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%%%%%%% Graded Greenlees-May Duality and the Cech Hull %%%%%%%
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%%%%%%% Ezra Miller %%%%%%%
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\begin{document}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mbox{}
\vspace{3.2in}
\title[Graded Greenlees--May Duality and the \v Cech hull]%
{Graded Greenlees-May Duality\\and the \v Cech Hull}
\author{Ezra Miller}
\address{Massachusetts Institute of Technology}
\email{ezra@math.mit.edu}
\begin{abstract}
\noindent
The duality theorem of Greenlees and May relating local cohomology with
support on an ideal $I$ and the left derived functors of $I$-adic
completion \cite{gm} holds for rather general ideals in commutative
rings. Here, simple formulas are provided for both local cohomology and
derived functors of $\ZZ^n$-graded completion, when $I$ is a monomial
ideal in the $\ZZ^n$-graded polynomial ring $k[x_1, \ldots, x_n]$.
Greenlees-May duality for this case is a consequence. A key construction
is the combinatorially defined \v Cech hull operation on $\ZZ^n$-graded
modules \cite{Mil1, Mil2, YanBass}. A simple self-contained proof of GM
duality in the derived category is presented for arbitrarily graded
noetherian rings, using methods motivated by the \v Cech~hull.
%\vskip 1ex
%\noindent
%{{\it AMS Classification:} ; }
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:intro}
Let $S = k[x_1, \ldots, x_n]$ be a polynomial ring over a field $k$, and
$I = \ \subset S$ a {\em Stanley-Reisner ideal}
generated by squarefree monomials. Local cohomology $H^\spot_\mm(S/I)$
with support on the maximal ideal $\mm = \$ of the
{\em Stanley-Reisner ring} $S/I$ has been familiar to combinatorialists
and algebraists ever since the fundamental work of Hochster and Stanley
(see \cite{Hoc,Sta}) as well as Gr\"abe \cite{Gra} relating these objects
to simplicial complexes, and Reisner's discovery of a simplicial
criterion for $S/I$ to be Cohen-Macaulay \cite{Rei}. Beginning with
Lyubeznik \cite{LyuMonomial} and continuing with a series of recent
papers \cite{TerLocalCoh, Mus1, YanBass, Mil2}, increasing attention has
been paid to properties of the local cohomology $H^\spot_I(M)$ of modules
$M$ with support on a Stanley-Reisner ideal~$I$.
\vfill
One consistent feature of the recent investigations into $H^\spot_I$ is
the presence of some sort of duality. The duality ranges in character,
from the topological duality between homology and cohomology, to a more
combinatorial duality in posets (Alexander duality, if the poset is
boolean), to algebraic dualities such as Matlis duality and local
duality. Elsewhere in this volume is a full treatment of the rather
general {\em Greenlees-May duality} \cite{gm,AJL}, an adjointness between
$H^\spot_I$ and the left derived functors $L^I_\spot$ of $I$-adic
completion in commutative rings (or better yet, for sheaves of ideals on
schemes), that as yet has not appeared explicitly in combinatorial
studies.
\vfill
The purpose of this paper is to compute simple formulas for both sides of
the $\ZZ^n$-graded Greenlees-May isomorphism over $S$, and to show how
the computation can be rephrased to give an easy proof of GM duality for
arbitrary noetherian rings graded by commutative semigroups. The
isomorphism between graded local homology and the left derived functors
of graded completion is an easy consequence. The $\ZZ^n$-graded local
duality theorem with monomial support and the Alexander duality between
$H^\spot_\mm(S/I)$ and $H^{n-\spot}_I(S)$, whose original proofs were
independent of GM duality, illustrate the theory as combinatorial
examples.
\vfill
The key construction for the $\ZZ^n$-graded case is the \v Cech hull, a
combinatorial operation on $\ZZ^n$-graded modules. Although it was first
defined in the context of Alexander duality quite independently of local
cohomological considerations \cite{Mil1}, the \v Cech hull turns out to
be a natural tool for computing left derived functors $\uL^I_\spot$ of
$\ZZ^n$-graded $I$-adic completion. More precisely, Theorem~\ref{thm:L}
carries out the computation of $\uL^I_\spot$ in terms of the \v Cech hull
of the $\ZZ^n$-graded $\hom$ module $\hhom_S(\FF_\spot,\omega_S)$ for a
free resolution $\FF_\spot$ of~$I$. When altered slightly to avoid using
the $\ZZ^n$-grading, this method provides a similar approach to the
isomorphism between $L^I_\spot$ and local homology $H^I_\spot$ in graded
noetherian rings (the new proof of this known isomorphism is the
interesting part; the arbitrary grading comes for free).
