see
[Dwyer/Greenlees:2002, Proposition 6.15], or [Stacks, Tag091N]

The derived $I$-complete modules from a colocalizing subcategory of
$\operatorname{D}(R)$, and this means that for $N$ in $\operatorname{D}(R)$
the natural map

$\operatorname{L}\\!\varLambda^{I}(\operatorname{RHom}_{R}(N,M))\longrightarrow\operatorname{RHom}_{R}(N,\operatorname{L}\\!\varLambda^{I}(M))$

is a quasi-isomorphism. In particular, when $F$ is a perfect complex, we have
an isomorphim in $\operatorname{D}(R)$

(1.2.1)
$F\otimes_{R}^{\operatorname{L}}\operatorname{L}\\!\varLambda^{I}(M)\simeq\operatorname{L}\\!\varLambda^{I}(F\otimes_{R}^{\operatorname{L}}M).$

These isomorphisms will be useful in what follows. It is a fundamental fact,
proved by Greenlees and May [Greenlees/May:1992a], see also
[Dwyer/Greenlees:2002, Proposition 4.3] or [Lipman:1999, §4], that derived
local cohomology and derived completions are adjoint functors:

(1.2.2)
$\operatorname{RHom}_{R}(\operatorname{R}\\!\varGamma_{\\!I}(M),N)\simeq\operatorname{RHom}_{R}(M,\operatorname{L}\\!\varLambda^{I}(N))\,.$

One can take this as a starting point for defining derived completions, which
works better in the non-noetherian settings; see [Stacks]. This adjunction
implies that the natural maps are quasi-isomorphisms:

(1.2.3)
$\operatorname{L}\\!\varLambda^{I}(\operatorname{R}\\!\varGamma_{\\!I}(M))\xrightarrow{\
\simeq\
}\operatorname{L}\\!\varLambda^{I}(M)\quad\text{and}\quad\operatorname{R}\\!\varGamma_{\\!I}(M)\xrightarrow{\
\simeq\
}\operatorname{R}\\!\varGamma_{\\!I}(\operatorname{L}\\!\varLambda^{I}(M))\,.$

The result below, due to A.-M. Simon [Simon:1990, 1.4], is a version of
Nakayama’s Lemma for cohomology of complete modules. It is clear from the
proof that we only need $X$ to be derived $I$-complete; see [Stacks, Tag09b9].

######  Lemma 1.3.

For any $R$-complex $X$ consisting of $I$-complete modules, and integer $i$,
if $I\operatorname{H}_{i}(X)=\operatorname{H}_{i}(X)$, then
$\operatorname{H}_{i}(X)=0$.

###### Proof.

The point is that $Z_{i}$, the module of cycle in degree $i$, is a closed
submodule of the $I$-complete module $X_{i}$, and hence is also $I$-complete.
Moreover $\operatorname{H}_{i}(X)$ is the cokernel of the map $X_{i+1}\to
Z_{i}$, and a map between $I$-complete modules is zero if and only if its
$I$-adic completion is zero. This translates to the desired result. ∎

###  1.4. Koszul complexes

Given a sequence of elements $\underline{r}\colonequals r_{1},\dots,r_{n}$ in
the ring $R$, and an $R$-complex $M$, we write $K(\underline{r};M)$ for the
Koszul complex on $\underline{r}$ with coefficients in $M$, namely

$K(\underline{r};M)\colonequals K(\underline{r};R)\otimes_{R}M\,.$

Its homology is denoted $\operatorname{H}_{*}(\underline{r};M)$. For a single
element $r\in R$, the complex $K(r;M)$ can be constructed as the mapping cone
of the homothety map $M\xrightarrow{r}M$. In particular, one has an exact
sequence

(1.4.1) $0\longrightarrow M\longrightarrow
K(r;M)\longrightarrow\mathsf{\Sigma}M\longrightarrow 0$

of $R$-complexes. The Koszul complex on a sequence can thus be constructed as
an iterated mapping cone. From Lemma 1.3 one gets the result below. Recall
that $\sup\operatorname{H}_{*}(-)$ denotes the supremum, in lower grading.

######  Lemma 1.5.

Let $R$ be a noetherian ring and $X$ a derived $I$-complete $R$-complex. For
any sequence $\underline{r}\colonequals r_{1},\dots,r_{n}$ in $I$ one has

$\sup\operatorname{H}_{*}(\underline{r};X)\geq\sup\operatorname{H}_{*}(X)\,.$

###### Proof.

When $X$ is derived $I$-complete so is $K(r;X)$ for any $r\in I$. It thus
suffices to verify the desired claim for $n=1$. Replacing $X$ by
$\varLambda^{I}{P}$, where $P$ is a semi-projective resolution of $X$, we can
assume $X$, and hence also $K(r;X)$, consists of $I$-complete modules. The
desired inequality is then immediate from the standard long exact sequence in
homology

$\cdots\longrightarrow\operatorname{H}_{i}(X)\xrightarrow{\ r\
}\operatorname{H}_{i}(X)\longrightarrow\operatorname{H}_{i}(r,X)\longrightarrow\operatorname{H}_{i-1}(X)\longrightarrow\cdots$

arising from the mapping cone sequence 1.4.1 and Lemma 1.3. ∎

To wrap up this section we recall the notion of depth for complexes.

###  1.6. Depth

The _$I$ -depth_ of an $R$-complex $M$ is

$0pt_{R}(I,M)\colonequals\inf\\{i\mid\operatorname{H}^{i}_{I}(M)\neq 0\\}.$

In particular, $0pt_{R}(I,M)=\infty$ if $\operatorname{H}_{I}(M)=0$. When the
ring $R$ is local, with maximal ideal $\mathfrak{m}$, the _depth_ of $M$
refers to the $\mathfrak{m}$-depth of $M$.

Depth can also be computed using Ext and Koszul homology:

$0pt_{R}(I,M)=\inf\\{i\mid\operatorname{Ext}^{i}_{R}(R/I,M)\neq 0\\}\,,$

and if a sequence $\underline{r}\colonequals r_{1},\dots,r_{n}$ generates $I$,
then

$0pt_{R}(I,M)=n-\sup\\{i\mid\operatorname{H}_{i}(\underline{r};M)\neq 0\\}$

This last equality can be expressed in terms of Koszul cohomology. All these
results are from [Foxby/Iyengar:2001], though special cases (for example, when
$M$ is an $R$-module) had been known for much longer.

######  Remark 1.7.

Let $R$ be a commutative ring, $I$ an ideal in $R$, and $M$ an $R$-complex.
Set $s=\sup\operatorname{H}_{*}(M)$.

  1. (1)

$0pt_{R}(I,M)\geq-s$ and equality holds if
$\varGamma_{I}{(\operatorname{H}_{s}(M))}\neq 0$.

  2. (2)

When $R$ is local and $F$ is a finite free complex, one has

$0pt_{R}(F\otimes_{R}M)=0pt_{R}M-\operatorname{proj\,dim}_{R}F$

For part (1) see [Foxby/Iyengar:2001, 2.7]. When $F$ is the resolution of a
module and $M=R$, part (2) is nothing but the equality of Auslander and
Buchsbaum. For a proof in the general case see, for example,
[Foxby/Iyengar:2001, Theorem 2.4].

##  2\. Complexes of maximal depth and the intersection theorems

In this section we introduce a notion of “maximal depth” for complexes over
local rings. The gist of the results presented here is that their existence
implies the Improved New Intersection Theorem, and hence a whole slew of
“homological conjectures”, most of which have been recently settled by André
[Andre:2018].

A _module_ of maximal depth is nothing but a big Cohen-Macaulay module and
Hochster proved, already in [Hochster:1975], that their existence implies the
homological conjectures mentioned above. On the other hand, the Canonical
Element Conjecture, now theorem, implies that $R$ has a _complex_ of maximal
depth, even one with finitely generated homology. This will be one of the
outcomes of the discussion in the next section; see Remark 3.15. No such
conclusion can be drawn about big Cohen-Macaulay modules.

###  2.1. Complexes of maximal depth

Throughout $(R,\mathfrak{m},k)$ will be a local ring, with maximal ideal
$\mathfrak{m}$ and residue field $k$. We say that an $R$-complex $M$ has
_maximal depth_ if the following conditions hold:

  1. (1)

$\operatorname{H}(M)$ is bounded;

  2. (2)

$\operatorname{H}_{0}(M)\to\operatorname{H}_{0}(k\otimes_{R}^{\operatorname{L}}M)$
is nonzero; and

  3. (3)

$0pt_{R}M=\operatorname{dim}R$.

The nomenclature is based on that fact that $0pt_{R}M\leq\operatorname{dim}R$
for any complex $M$ that satisfies condition (2) above. This inequality
follows from Lemma 2.2 applied with $F\colonequals K$, the Koszul complex on a
system of parameters for $R$. Condition (3) can be restated as

(2.1.1) $\operatorname{H}^{i}_{\mathfrak{m}}(M)=0\quad\text{for
$i<\operatorname{dim}R$}\quad\text{and}\quad\operatorname{H}^{\operatorname{dim}R}_{\mathfrak{m}}(M)\neq
0\,.$

Clearly when $M$ is a module it has maximal depth precisely when it is big
Cohen-Macaulay; condition (2) says that $M\neq\mathfrak{m}M$. Note also that
if a complex $M$ has maximal depth then so does $M\oplus\mathsf{\Sigma}^{-n}N$
for any $R$-module $N$ and integer $n\geq\operatorname{dim}R$.

######  Lemma 2.2.

Let $M$ be an $R$-complex with the natural map
$\operatorname{H}_{0}(M)\to\operatorname{H}_{0}(k\otimes_{R}^{\operatorname{L}}M)$
nonzero. For any $R$-complex $F$ with $\operatorname{H}_{i}(F)=0$ for $i<0$,
if $\operatorname{H}_{0}(F)\otimes_{R}k$ is nonzero, then so is
$\operatorname{H}_{0}(F\otimes_{R}^{\operatorname{L}}M)$.

###### Proof.

We can assume $M$ is semi-projective, so the functor
$-\otimes_{R}^{\operatorname{L}}M$ is represented by $-\otimes_{R}M$. By
hypothesis there exists a cycle, say $z$, in $M_{0}$ whose image in
$k\otimes_{R}M=M/\mathfrak{m}M$ is not a boundary. Consider the morphism $R\to
M$ of $R$-complexes, where $r\mapsto rz$. Its composition $R\to M\to
k\otimes_{R}M$ factors through the canonical surjection $R\to k$, yielding the
commutative square

${R}$${M}$${k}$${k\otimes_{R}M.}$

The dotted arrow is a left-inverse in $\operatorname{D}(R)$ of the induced
$k\to k\otimes_{R}M$. It exists because $k\to\operatorname{H}(k\otimes_{R}M)$
is nonzero, by the choice of $z$, and the complex $k\otimes_{R}M$ is quasi-
isomorphic to $\operatorname{H}(k\otimes_{R}M)$ in $\operatorname{D}(k)$, and
hence in $\operatorname{D}(R)$. Applying $F\otimes_{R}^{\operatorname{L}}-$ to
the diagram above yields the commutative square in $\operatorname{D}(R)$ on
the left:

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The commutative square on the right is obtained by applying
$\operatorname{H}_{0}(-)$ to the one on the left. In this square, the
hypotheses on $F$ imply that the vertical map on the left is nonzero, so hence
is its composition with the horizontal arrow. The commutativity of the square
then yields that $\operatorname{H}_{0}(F\otimes_{R}^{\operatorname{L}}M)$ is
nonzero. ∎

The following result is due to Hochster and Huneke for rings containing a
field, and due to André in the mixed characteristic case.

######  Theorem 2.3 (André [Andre:2018], Hochster & Huneke
[Hochster/Huneke:1992, Hochster/Huneke:1995]).

Each noetherian local ring possesses a big Cohen-Macaulay algebra. ∎

As has been said before, the existence of big Cohen-Macaulay algebras, and
hence big Cohen-Macaulay modules, implies many of the homological conjectures.
In particular, it can be used to give a quick proof of the New Intersection
Theorem, first proved in full generality by P. Roberts [Roberts:1987] using
intersection theory; see also [Piepmeyer/Walker:2009]. Here is a proof that
uses only the existence of a _complexes_ of maximal depth; the point being
that they are easier to construct than big Cohen-Macaulay modules. Our
argument is modeled on that of [Iyengar:1999, Theorem 2.5], which uses big
Cohen-Macaulay modules.

######  Theorem 2.4.

Let $R$ be a local ring. Any finite free $R$-complex

$F\colonequals 0\to F_{n}\to\cdots\to F_{0}\to 0$

with $\operatorname{H}_{0}(F)\neq 0$ and
$\operatorname{length}_{R}\operatorname{H}_{i}(F)$ finite for each $i$
satisfies $n\geq\operatorname{dim}R$.

###### Proof.

Let $M$ be an $R$-complex of maximal depth. As $\operatorname{H}(F)$ is of
finite length, the $R$-module $\operatorname{H}(F\otimes_{R}M)$ is
$\mathfrak{m}$-power torsion, so 1.7(1) yields the second equality:

$\displaystyle\operatorname{proj\,dim}_{R}F$
$\displaystyle=0pt_{R}M-0pt_{R}(F\otimes_{R}M)$
$\displaystyle=0pt_{R}M+\sup\operatorname{H}_{*}(F\otimes_{R}M)$
$\displaystyle\geq 0pt_{R}M$ $\displaystyle=\operatorname{dim}R$

The first one is by 1.7(2). The inequality is by Lemma 2.2, noting that
$\operatorname{H}_{0}(F)\otimes_{R}k$ is nonzero by Nakayama’s lemma. ∎

One can deduce also the Improved New Intersection Theorem 2.7 from the
existence of complexes of maximal depth, but the proof takes some more
preparation.

######  Lemma 2.5.

Let $R$ be a local ring and $M$ an $R$-complex. If $M$ has maximal depth, then
so does $\operatorname{L}\\!\varLambda^{I}(M)$ for any ideal $I\subset R$.

###### Proof.

Condition (1) for maximal depth holds because $\operatorname{H}(M)$ bounded
implies $\operatorname{H}(\operatorname{L}\\!\varLambda^{I}(M))$ is bounded;
this follows, for example, from (1.2.2) and the observation
$\operatorname{R}\\!\varGamma_{\\!I}(R)$ has finite projective dimension. As
to the other conditions, the main point is that for any $R$-complex $X$ such
that $\operatorname{H}(X)$ is $I$-power torsion, the canonical map
$M\to\operatorname{L}\\!\varLambda^{I}(M)$ in $\operatorname{D}(R)$ induces a
quasi-isomorphism

$X\otimes_{R}^{\operatorname{L}}M\xrightarrow{\ \simeq\
}X\otimes_{R}^{\operatorname{L}}\operatorname{L}\\!\varLambda^{I}(M).$

This can be deduced from (1.2.1) and (1.2.3). In particular, taking
$X=\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(R)$, where $\mathfrak{m}$ is
the maximal ideal of $R$, yields

$\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(M)\simeq\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(\operatorname{L}\\!\varLambda^{I}(M))\,,$

so that $0pt_{R}M=0pt_{R}\operatorname{L}\\!\varLambda^{I}(M)$. Moreover,
taking $X=k$ gives the isomorphism in the following commutative diagram in
$\operatorname{D}(R)$:

${M}$${k\otimes_{R}^{\operatorname{L}}M}$${\operatorname{L}\\!\varLambda^{I}(M)}$${k\otimes_{R}^{\operatorname{L}}\operatorname{L}\\!\varLambda^{I}(M)}$$\scriptstyle{\simeq}$

that is induced by the morphism $M\to\operatorname{L}\\!\varLambda^{I}(M)$.
Since $M$ has maximal depth, the map in the top row is nonzero when we apply
$\operatorname{H}_{0}(-)$, and so the same holds for the map in the bottom
row. Thus $\operatorname{L}\\!\varLambda^{I}(M)$ has maximal depth. ∎

######  Lemma 2.6.

Let $(R,\mathfrak{m},k)$ is a local ring and $M$ a derived
$\mathfrak{m}$-complete $R$-complex of maximal depth. Set
$d\colonequals\operatorname{dim}R$. The following statements hold:

  1. (1)

For any system of parameters $r_{1},\dots,r_{d}$ for $R$, one has

$0pt_{R}(K(r_{1},\dots,r_{n};M))=n\quad\text{for each $1\leq n\leq d$.}$

In words, the depth of $M$ with respect to the ideal $(r_{1},\dots,r_{n})$ is
$n$.