\vfill
The organization is as follows. The \v Cech hull and $\ZZ^n$-graded
construction of Matlis duality are reviewed in Section~\ref{sec:basics}.
The relation between the \v Cech hull and local cohomology is
demonstrated in Section~\ref{sec:cech}. The left derived functors
$\uL^I_\spot$ of graded \hbox{$I$-adic} completion are introduced in
Section~\ref{sec:completion}, and computed over the polynomial ring via
the \v Cech hull. Section~\ref{sec:gm} treats $\ZZ^n$-graded
Greenlees-May duality (and its consequences) over~$S$. Finally,
Section~\ref{sec:graded} covers the reformulation of the methods to give
a conceptually easy proof for noetherian graded rings.
\vfill
The reader interested only in the (short) proof of Greenlees-May duality
for graded noetherian rings is advised to read
Sections~\ref{sub:completion}--\ref{sub:TelMic} and
Definition~\ref{defn:localhom} before continuing on to
Section~\ref{sec:graded}, which requires no other results.
This paper is intended to be accessible to those unfamiliar with
Greenlees-May duality as well as to those unfamiliar with the
combinatorial side of local cohomology and $\ZZ^n$-gradings. Therefore
many parts are longer than strictly necessary, and have a decidedly
expository feel. This seems to be in the spirit of the ``first ever''
conference on local cohomology, as well as this workshop proceedings.
\subsection*{Acknowledgements}
I am grateful to Joe Lipman and John Greenlees for enlightening
discussions (and courses) at the Guanajuato conference on local
cohomology.
% Preliminary versions of some of the material here made their way into
% Chapter~6 of the author's Ph.D.\ thesis \cite{Mil3}.
Funding during various stages of this project was provided by an Alfred
P. Sloan Foundation Doctoral Dissertation Fellowship and a National
Science Foundation Postdoctoral Research Fellowship.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic constructions}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:basics}
\subsection{Matlis duality}\label{sub:matlis}%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $S = k[x_1, \ldots, x_n]$ be a polynomial ring over a field $k$, and
set $\xx^\aa = x_1^{a_1} \cdots x_n^{a_n}$ for $\aa \in \NN^n$. An
$S$-module $J$ is {\em $\ZZ^n$-graded} if $J = \bigoplus_{\bb \in \ZZ^n}
J_\bb$ and $\xx^\aa J_\bb \subseteq J_{\aa+\bb}$ for all monomials
$\xx^\aa \in S$. Thus the polynomial ring $S$ is $\ZZ^n$-graded, as is
the localization $S[x_i\inv \mid i \in F]$ for any subset $F \subseteq
\{1,\ldots,n\}$.
\begin{defn} \label{defn:matlis}
Define the {\em Matlis dual} $J^\vee$ of $J$ by
$$
(J^\vee)_{-\bb} = \hom_k(J_\bb,k),
$$
with $S$-module structure determined by letting $\xx^\aa : (J^\vee)_\bb
\to (J^\vee)_{\aa+\bb}$ be the transpose of $\xx^\aa : J_{-\aa-\bb} \to
J_{-\bb}$.
\end{defn}
It is obvious from this definition that Matlis duality is an exact
contravariant functor on $\ZZ^n$-graded modules, and that $(J^\vee)^\vee
= J$ if $\dim_k J_\bb < \infty$ for all $\bb \in \ZZ^n$ (such a module
$J$ is said to be {\em $\ZZ^n$-finite}). To orient the reader, this
Matlis duality restricts to the usual one between finitely generated and
artinian $\ZZ^n$-graded modules (see Lemma~\ref{lemma:vee}), although
this fact won't arise here.
\begin{example} \label{ex:vees}
The Matlis dual of an $S$-module that is expressed as
$$
J = k[x_i \mid i \in F][x_j\inv \mid j \in G] \quad \hbox{is} \quad
J^\vee = k[x_i\inv \mid i \in F][x_j \mid j \in G].
$$
It is not necessary that $F$ and $G$ be disjoint, or that $F \supseteq
G$, or that their union be $\{1,\ldots,n\}$; when $J_\bb \neq 0$ but
$J_{\aa+\bb} = 0$, it is understood that $\xx^\aa J_\bb = 0$. The
easiest and most important example along these lines is when $J = S$, so
that $J^\vee = S^\vee = k[x_1\inv, \ldots, x_n\inv]$ is the {\em
injective hull} of~$k$.