  2. (2)

For any $\mathfrak{p}\in\operatorname{Spec}R$ one has

$0pt_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\geq\operatorname{dim}R_{\mathfrak{p}}\,,$

and equality holds when the map
$\operatorname{H}_{0}(M)\to\operatorname{H}_{0}(k(\mathfrak{p})\otimes_{R}^{\operatorname{L}}M)$
is nonzero, in which case the $R_{\mathfrak{p}}$-complex $M_{\mathfrak{p}}$
has maximal depth.

###### Proof.

(1) Set $\underline{r}=r_{1},\dots,r_{d}$. The hypothesis that $M$ has maximal
depth and the depth sensitivity of the Koszul complex $K(\underline{r};R)$
yield $\operatorname{H}_{i}(\underline{r};M)=0$ for $i\geq 1$. One has an
isomorphism of $R$-complexes

$K(\underline{r};M)\cong K(r_{n+1},\dots,r_{d};K(r_{1},\dots,r_{n};M))\,.$

Since $M$ is derived complete with respect to $\mathfrak{m}$, it follows from
Lemma 1.5, applied to the sequence $r_{n+1},\dots,r_{d}$ and $X\colonequals
K(r_{1},\dots,r_{n};M)$, that

$\operatorname{H}_{i}(K(r_{1},\dots,r_{n};M))=0\quad\text{for $i\geq 1$.}$

On the other hand, since the natural map
$\operatorname{H}_{0}(M)\to\operatorname{H}_{0}(k\otimes_{R}^{\operatorname{L}}M)$
is nonzero, Lemma 2.2 applied with $F=K(r_{1},\dots,r_{n};R)$, yields

$\operatorname{H}_{0}(K(r_{1},\dots,r_{n};M))\neq 0\,.$

Thus the depth sensitivity of $K(r_{1},\dots,r_{n};M)$ yields the equality in
(1).

(2) Set $h\colonequals 0pt\mathfrak{p}$ and choose a system of parameters
$\underline{r}\colonequals r_{1},\dots,r_{d}$ for $R$ such that the elements
$r_{1},\dots,r_{h}$ are in $\mathfrak{p}$. One has

$0pt_{R_{\mathfrak{p}}}\\!M_{\mathfrak{p}}\geq
0pt_{R}((r_{1},\dots,r_{h}),M)\geq h\,.$

where the first inequality is clear and the second one holds by (1). The
natural map $M\to k(\mathfrak{p})\otimes_{R}^{\operatorname{L}}M$ factors
through $M_{\mathfrak{p}}$, so under the additional hypothesis Lemma 2.2
implies $0pt_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\leq h$. We conclude that
$M_{\mathfrak{p}}$ has maximal depth. ∎

Given the preceding result, we argue as in the proof of [Iyengar:1999, Theorem
3.1] to deduce the Improved New Intersection Theorem:

######  Theorem 2.7.

Let $R$ be a noetherian local ring and $F\colonequals 0\to F_{n}\to\cdots\to
F_{0}\to 0$ a finite free $R$-complex with $\operatorname{H}_{0}(F)\neq 0$ and
$\operatorname{length}\operatorname{H}_{i}(F)$ finite for each $i\geq 1$. If
an ideal $I$ annihilates a minimal generator of $\operatorname{H}_{0}(F)$,
then $n\geq\operatorname{dim}R-\operatorname{dim}(R/I)$.

###### Proof.

Let $M$ be an $R$-complex of maximal depth. By Lemma 2.5, we can assume $M$ is
derived $\mathfrak{m}$-complete, so Lemma 2.6 applies. Set
$s\colonequals\sup\operatorname{H}_{*}(F\otimes_{R}M)$ and note that $s\geq
0$, by Lemma 2.2.

Fix $\mathfrak{p}$ in
$\operatorname{Ass}_{R}\operatorname{H}_{s}(F\otimes_{R}M)$, so that
$0pt_{R_{\mathfrak{p}}}\operatorname{H}_{s}(F\otimes_{R}M)_{\mathfrak{p}}=0$.
The choice of $\mathfrak{p}$ implies that
$\operatorname{H}(F\otimes_{R}M)_{\mathfrak{p}}$ is nonzero, and hence
$\operatorname{H}(F)_{\mathfrak{p}}$ and $\operatorname{H}(M)_{\mathfrak{p}}$
are nonzero as well. Therefore one gets

(2.7.1)
$\displaystyle\operatorname{proj\,dim}_{R_{\mathfrak{p}}}F_{\mathfrak{p}}$
$\displaystyle=0pt_{R_{\mathfrak{p}}}\\!M_{\mathfrak{p}}-0pt_{R_{\mathfrak{p}}}(F\otimes_{R}M)_{\mathfrak{p}}$
$\displaystyle=0pt_{R_{\mathfrak{p}}}\\!M_{\mathfrak{p}}+s$
$\displaystyle\geq\operatorname{dim}R_{\mathfrak{p}}+s$

The equalities are by 1.7 and the inequality is by Lemma 2.6(2).

Suppose $s\geq 1$. We claim that $\mathfrak{p}=\mathfrak{m}$, the maximal
ideal of $R$, so (2.7.1) yields

$\operatorname{proj\,dim}_{R}F\geq\operatorname{dim}R\,,$

which implies the desired inequality.

Indeed if $\mathfrak{p}\neq\mathfrak{m}$, then since
$\operatorname{length}_{R}\operatorname{H}_{i}(F)$ is finite for $i\geq 1$,
one gets that
$F_{\mathfrak{p}}\simeq{\operatorname{H}_{0}(F)}_{\mathfrak{p}}$, which
justifies the equality below:

$0ptR_{\mathfrak{p}}\geq\operatorname{proj\,dim}_{R_{\mathfrak{p}}}{\operatorname{H}_{0}(F)}_{\mathfrak{p}}=\operatorname{proj\,dim}_{R_{\mathfrak{p}}}F_{\mathfrak{p}}\geq\operatorname{dim}R_{\mathfrak{p}}+s$

The first inequality is a consequence of the Auslander-Buchsbaum equality
1.7(2), the second one is from (2.7.1). We have arrived at a contradiction for
$s\geq 1$.

It remains to consider the case $s=0$. Set $X\colonequals F\otimes_{R}M$.
Since $\operatorname{H}_{0}(F)$ is finitely generated, Nakayama’s Lemma and
Lemma 2.2 imply that each minimal generator of $\operatorname{H}_{0}(F)$ gives
a nonzero element in $\operatorname{H}_{0}(X)$. One of these is thus
annihilated by $I$, by the hypotheses. Said otherwise,
$\varGamma_{I}{\operatorname{H}_{0}(F)}\neq 0$. Since
$\sup\operatorname{H}_{*}(X)=0$, this implies $0pt_{R}(I,X)=0$, by Remark 1.7,
and hence one gets the equality below

$0pt_{R}X\leq 0pt_{R}(I,X)+\operatorname{dim}(R/I)=\operatorname{dim}(R/I)$

The inequality can be verified by arguing as in the proof of [Iyengar:1999,
Proposition 5.5(4)]: Let $\underline{a}\colonequals a_{1},\dots,a_{l}$ be a
set of generators for the ideal $I$, and let $\underline{b}\colonequals
b_{1},\dots,b_{n}$ be elements in $R$ whose residue classes in $R/I$ form a
system of parameters. Since $M$ is derived $\mathfrak{m}$-complete, so is $X$
and hence also $K(\underline{a};X)$. Then Lemma 1.5 applied to the sequence
$\underline{b}$ and complex $K(\underline{a};X)$ yields

$\sup\operatorname{H}_{*}(\underline{a},\underline{b};X)\geq\sup\operatorname{H}_{*}(\underline{a};X)\,;$

this gives the desired inequality. Finally it remains to invoke the Auslander-
Buchsbaum equality once again to get

$\operatorname{proj\,dim}_{R}F=0pt_{R}M-0pt_{R}X\geq\operatorname{dim}R-\operatorname{dim}(R/I)\,.$

This completes the proof. ∎

##  3\. MCM complexes

In this section we introduce two strengthenings of the notion of complexes of
maximal depth, and discuss various constructions that yield such complexes. As
before let $(R,\mathfrak{m},k)$ be a local ring, of Krull dimension $d$.

###  3.1. Big Cohen-Macaulay complexes

We say that an $R$-complex $M$ is _big Cohen-Macaulay_ if the following
conditions hold:

  1. (1)

$\operatorname{H}(M)$ is bounded;

  2. (2)

$\operatorname{H}^{0}(M)\to\operatorname{H}^{0}(k\otimes_{R}^{\operatorname{L}}M)$
is nonzero.

  3. (3)

$\operatorname{H}^{i}_{\mathfrak{m}}(M)=0$ for $i\neq\operatorname{dim}R$;

If in addition $\operatorname{H}(M)$ is finitely generated, $M$ is _maximal
Cohen-Macaulay_ ; usually abbreviated to MCM. Condition (2) implies in
particular that $\operatorname{H}^{0}(k\otimes_{R}^{\operatorname{L}}M)$ is
nonzero, and from this it follows that
$\operatorname{H}^{i}_{\mathfrak{m}}(M)\neq 0$ for some $i$. Thus condition
(3) implies $0pt_{R}M=\operatorname{dim}R$; in particular, a big Cohen-
Macaulay complex has maximal depth, in the sense of 2.1 and
$\operatorname{H}^{\operatorname{dim}R}_{\mathfrak{m}}(M)\neq 0$. However (3)
is more restrictive, as the following observation shows.

######  Lemma 3.2.

If $M$ is an MCM $R$-complex, then $\operatorname{H}^{i}(M)=0$ for
$i\not\in[0,\operatorname{dim}R]$; moreover, $\operatorname{H}^{0}(M)\neq 0$.

###### Proof.

The last part of the statement is immediate from condition (2).

Set $d=\operatorname{dim}R$. Let $K$ be the Koszul complex on a system of
parameters for $R$. Then one has isomorphisms

$K\otimes_{R}M\simeq
K\otimes_{R}^{\operatorname{L}}\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(M)\simeq
K\otimes_{R}^{\operatorname{L}}\mathsf{\Sigma}^{-d}\operatorname{H}^{d}_{\mathfrak{m}}(M)$

where the first one is from (1.1.1), since $K\otimes_{R}M$ is
$\mathfrak{m}$-power torsion, and the second isomorphism holds by the defining
property (3) of a big Cohen-Macaulay complex. Hence

$\inf\operatorname{H}^{*}(K\otimes_{R}M)\geq
0\quad\text{and}\quad\sup\operatorname{H}^{*}(K\otimes_{R}M)\leq d\,.$

By our hypotheses, the $R$-module $\operatorname{H}^{i}(M)$ is finitely
generated for each $i$, and since $K$ is a Koszul complex on $d$ elements, a
standard argument leads to the desired vanishing of $\operatorname{H}^{i}(M)$.
∎

Any nonzero MCM $R$-module is MCM when viewed as complex. However, even over
Cohen-Macaulay rings, which are not fields, there are MCM complexes that are
not modules; see the discussion in (2.1). In the rest of this section we
discuss various ways MCM complexes can arise, or can be expected to arise. It
turns out that often condition (2) is the one that is hardest to verify. Here
is one case when this poses no problem; see 3.18 for an application. The main
case of interest is where $A$ is a dg (=differential graded) $R$-algebra.

######  Lemma 3.3.

Let $A$ be an $R$-complex with a unital (but not necessarily associative)
multiplication rule such that the Leibniz rule holds and
$i\colonequals\inf\operatorname{H}_{*}(A)$ is finite. If
$\operatorname{H}_{i}(A)$ is finitely generated, then the identity element of
$A$ is nonzero in $\operatorname{H}_{0}(A\otimes_{R}^{\operatorname{L}}k)$.

###### Proof.

One has
$\operatorname{H}_{i}(A\otimes_{R}^{\operatorname{L}}k)\cong\operatorname{H}_{i}(A)\otimes_{R}k$
and the latter module is nonzero, by Nakayama’s lemma and the finite
generation hypothesis. We have
$A\otimes_{R}^{\operatorname{L}}k=A\otimes_{R}T$ where $T$ is a Tate
resolution of $k$; see [Tate:1957]. So $A\otimes_{R}T$ is also a (possibly
non-associative) dg algebra. Thus if the identity element were trivial in
$\operatorname{H}(A\otimes_{R}T)$, then $\operatorname{H}(A\otimes_{R}T)=0$
holds, contradicting
$\operatorname{H}_{i}(A\otimes_{R}^{\operatorname{L}}k)\neq 0$. ∎

The MCM property for complexes has a simple interpretation in terms of their
duals with respect to dualizing complexes.

###  3.4. Dualizing complexes

Let $D$ be a dualizing complex for $R$, normalized111In [Hartshorne:1966,
Roberts:1980], a dualizing complex is normalized to be nonzero in $[-d,0]$. so
$D^{i}$ is nonzero only in the range $[0,d]$, where
$d\colonequals\operatorname{dim}R$ and always with nonzero cohomology in
degree 0. Thus $D$ is an $R$-complex with $\operatorname{H}(D)$ finitely
generated, and
$\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(D)\simeq\Sigma^{-d}E$, where
$E$ is the injective hull of $k$; see [Roberts:1980, Chapter 2, §3]. For any
$R$-complex $M$ set

${M}^{\dagger}\colonequals\operatorname{RHom}_{R}(M,D)\,.$

One version of the local duality theorem is that the functor
$M\mapsto{M}^{\dagger}$ is a contravariant equivalence when restricted to
$\mathrm{D^{b}}(\mathrm{mod}\,R)$, the bounded derived category of finitely
generated $R$-modules; see [Roberts:1980, Chapter 2, Theorem 3.5]. For $M$ in
this subcategory, this gives the last of following quasi-isomorphisms:

$\displaystyle\operatorname{RHom}_{R}({M}^{\dagger},E)$
$\displaystyle=\operatorname{RHom}_{R}(\operatorname{RHom}_{R}(M,D),E)$
$\displaystyle\simeq\operatorname{RHom}_{R}(\operatorname{RHom}_{R}(M,D),\Sigma^{d}\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(D))$
$\displaystyle\simeq\Sigma^{d}\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(\operatorname{RHom}_{R}(\operatorname{RHom}_{R}(M,D),D))$
$\displaystyle\simeq\Sigma^{d}\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(M)$

The rest are standard. Passing to cohomology yields the usual local duality:

(3.4.1)
$\operatorname{Hom}_{R}(\operatorname{H}^{i}({M}^{\dagger}),E)\cong\operatorname{H}^{d-i}_{\mathfrak{m}}(M)\quad\text{for
each $i$.}$

When $R$ is $\mathfrak{m}$-adically complete, one can apply Matlis duality to
express $\operatorname{H}^{i}({M}^{\dagger})$ as a dual of
$\operatorname{H}^{d-i}_{\mathfrak{m}}(M)$.

We also need to introduce a class of maps that will play an important role in
the sequel: For any $R$-module $N$ let $\zeta^{i}_{N}$ denote the composition
of maps

(3.4.2) $\operatorname{Ext}^{i}_{R}(k,N)\xrightarrow{\ \cong\
}\operatorname{Ext}^{i}_{R}(k,\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(N))\longrightarrow\operatorname{Ext}^{i}_{R}(R,\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(N))\cong\operatorname{H}^{i}_{\mathfrak{m}}(N)$

where the one in the middle is induced by the surjection $R\to k$. We will be
particular interested in $\zeta^{d}_{N}$. If this map is nonzero, then
$\operatorname{dim}_{R}N=\operatorname{dim}R$, but the converse does not hold.

######  Proposition 3.5.

With $D$ as above and $M$ an $R$-complex with $\operatorname{H}(M)$ finitely
generated, set $N\colonequals\operatorname{H}^{0}({M}^{\dagger})$. Then $M$ is
MCM if and only if ${M}^{\dagger}\simeq N$ and the map $\zeta^{d}_{N}$ is
nonzero, for $d=\operatorname{dim}R$.