\end{example}
Let $R$ be any $\ZZ^n$-graded ring (we will use only $S$ and its
$\ZZ^n$-graded subring $k$ concentrated in degree~$\0$). A map $\phi : M
\to N$ of $\ZZ^n$-graded $R$-modules is called {\em homogeneous of degree
$\bb \in \ZZ^n$} (or just {\em homogeneous} when $\bb = \0$) if
$\phi(M_\cc) \subseteq N_{\bb+\cc}$. When $R = S$ and $\bb$ is fixed,
the set of such maps is a $k$-vector space denoted
$$
\hhom_S(M,N)_\bb = \hbox{ degree $\bb$ homogeneous maps } M \to N.
$$
% (The case $R = k$ will come up in the proofs of Lemma~\ref{lemma:vee}
% and Proposition~\ref{prop:flinj}.)
As the notation suggests,
$$
\hhom_S(M,N) = \bigoplus_{\bb \in \ZZ^n} \hhom_S(M,N)_\bb
$$
is a $\ZZ^n$-graded $S$-module, with $\xx^\aa \phi$ defined by $(\xx^\aa
\phi)(m) = \xx^\aa(\phi m) = \phi(\xx^\aa m)$.
% This action of $S$ is well-defined because $\phi$ is a homomorphism of
% $S$-modules.
Matlis duality can now be expressed without resorting to degree-by-degree
vector space duals.
% and leaving the category of $S$-modules.
\begin{lemma} \label{lemma:vee}
For any $\ZZ^n$-graded modules $J$ and $M$,
$$
\hhom_S(J,M^\vee) = (M \otimes_S J)^\vee = \hhom_S(M, J^\vee).
$$
In particular, $J^\vee = \hhom_S(J, S^\vee)$.
\end{lemma}
\begin{proof}
$J^\vee$ can be expressed as the $\ZZ^n$-graded module $\hhom_k(J,k)$,
because a $k$-vector space homomorphism $J \to k$ that is homogeneous of
degree $\bb$ is the same thing as a vector space map $J_{-\bb} \to k$,
the $k$ being concentrated in degree~$\0$. The result is a consequence
of the adjointness between $\hhom$ and $\otimes$ that holds for arbitrary
$\ZZ^n$-graded $k$-algebras $S$ and $S$-modules $J$, $M$:
$$
\hhom_k(M \otimes J,k) = \hhom_S(M,\hhom_k(J,k)) = \hhom_S(M,J^\vee).
$$
That $M$ and $J$ can be switched is by the symmetry of $\otimes$.%
%\cite[A5.2.2(e)]{Eis}.
\end{proof}
Matlis duality switches flat and injective modules in the category of
$\ZZ^n$-graded modules.
\begin{lemma} \label{lemma:flinj}
$J \in \MM$ is flat if and only if $J^\vee$ is injective.
\end{lemma}
\begin{proof}
The functor $\hhom_S(-,J^\vee)$ on the right in
$$
\hhom_k(- \otimes_S J,k) = \hhom_S(-,\hhom_k(J,k))\,.
$$
is exact $\Leftrightarrow$ the functor on the left is exact
$\Leftrightarrow \,- \otimes_S J$ is, because $k$ is a field.%
\end{proof}
\begin{example} \label{ex:dualizing}
The $\ZZ^n$-graded dualizing complex for $S$ is the Matlis dual of the \v
Cech complex on $x_1, \ldots, x_n$, appropriately shifted homologically.%
\end{example}
See \cite{GWii,Sta,Mil2,MP} for more on the category of $\ZZ^n$-graded
modules and its relation to combinatorial commutative algebra.