###### Proof.

Given the hypothesis on the local cohomology on $M$, it follows that
$\operatorname{H}^{i}({M}^{\dagger})$ is nonzero for $i\neq 0$ and hence
${{M}^{\dagger}}\simeq N$. Moreover, this quasi-isomorphism yields

$\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(N)\simeq\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(\operatorname{RHom}_{R}(M,D))\simeq\operatorname{RHom}_{R}(M,\operatorname{R}\\!\varGamma_{\\!\mathfrak{m}}(D))\simeq\Sigma^{-d}\operatorname{Hom}_{R}(M,E)\,.$

Therefore the map (3.4.2) is induced by (to be precise, the degree $0$
component of the map in cohomology induced by) the map

$\operatorname{RHom}_{R}(k,\operatorname{Hom}_{R}(M,E))\longrightarrow\operatorname{Hom}_{R}(M,E)$

By adjunction, the map above is

$\operatorname{RHom}_{R}(k\otimes_{R}^{\operatorname{L}}M,E)\longrightarrow\operatorname{Hom}_{R}(M,E)$

That is to say, (3.4.2) is the Matlis dual of the map
$\operatorname{H}^{0}(M)\to\operatorname{H}^{0}(k\otimes_{R}^{\operatorname{L}}M)$.
This justifies the claims.

Clearly, these steps are reversible: if $N$ is a finitely generated $R$-module
such that the map (3.4.2) is nonzero, the $R$-complex
$\operatorname{RHom}_{R}(N,D)$ is MCM. ∎

Here then is a way (and the only way) to construct MCM complexes when $R$ has
a dualizing complex: Take a finitely generated $R$-module $N$ for which
$\zeta^{d}_{N}$ is nonzero; then the complex $\operatorname{RHom}_{R}(N,D)$ is
MCM. It thus becomes important to understand the class of finitely generated
$R$-modules for which the map $\zeta^{d}_{N}$ is nonzero.

To that end let $F$ be a minimal free resolution of $k$, and set

$\Omega\colonequals\operatorname{Coker}(F_{d+1}\to F_{d})\,;$

this is the $d$th syzygy module of $k$. Since minimal free resolutions are
isomorphic as complexes, this $\Omega$ is independent of the choice of
resolution, up to an isomorphism. The canonical surjection $F\to F_{\geqslant
d}$ gives a morphism in $\operatorname{D}(R)$:

(3.5.1) $\varepsilon\colon k\longrightarrow\mathsf{\Sigma}^{d}\Omega\,.$

We view it as an element in $\operatorname{Ext}^{d}_{R}(k,\Omega)$. The map
$\zeta^{d}_{\Omega}$ below is from (3.4.2).

######  Lemma 3.6.

One has $\zeta^{d}_{\Omega}(\varepsilon)=0$ if and only if
$\zeta^{d}_{\Omega}=0$ if and only if $\zeta^{d}_{N}=0$ for all $R$-modules
$N$.

###### Proof.

Fix an $R$-module $N$. Any map $f$ in $\operatorname{Hom}_{R}(\Omega,N)$
induces a map

$f_{*}\colon\operatorname{Ext}^{d}_{R}(k,\Omega)\longrightarrow\operatorname{Ext}^{d}_{R}(k,N)\,.$

Let $F$ be a resolution of $k$ as above, defining $\Omega$. Any map
$k\to\mathsf{\Sigma}^{d}N$ in $\operatorname{D}(R)$ is represented by a
morphism of complexes $F\to\mathsf{\Sigma}^{d}N$, and hence factors through
the surjection $F\to F_{\geqslant d}$, that is to say, the morphism
$\varepsilon$. We deduce that any element of $\operatorname{Ext}^{d}_{R}(k,N)$
is of the form $f_{*}(\varepsilon)$, for some $f$ in
$\operatorname{Hom}_{R}(\Omega,N)$.

In particular, $\operatorname{Ext}^{d}_{R}(k,\Omega)$ is a generated by
$\varepsilon$ as a left module over $\operatorname{End}_{R}(\Omega)$. This
observation, and the linearity of the $\zeta^{d}_{\Omega}$ with respect to
$\operatorname{End}_{R}(\Omega)$, yields $\zeta^{d}_{\Omega}=0$ if and only if
$\zeta^{d}_{\Omega}(\varepsilon)=0$. Also each $f$ in
$\operatorname{Hom}_{R}(\Omega,N)$ induces a commutative square

${\operatorname{Ext}^{d}_{R}(k,\Omega)}$${\operatorname{H}^{d}_{\mathfrak{m}}(\Omega)}$${\operatorname{Ext}^{d}_{R}(k,N)}$${\operatorname{H}^{d}_{\mathfrak{m}}(N)}$$\scriptstyle{f_{*}}$$\scriptstyle{\zeta^{d}_{\Omega}}$$\scriptstyle{\operatorname{H}^{d}_{\mathfrak{m}}(f)}$$\scriptstyle{\zeta^{d}_{N}}$

Thus if $\zeta^{d}_{\Omega}=0$ we deduce that
$\zeta^{d}_{N}(f_{*}{\varepsilon})=0$. By varying $f$ we conclude from the
discussion above that $\zeta^{d}_{N}=0$. ∎

We should record the following result immediately. It is one formulation of
the Canonical Element Theorem; see [Hochster:1983, (3.15)]. The “canonical
element” in question is $\zeta^{d}_{\Omega}(\varepsilon)$; see Lemma 3.6.

######  Theorem 3.7.

For any noetherian local ring $R$, one has $\zeta^{d}_{\Omega}\neq 0$. ∎

Here then is first construction of an MCM $R$-complex.

######  Corollary 3.8.

If $R$ has a dualizing complex the $R$-complex ${\Omega}^{\dagger}$ is MCM.∎

######  Remark 3.9.

Suppose $R$ has a dualizing complex. Given Proposition 3.5 and Lemma 3.6 it
follows that ${\Omega}^{\dagger}$ is MCM if and only there exists _some_
$R$-complex $M$ that is MCM. Therefore, the Canonical Element Theorem, in all
its various formulations [Hochster:1983], is equivalent to the statement that
$R$ has an MCM R-complex!

We now describe another way to construct an MCM complex. Let $D$ be a
dualizing complex for $R$ and set
$\omega_{R}\colonequals\operatorname{H}^{0}(D)$; this is the _canonical
module_ of $R$.

######  Lemma 3.10.

One has $\zeta^{d}_{\Omega}\neq 0$ if and only if $\zeta^{d}_{\omega_{R}}\neq
0$.

###### Proof.

We write $\omega$ for $\omega_{R}$. Given Lemma 3.6 we have to verify that if
$\zeta^{d}_{\Omega}\neq 0$, then $\zeta^{d}_{\omega}\neq 0$. Let $E$ be an
injective hull of $k$, the residue field of $R$. Since this is a faithful
injective, there exists a map
$\alpha\colon\operatorname{H}^{d}_{\mathfrak{m}}(\Omega)\to E$ such that
$\alpha\circ\zeta^{d}_{\Omega}\neq 0$.

It follows from local duality 3.4.1, applied to $M=\Omega$, that $\alpha$ is
induced by a morphism $f\colon\Omega\to D$; equivalently, an $R$-linear map
$f\colon\Omega\to\omega$. This gives the following commutative diagram

${\operatorname{Ext}^{d}_{R}(k,\Omega)}$${\operatorname{H}^{d}_{\mathfrak{m}}(\Omega)}$${\operatorname{Ext}^{d}_{R}(k,N)}$${\operatorname{H}^{d}_{\mathfrak{m}}(\omega)}$${E}$$\scriptstyle{f_{*}}$$\scriptstyle{\zeta^{d}_{\Omega}}$$\scriptstyle{\operatorname{H}^{d}_{\mathfrak{m}}(f)}$$\scriptstyle{\alpha}$$\scriptstyle{\zeta^{d}_{\omega}}$

Since $\alpha\circ\zeta^{d}_{\Omega}\neq 0$ we conclude that
$\zeta^{d}_{\omega}\neq 0$, as desired. ∎

######  Corollary 3.11.

If $R$ has a dualizing complex, the $R$-complex ${\omega_{R}}^{\dagger}$ is
MCM. ∎

The preceding result prompts a natural question.