\subsection{The \v Cech hull}\label{sub:cech}%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{defnlabeled}{\cite{Mil2}} \label{defn:cechHull}
The {\em \v Cech hull} of a $\ZZ^n$-graded module $M$ is the
$\ZZ^n$-graded module $\cech M$ whose degree $\bb$ piece is
$$
(\cech M)_\bb = M_{\bb_+} \quad {\rm where} \quad \bb_+ = \sum_{b_i
\geq 0} b_i \ee_i
$$
is the positive part of $\bb$. Equivalently,
$$
\cech M = \bigoplus_{\bb \in \NN^n} M_\bb \otimes_k k[x_i\inv \mid b_i
= 0].
$$
If $\ee_i \in \ZZ^n$ is the $i^\th$ standard basis vector, the action of
multiplication by $x_i$ is
$$
\cdot x_i : (\cech M)_\bb \to (\cech M)_{\ee_i + \bb} =
\left\{
\begin{array}{@{}l@{\ }l@{}}
{\rm identity} & {\rm if\ } b_i < 0 \cr
\cdot x_i : M_{\bb_+} \to M_{\ee_i + \bb_+} &
{\rm if\ } b_i \geq 0 \end{array}\right.,
$$
Note that $(\ee_i + \bb)_+ = \bb_+$ whenever $b_i < 0$, and $(\ee_i +
\bb)_+ = \ee_i + \bb_+$ whenever $b_i \geq 0$.
\end{defnlabeled}
Heuristically, the first description of $\cech M$ in the definition says
that if you want to know what $\cech M$ looks like in degree $\bb \in
\ZZ^n$, then check what $M$ looks like in the nonnegative degree closest
to $\bb$; the second description says that the vector space $M_\aa$ for
$\aa \in \NN^n$ is copied into all degrees $\bb$ such that $\bb_+ = \aa$.
The \v Cech hull ``forgets'' everything about $M$ that isn't in $\NN^n$.
The \v Cech hull can just as well be applied to a homogeneous map of
degree $\0$ between two modules, by copying the maps in the
$\NN^n$-graded degrees as prescribed. Some properties of $\cech$ are now
immediate from checking things degree by degree.
\begin{lemma} \label{lemma:exact}
The \v Cech hull is an exact covariant functor on $\ZZ^n$-graded modules.
\end{lemma}
The utility of the \v Cech hull stems from its ability to localize free
modules and take injective hulls of quotients by primes simultaneously.
For instance, $\cech S = S[x_1\inv, \ldots, x_n\inv]$, while
$\cech(S/\mm) = S^\vee$; see \cite[Example~2.8]{Mil2} for more details.
Here, the applications require only the localization property.
The notation henceforth is as follows. For $\bb \in \ZZ^n$, the
$\ZZ^n$-graded shift $M(\bb)$ is the module satisfying $M(\bb)_\cc =
M_{\bb+\cc}$. Thus, for $\aa \in \NN^n$, the free module of rank $1$
generated in degree $\aa$ is $S(-\aa)$. If $F \in \{0,1\}^n$, the
localization $M[\xx^{-F}]$ is $M \otimes_S S[x_i\inv \mid F_i = 1]$.
Setting $\1 = (1,\ldots,1)$, the next lemma is straightforward.
\begin{lemma} \label{lemma:cech}
If $F \in \{0,1\}^n$ then $\cech(S(F-\1)) \cong S(F-\1)[\xx^{-F}]$.
\end{lemma}
Lemma~\ref{lemma:frob} will clarify the pertinence of the next
definition.
\begin{defn} \label{defn:frob}
Define the {\em $t^\th$ Frobenius power} of a monomial ideal $I = \$ to be $I^{[t]} = \$ for $1 \leq t
\in \ZZ$. Any direct sum $\FF = \bigoplus_j S(-F_j)$ with each $F_j \in
\NN^n$ is isomorphic to a direct sum $\bigoplus_j \<\xx^{F_j}\>$ of
ideals, so $\FF^{[t]} = \bigoplus_j S(-tF_j)$ is similarly defined.
\end{defn}
The advantage to Frobenius powers is that their free resolutions are all
related (no assumption is required on the characteristic of $S$). Let
$\FF_\spot$ be a free resolution of $S/I$. Then there is an induced free
resolution $\FFlspotT$ of $S/I^{[t]}$: choose matrices for the
differentials in $\FF_\spot$ and replace every occurrence of $x_i$ with
$x_i^t$, for all $i$. Equivalently, if $\varphi^{[t]}$ is the
$k$-algebra isomorphism mapping $S = k[x_1, \ldots, x_n]$ to $S^{[t]} =
k[x_1^t, \ldots, x_n^t]$ via $x_i \mapsto x_i^t$ for all $i$, then
$\FF_\spot$ can be considered as a complex $\varphi^{[t]}(\FF_\spot)$ of
$S^{[t]}$-modules, and $\FFlspotT = S \otimes_{S^{[t]}}
\varphi^{[t]}(\FF_\spot)$ (the tensor product over $S^{[t]}$ is via the
inclusion $S^{[t]} \into S$). This implies that $\FFlspotT$ is indeed
acyclic, since $S$ is a free (and hence flat) $S^{[t]}$-module. For the
$\ZZ^n$-graded shifts of the summands in $\FF_\spot$ to work out
properly, $S^{[t]}$ must be graded by the sublattice $t\ZZ^n \subseteq
\ZZ^n$ if $S$ insists on being graded by $\ZZ^n$.
% What can be said about the general characteristic p case when, say, A
% is flat over A^p? HOW OFTEN DOES THIS OCCUR? Shouldn't it then be
% that Frobenius powers of resolutions are resolutions by the argument
% here? Are there any applications of this to local cohomology, or the
% limits therein? Does one need the ring to be regular for any of this
% to work? Ask somebody...
For each $t \geq 1$ there is an inclusion $\FFlspotTT \to \FFlspotT$ of
complexes via the homogeneous degree $\0$ maps $S(-(t+1)F) \to S(-tF)$
sending $1 \mapsto \xx^F$. This makes $\{\FFlspotTT\}$ into an inverse
system of complexes. Just as with Koszul complexes, setting $\omega_S =
S(\1)$ and applying $\hhom_S(-,\omega_S)$ to $\{\FFlspotT\}$ yields a
directed system $\{\FFhspotT\}$, in which $S(tF-\1) \to S((t+1)F-\1)$ via
the natural inclusion $1 \mapsto \xx^F$.
Assuming now that $I$ is squarefree, $\FF_\spot$ can be chosen so that
all of the summands $S(-F_j)$ of Definition~\ref{defn:frob} are in {\em
squarefree degrees} $F_j \in \{0,1\}^n$. (This may not be obvious, but
will follow from Lemma~\ref{lemma:[tay]}.) In this case $\FF^\spot =
\hhom_S(\FF_\spot,\omega_S)$ also has all of its summands in squarefree
degrees, since $\hhom_S(S(-F),\omega_S) = S(F-\1)$. The reason for
introducing Frobenius powers is the next lemma, to be applied in
Section~\ref{sec:completion}.
\begin{lemma} \label{lemma:frob}
Let $\FF_\spot$ be a minimal free resolution of $S/I$ and $\omega_S =
S(\1)$. If $\FFhspotT = \hhom_S(\FFlspotT,\omega_S)$ then $\dirlim t\,
\FFhspotT = \cech\FF^\spot$.
\end{lemma}
\begin{proof}
$\FF^\spot$ is composed of maps $S(G-\1) \to S(F-\1)$ between free
modules generated in squarefree degrees. Since $\dirlimt S(tG-\1) =
S(G-\1)[\xx^{-G}]$ and $\dirlimt S(tF-\1) = S(F-\1)[\xx^{-F}]$ are the
corresponding localizations, it follows that $\dirlimt \FFhspotT$
is composed of natural inclusions $S(G-\1)[\xx^{-G}] \to
S(F-\1)[\xx^{-F}]$. Now use Lemma~\ref{lemma:cech}.%
\end{proof}
All of the $\ZZ^n$-graded shifts by $\1$ in the preceding are essential,
because taking \v Cech hulls rarely commutes with such shifts. However,
the restriction of minimality in Lemma~\ref{lemma:frob} is unnecessary,
as is clear from the proof: being generated in squarefree degrees will
do.
%, so (for instance) the Taylor resolution of Section~\ref{sec:cech}
%could be used instead.
See Proposition~\ref{prop:taylor} and Example~\ref{ex:maxcech} for
examples.
\begin{remark} \label{rk:dirlim}
The colimit $\dirlim t J_t$ of a directed system of $\ZZ^n$-graded
modules is a quotient of $\bigoplus_t J_t$ by a $\ZZ^n$-graded submodule.
It is therefore naturally $\ZZ^n$-graded.