######  Question 3.12.

When is the dualizing complex itself an MCM complex?

Let $R$ be a local ring with a dualizing complex $D$, normalized as in 3.4.
The local cohomology of $D$ has the right properties, so, by Proposition 3.5,
the $R$-complex $D$ is MCM precisely when $\zeta^{d}_{R}$ is nonzero. Easy
examples involving non-domains show that this is not always the case; Dutta
[Dutta:1994a] asked: Is $\zeta^{d}_{R}$ nonzero whenever $R$ is a complete
normal domain? Recently, Ma, Singh, and Walther [Ma/Singh/Walther:2020]
constructed counterexamples.

On the other hand, when $R$ is _quasi-Gorenstein_ , that is to say, when
$\omega_{R}$ is free, it follows from Corollary 3.11 that $D$ is MCM.

Here is a broader question, also of interest, concerning the maps
$\zeta^{i}_{N}$: It is easy to check that this is nonzero when $i=0pt_{R}N$.
What conditions on $N$ ensure that this is the only $i$ for which it is true?
By taking direct sums of modules of differing depths, we obtain modules $N$
with $\zeta^{i}_{N}$ nonzero for more than a single $i$.

######  Example 3.13.

When $(R,\mathfrak{m},k)$ is a regular local ring and $N$ is a finitely
generated $R$-module, then $N$ is Buchsbaum if and only if $\zeta^{i}_{N}$ is
surjective for each $i<\operatorname{dim}_{R}N$. So any non-CM Buchsbaum
$R$-module would give an example.

######  Remark 3.14.

Let $F\simeq k$ be a free resolution of $k$ and $\underline{r}\colonequals
r_{1},\dots,r_{n}$ elements such that $(\underline{r})$ is primary to the
maximal ideal. The canonical surjection $R/(\underline{r})\to k$ lifts to a
morphism of complexes $K(\underline{r};R)\to F$. Applying
$\operatorname{Hom}_{R}(-,N)$ induces maps

$\operatorname{Ext}^{i}_{R}(k,N)\longrightarrow\operatorname{H}^{i}(\underline{r};N)$

It is easy to verify that $\zeta^{i}_{N}$ factors through this map. What is
more, if $\underline{s}$ is another sequence of elements such that
$\underline{r}\in(\underline{s})$, then the map above factors as

$\operatorname{Ext}^{i}_{R}(k,N)\longrightarrow\operatorname{H}^{i}(\underline{s};N)\longrightarrow\operatorname{H}^{i}(\underline{r};N)$

Thus if any of maps above are zero, so is $\zeta^{i}_{N}$.

We would like to record a few more observations about MCM complexes.

######  Remark 3.15.

Let $(R,\mathfrak{m},k)$ be an arbitrary noetherian local ring. Then its
$\mathfrak{m}$-completion, $\widehat{R}$, has a dualizing complex, and hence
an MCM $\widehat{R}$-complex, as discussed above. Since any MCM
$\widehat{R}$-complex is a big Cohen-Macaulay complex over $R$, we conclude
that $R$ has a big Cohen-Macaulay complex, and, in particular, a complex of
maximal depth.

######  Remark 3.16.

Assume $R$ has a dualizing complex and that $M$ is an MCM $R$-complex. It is
easy to check using Proposition 3.5 that $M_{\mathfrak{p}}$ is an MCM
$R_{\mathfrak{p}}$-complex for $\mathfrak{p}$ in $\operatorname{Spec}R$, as
long as condition (2) defining MCM complexes holds at $\mathfrak{p}$. For
example, if $A$ is dg $R$-algebra that is MCM as an $R$-complex, then since
$A_{\mathfrak{p}}$ is a dg $R_{\mathfrak{p}}$-algebra, Lemma 3.3 implies that
it is an MCM $R_{\mathfrak{p}}$-complex.

######  Remark 3.17.

While MCM complexes have their uses, as the discussion in Section 2 makes
clear, they are not always a good substitute for MCM _modules_. Indeed, in
[Hochster:1975, §3] Hochster proves if every local ring has an MCM module,
then the Serre positivity conjecture on multiplicities is a consequence of the
vanishing conjecture; see also [Hochster:2017, §4]. Hochster’s arguments
cannot be carried out with MCM complexes in place of modules. The basic
problem is this: Given a finite free complex $F$, over a local ring $R$, with
homology of finite length, if $M$ is an MCM $R$-module, then
$\operatorname{H}(F\otimes_{R}M)$ is concentrated in at most one degree; this
need not be the case when $M$ is an MCM complex. Indeed this is clear from
Iversen’s Amplitude inequality [Iversen:1977], which is a reformulation of the
New Intersection Theorem, and reads:

$\mathrm{amp}(F\otimes_{R}^{\operatorname{L}}X)\geq\mathrm{amp}(X)$

where $F$ is any finite free complex with $\operatorname{H}(F)\neq 0$ and $X$
is an $R$-complex with $\operatorname{H}(X)$ bounded. Here
$\mathrm{amp}(X)\colonequals\sup\operatorname{H}_{*}(X)-\inf\operatorname{H}_{*}(X)$,
the _amplitude_ of $X$. By the way, the Amplitude Inequality holds even when
$\operatorname{H}(X)$ is unbounded [Foxby/Iyengar:2001].

###  3.18. Via resolution of singularities

The constructions of MCM complexes described above are independent of the
characteristic of the ring, but proving that they are MCM is a non-trivial
task, for it depends on knowing that one has MCM complexes to begin with; see
Remark 3.9. Next, we describe a complex that arises from a completely
different source that one can prove is MCM independently. The drawback is that
it is restricted to algebras essentially of finite type and containing the
rationals. We first record a well-known observation about proper maps.

######  Lemma 3.19.

Let $R$ be any commutative noetherian ring and $\pi\colon
X\to\operatorname{Spec}(R)$ a proper map from a noetherian scheme $X$. Viewed
as an object in $\operatorname{D}(R)$ the complex
$\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}$ is equivalent to a dg algebra
with cohomology graded-commutative and finitely generated. When $R$ contains a
field of characteristic zero, the dg algebra itself can be chosen to be
graded-commutative.

###### Proof.

By [Grothendieck:ega3a, Theorem 3.2.1], since $\mathcal{O}_{X}$ is coherent
and $\pi$ is proper, $\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}$ is coherent
and hence its cohomology is finitely generated. Next, we explain why this
complex is equivalent, in $\operatorname{D}(R)$, to a dg algebra. The idea is
that $\mathcal{O}_{X}$ is a ring object in $\operatorname{D}(X)$ and there is
a natural morphism

$\operatorname{R}\\!{\pi}_{*}\mathcal{F}\otimes_{R}^{\operatorname{L}}\operatorname{R}\\!{\pi}_{*}{\mathcal{G}}\longrightarrow\operatorname{R}\\!{\pi}_{*}{(\mathcal{F}\otimes_{X}^{\operatorname{L}}{\mathcal{G}})}$

so $\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}$ is ring object in
$\operatorname{D}(R)$. One can realize this concretely as follows.

Let $\\{U_{i}\\}_{i=1}^{n}$ be an affine cover of $X$. Then the Čech complex
computing $\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}$ is equivalent to the
total complex associated to the co-simplicial commutative ring

${0}$${\prod_{i}\Gamma(X,U_{i})}$${\prod_{i,j}\Gamma(X,U_{i}\cap
U_{j})}$${\prod_{i,j,k}\Gamma(X,U_{i}\cap U_{j}\cap U_{k})}$

It remains to point out that the Alexander-Whitney map makes the normalization
of a co-simplicial ring a dg algebra, with graded-commutative cohomology.
Moreover, over a field of characteristic zero, it is even quasi-isomorphic to
a graded-commutative dg algebra. ∎

The statement of the next result, which is due to Roberts [Roberts:1980],
invokes the resolution of singularities in characteristic zero, established by
Hironaka. The proof uses Grothendieck duality for projective maps
[Hartshorne:1966] and the theorem of Grauert and Riemenschneider
[Grauert/Riemenschneider:1970] on the vanishing of cohomology. Given these,
the calculation that is needed is standard; see the proof of
[Hartshorne/Ogus:1974, Proposition 2.2 ] due to Hartshorne and Ogus. It will
clear from the argument that it can be carried out also in any context where
one has sufficient vanishing of cohomology; see [Roberts:1980, Theorem 3.3].

######  Proposition 3.20.

Let $(R,\mathfrak{m},k)$ be a noetherian local ring essentially of finite over
a field of characteristic zero, and $\pi\colon X\to\operatorname{Spec}(R)$ a
resolution of singularities. The $R$-complex
$\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}$ is MCM and equivalent to a
graded-commutative dg algebra.

###### Proof.

Given Lemmas 3.19 and 3.3 it remains to verify that
$\operatorname{H}^{j}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X})=0$
for $j\neq d$, where $d\colonequals\operatorname{dim}R$. Let
$\kappa\colon\operatorname{Spec}(R)\to\operatorname{Spec}(k)$ be the structure
map and set $D\colonequals\kappa^{!}(\mathcal{O}_{k})$; this is then a
dualizing complex for $R$, and
$\pi^{!}D=\pi^{!}\kappa^{!}(\mathcal{O}_{k})\cong\omega_{X}$, the dualizing
sheaf for $X$. Since the $R$-complex
$\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}$ has finitely generated
cohomology, local duality 3.4.1 yields the first isomorphism below

$\displaystyle\operatorname{H}^{j}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}O_{X})$
$\displaystyle\cong{\operatorname{Ext}^{d-j}_{R}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X},D)}^{\vee}$
$\displaystyle\cong\operatorname{Ext}^{d-j}_{X}(\mathcal{O}_{X},\pi^{!}D)^{\vee}$
$\displaystyle={\operatorname{H}^{d-j}(X,\omega_{X})}^{\vee}$

The second isomorphism is by coherent Grothendieck duality [Hartshorne:1966].
It remains to invoke the Grauert-Riemenschneider vanishing theorem
[Grauert/Riemenschneider:1970] to deduce that the last module in the display
is $0$ for all $j\neq d$. ∎

Here is a natural question, growing out of Proposition 3.20. A positive answer
might have a bearing on the theory of multiplier ideals; see Theorem 4.3.

######  Question 3.21.

When $R$ contains a field of positive characteristic, or is of mixed
characteristic, does it have an MCM $R$-complex that is also a dg algebra?
What about a graded-commutative dg algebra?

##  4\. Applications to birational geometry

In this section we prove two celebrated results in birational geometry using
MCM complexes constructed via Proposition 3.20. The first one generalizes
Boutot’s theorem on rational singularities [Boutot:1987]; the argument is only
a slight reworking of Boutot’s proof, emphasizing the role of the derived
push-forward as an MCM complex. Related circles of ideas can be found in the
work of Bhatt, Kollár, Kovács, and Ma [Bhatt:2012, KollarShaferavich,
Kovacs:2000, Ma:2018].

######  Theorem 4.1.

Let $\rho\colon Z\to\operatorname{Spec}R$ be a map of schemes essentially of
finite type over a field of characteristic $0$ such that
$R\to\operatorname{R}\\!{\rho}_{*}\mathcal{O}_{Z}$ splits in
$\operatorname{D}(R)$. If $Z$ has rational singularities, then so does $R$.

###### Proof.

We may assume $(R,\mathfrak{m})$ is local. Note that the condition implies
$R\to\pi_{*}O_{X}$ is injective so in particular $R$ is reduced (as $X$ is
reduced). Thus we can take $\pi\colon X\to\operatorname{Spec}R$ to be a
resolution of singularities. Then there is a (reduced) subscheme of
$X\times_{\operatorname{Spec}R}Z$ that is birational over $Z$ for each
irreducible component of $Z$. Let $Y$ be a resolution of singularities of that
subscheme. Thus there is a commutative diagram:

${Y}$${X}$${Z}$${\operatorname{Spec}R.}$$\scriptstyle{\sigma}$$\scriptstyle{\pi}$$\scriptstyle{\rho}$

This induces a commutative diagram

${R}$${\operatorname{R}\\!{\rho}_{*}\mathcal{O}_{Z}}$${\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}}$${\operatorname{R}\\!{\rho}_{*}\operatorname{R}\\!{\sigma}_{*}\mathcal{O}_{Y}.}$$\scriptstyle{\cong}$

The right vertical map is an isomorphism since $Z$ has rational singularities.
Now since $R\to\operatorname{R}\\!{\rho}_{*}\mathcal{O}_{Z}$ splits in $D(R)$,
chasing the diagram shows that
$R\to\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}$ splits in
$\operatorname{D}(R)$. In particular, we know that the induced map

$\operatorname{H}^{i}_{\mathfrak{m}}(R)\hookrightarrow\operatorname{H}^{i}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X})$

is split-injective for all $i$. Because
$\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}$ is a MCM complex, by Proposition
3.20, it follows that $\operatorname{H}^{i}_{\mathfrak{m}}(R)=0$ for $i<d$,
that is to say, $R$ is Cohen-Macaulay. Finally, the Matlis dual of the
injection above yields a surjective map
$\pi_{*}\omega_{X}\twoheadrightarrow\omega_{R}$. Therefore
$\pi_{*}\omega_{X}\cong\omega_{R}$ since $X\to\operatorname{Spec}R$ is
birational.

Putting these together yields
$\omega_{R}^{\bullet}\cong\operatorname{R}\\!{\pi}_{*}\omega_{X}^{\bullet}$,
where $\omega_{R}^{\bullet}$ and $\omega_{X}^{\bullet}$ are the normalized
dualizing complex of $R$ and $X$ respectively. Applying
$\operatorname{RHom}_{R}(-,\omega_{R}^{\bullet})$ and using Grothendieck
duality yields $R\cong\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}$. Thus $R$
has rational singularities. ∎

Here is an application.

######  Corollary 4.2.

Suppose $(R,\Delta)$ is KLT, then $R$ has rational singularities.

###### Proof.

Let $\pi\colon Y\to X=\operatorname{Spec}R$ be a log resolution of
$(R,\Delta)$. Since $(R,\Delta)$ is KLT, we know that $\lceil
K_{Y}-\pi^{*}(K_{X}+\Delta)\rceil$ is effective and exceptional, thus

$R=\pi_{*}\mathcal{O}_{Y}(\lceil
K_{Y}-\pi^{*}(K_{X}+\Delta)\rceil)=\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{Y}(\lceil
K_{Y}-\pi^{*}(K_{X}+\Delta)\rceil)\,,$

where the second equality follows from relative Kawamata-Viehweg vanishing
[Lazarsfeld:2004, Theorem 9.4.1]. Now we have

$R\to\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{Y}\to\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{Y}(\lceil
K_{Y}-\pi^{*}(K_{X}+\Delta)\rceil)\cong R\,,$

that is to say, the map $R\to\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{Y}$
splits in $\operatorname{D}(R)$. Theorem 4.1 then implies $R$ has rational
singularities. ∎

Our second application is a new proof of the subadditivity property of
multiplier ideals [Lazarsfeld:2004]. Our idea of using the MCM property of
$\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}$ to prove this comes from the
analogous methods in positive and mixed characteristic [Ma/Schwede:2018,
Takagi:2006].

######  Theorem 4.3.

Let $(A,\mathfrak{m})$ be a noetherian regular local ring essentially of
finite type over a field of characteristic $0$. Given ideals
$\mathfrak{a},\mathfrak{b}$ in $A$ and numbers $s,t\in\mathbb{Q}_{\geq 0}$,
one has $J(A,\mathfrak{a}^{s}\mathfrak{b}^{t})\subseteq
J(A,\mathfrak{a}^{s})J(A,\mathfrak{b}^{t})$.

###### Proof.

We first claim that we may assume that $\mathfrak{a},\mathfrak{b}$ are both
principal ideals. For this we need an analog, for _mixed_ multiplier ideals,
[Lazarsfeld:2004, Proposition 92.26]. The argument is the same and we now
sketch it. Indeed, fix general elements $f_{1},\dots,f_{k}$ in $\mathfrak{a}$
and $g_{1},\dots,g_{l}$ in $\mathfrak{b}$ for $k>s$ and $l>t$ and set

$\displaystyle D_{1}$ $\displaystyle={1\over k}\sum\mathrm{Div}(f_{i})={1\over
k}\mathrm{Div}(\prod\mathrm{Div}(f_{i}))$ $\displaystyle D_{2}$
$\displaystyle={1\over l}\sum\mathrm{Div}(g_{i})={1\over
l}\mathrm{Div}(\prod\mathrm{Div}(g_{i}))\,.$

For a log resolution $\pi\colon X\to\mathrm{Spec}A$ of
$(A,\mathfrak{a},\mathfrak{b})$ with
$\mathcal{O}_{X}(-F)=\mathfrak{a}\cdot\mathcal{O}_{X}$ and
$\mathcal{O}_{X}(-G)=\mathfrak{b}\cdot\mathcal{O}_{X}$, we have that

$\pi^{-1}_{*}\mathrm{Div}(f_{i})+F_{\mathrm{exc}}=\pi^{*}\mathrm{Div}(f_{i})\;\;\text{
and
}\;\;\pi^{-1}_{*}\mathrm{Div}(g_{i})+G_{\mathrm{exc}}=\pi^{*}\mathrm{Div}(g_{i}).$

where $\pi^{-1}_{*}$ denotes the strict transform and $F_{\mathrm{exc}}$ and
$G_{\mathrm{exc}}$ are the $\pi$-exceptional parts of $F$ and $G$. Since the
$f_{i}$ and $g_{i}$ are generic, the associated divisors and their strict
transforms are reduced. A straightforward computation then shows that

$\lfloor sF\rfloor=\left\lfloor{s\over
k}\sum\pi^{*}\mathrm{Div}(f_{i})\right\rfloor\;\;\text{ and }\;\;\lfloor
tG\rfloor=\left\lfloor{t\over l}\sum\pi^{*}\mathrm{Div}(g_{i})\right\rfloor.$

Thus $J(A,\mathfrak{a}^{s}\mathfrak{b}^{t})=J(A,(\prod f_{i})^{s/k}(\prod
g_{i})^{t/l})$, and likewise $J(A,\mathfrak{a}^{s})=J(A,(\prod f_{i})^{s/k})$
and $J(A,\mathfrak{b}^{t})=J(A,(\prod g_{i})^{t/l})$. Therefore we may assume
that $\mathfrak{a}$ and $\mathfrak{b}$ are principal.

Now we assume $\mathfrak{a}=(f)$ and $\mathfrak{b}=(g)$. Let $R$ be the
normalization of $A[f^{1/d_{s}},g^{1/d_{t}}]$ where $d_{s}$ and $d_{t}$ are
the denominators of $s$ and $t$; thus $f^{s},g^{t}$ are elements in $R$. Let
$\pi\colon X\to\operatorname{Spec}R$ be a resolution of singularities. Thus
$X\to\operatorname{Spec}A$ is a regular alteration; we write $\pi$ also for
this map.

In what follows, to simplify notation, we write $E$ for
$\operatorname{H}^{d}_{\mathfrak{m}}(A)$. Given an element $r\in R$ let
$0^{r}_{E}$ for the kernel of the composite map

$E=\operatorname{H}^{d}_{\mathfrak{m}}(A)\longrightarrow\operatorname{H}^{d}_{\mathfrak{m}}(R)\xrightarrow{\
r\
}\operatorname{H}^{d}_{\mathfrak{m}}(R)\longrightarrow\operatorname{H}^{d}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X})$

Now suppose that a power $r^{m}$ of $r$ lives in $A$ (for instance $r=f^{s}$
or $r=g^{t}$). Then by [BlickleSchwedeTucker, Theorem 8.1] we have that
$\mathrm{Tr}(J(\omega_{R},r))=J(A,(r^{m})^{1/m})$. By local duality it is easy
to see that

$J(A,(r^{m})^{1/m})=\operatorname{ann}_{A}0^{r}_{E}\,.$

In particular, $J(A,f^{s})=\operatorname{ann}_{A}0^{f^{s}}_{E}$ and
$J(A,g^{t})=\operatorname{ann}_{A}0^{g^{t}}_{E}$.

We next claim that the following inclusion holds:

(4.3.1) $\\{\eta\in E\mid J(A,f^{s})\cdot\eta\subseteq
0^{g^{t}}_{E}\\}\subseteq 0^{f^{s}g^{t}}_{E}.$

Indeed, suppose $J(A,f^{s})\eta\subseteq 0^{g^{t}}_{E}$, then $J(A,f^{s})\cdot
g^{t}\eta=0$ in
$\operatorname{H}^{d}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X})$.
Note that $g^{t}\eta$ makes sense in
$\operatorname{H}^{d}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X})$
as the latter is a module over $R$. Thus

$g^{t}\eta\in\operatorname{ann}_{\operatorname{H}^{d}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X})}J(A,f^{s})\cong\operatorname{Hom}_{A}(A/J(A,f^{s}),\operatorname{H}^{d}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}))\,.$

Next, because

$h^{-i}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}\otimes_{A}^{\operatorname{L}}E)\cong\operatorname{H}^{d-i}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X})=0$

for all $i\geq 1$, we know that $g^{t}\eta$ is in the module

$\displaystyle\operatorname{Hom}_{A}(\frac{A}{J(A,f^{s})},\operatorname{H}^{d}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}))$
$\displaystyle\cong$ $\displaystyle
h^{0}\left(\operatorname{RHom}_{A}(\frac{A}{J(A,f^{s})},\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}\otimes_{A}^{\operatorname{L}}E)\right)$
$\displaystyle\cong$ $\displaystyle
h^{0}\left(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}\otimes_{A}^{\operatorname{L}}\operatorname{RHom}_{A}(\frac{A}{J(A,f^{s})},E)\right)$
$\displaystyle\cong$ $\displaystyle
h^{0}\left(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}\otimes_{A}^{\operatorname{L}}\operatorname{ann}_{E}J(A,f^{s})\right)$

where the second isomorphism follows from [Foxby:1977, Proposition 1.1 (4)],
noting that $A$ is regular thus every bounded complex is isomorphic to a
bounded complex of flat modules in $\operatorname{D}(A)$, and the third
isomorphism follows from the fact that $E$ is an injective $A$-module.

Consider following composite map; again, the second multiplication by $f^{s}$
map makes sense since we can view
$\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}$ as a complex over $R$ and not
merely over $A$:

$E\to
h^{0}\left(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}\otimes_{A}^{\operatorname{L}}{E}\right)\xrightarrow{\cdot
f^{s}}h^{0}\left(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}\otimes_{A}^{\operatorname{L}}{E}\right)$

Its kernel is $\operatorname{ann}_{E}J(A,f^{s})$, by Matlis duality. Thus the
composition of the natural induced maps

$\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}\otimes_{A}^{\operatorname{L}}\operatorname{ann}_{E}J(A,f^{s})\to\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}\otimes_{A}^{\operatorname{L}}{E}\xrightarrow{\cdot
f^{s}}\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}\otimes_{A}^{\operatorname{L}}{E}$

is zero in $h^{0}$. In particular, since $g^{t}\eta$ is in $h^{0}$ of the
source of this composite map, we deduce that, viewed as an element in target,
namely in

$h^{0}\left(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X}\otimes_{A}^{\operatorname{L}}{E}\right)\cong\operatorname{H}^{d}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X})$

it is killed by $f^{s}$. Therefore $f^{s}g^{t}\eta=0$ in
$\operatorname{H}^{d}_{\mathfrak{m}}(\operatorname{R}\\!{\pi}_{*}\mathcal{O}_{X})$
and hence $\eta\in 0^{f^{s}g^{t}}_{E}$.

This justifies (4.3.1).

Finally, for any $z\in\operatorname{ann}_{E}J(A,f^{s})J(A,g^{t})$, we have
$J(A,f^{s})z\subseteq 0^{g^{t}}_{E}$ and thus $z\in 0^{f^{s}g^{t}}_{E}$ by
(4.3.1). Therefore

$\operatorname{ann}_{E}J(A,f^{s})J(A,g^{t})\subseteq 0^{f^{s}g^{t}}_{E}$

and hence by Matlis duality $J(A,f^{s}g^{t})\subseteq J(A,f^{s})J(A,g^{t})$. ∎

## References