\end{remark}
\begin{remark} \label{rk:straight}
In the context of Lemma~\ref{lemma:cech}, Yanagawa's notion of {\em
straight hull} \cite{YanBass} coincides with the \v Cech hull. The \v
Cech hull is generalized in \cite{HelM} to algebras graded by arbitrary
semigroups, where it isn't exact. Its derived functors yield a spectral
sequence of Ext modules converging to local cohomology, and are used to
prove the finiteness for Bass numbers of graded local cohomology of
finitely generated modules over simplicial semigroup rings. It is also
shown that the local cohomology of some finitely generated module has
infinite Bass numbers if the semigroup is not simplicial.
\end{remark}
\vfill
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The \v Cech hull and local cohomology}%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:cech}
\subsection{Generalized \v Cech complexes}\label{sub:generalized}%%%%%%%%
This section reviews the connection between the \v Cech hull and local
cohomology \cite[Section~6]{Mil2} from a new point of view.
Let $\omega_S = S(-\1)$ be the canonical module of $S$.
\begin{defn} \label{defn:cech}
If $\FF_\spot$ is a free resolution of $S/I$ and $\FF^\spot =
\hhom(\FF_\spot,\omega_S)$, define
$$
\CC^\spot_\FF = (\cech\FF^\spot)(\1)
$$
to be the {\em generalized \v Cech complex} determined by $\FF_\spot$.
When $\FF_\spot$ is minimal, $\CC^\spot_\FF$ is called the {\em canonical
\v Cech complex} for $I$, and is denoted by $\CC^\spot_I$.
\end{defn}
The first task, Proposition~\ref{prop:taylor}, is to identify the usual
\v Cech complex on generators for $I$ as the generalized \v Cech complex
for a certain (usually far from minimal) free resolution. Suppose
$\xx^{F_1}, \ldots, \xx^{F_r} \in S$ are monomials (they don't need to be
squarefree, yet). Given a subset $X \subseteq \{1, \ldots, r\}$, define
$F_X$ so that $\xx^{F_X}$ is the least common multiple of the monomials
$\xx^{F_j}$ for $j \in X$. The {\em Taylor resolution} $\TT_\spot$ on
$\xx^{F_1}, \ldots, \xx^{F_r}$ is
$$
0 \from S \from \bigoplus_{j=1}^r S(-F_j) \from \cdots \from
\bigoplus_{|X| = \ell} S(-F_X) \from \cdots \from
S(-F_{\{1,\ldots,r\}}) \from 0,
$$
where the map $S(-F_{X \setminus j}) \from S(-F_X)$ is $(-1)^{s-1}$ times
the natural inclusion if $j$ is the $s^\th$ element of $X$.
\begin{lemma}[\cite{Tay}] \label{lemma:[tay]}
$\TT_\spot$ is a free resolution of $S/\<\xx^{F_1}, \ldots, \xx^{F_r}\>$.
\end{lemma}
\begin{proof}
The $\ZZ^n$-degree $\bb$ piece of $\TT_\spot$ is zero unless $\bb \in
\NN^n$, in which case $(\TT_\spot)_\bb$ is the reduced chain complex of
the simplex whose vertices are $\{j \mid F_j \preceq \bb\}$, with
$\nothing$ in homological degree $0$. This chain complex has no homology
unless $\xx^\bb \not\in \<\xx^{F_1}, \ldots, \xx^{F_r}\>$, when the only
homology is $k$ in homological degree $0$. Thus $\TT_\spot$ is a free
resolution of something. The image of the last map (to $S$) in
$\TT_\spot$ is obviously $\<\xx^{F_1}, \ldots, \xx^{F_r}\>$.%
\end{proof}
The next result is needed for Proposition~\ref{prop:cech->taylor}. It
identifies the \v Cech complex not just as a limit of Koszul cochain
complexes on generators for $I^{[t]}$, whose duals $\KK_\spot^{[t]}$ may
not be acyclic, but as the result of applying the \v Cech hull and shift
by $\1$ operations, which are exact, to a complex $\TT^\spot =
\hhom_S(\TT_\spot,\omega_S)$ whose dual is a {\em resolution} $\TT_\spot$
of~$S/I$.
\begin{prop} \label{prop:taylor}
If $I = \<\xx^{F_1}, \ldots, \xx^{F_r}\>$ is generated by squarefree
monomials, so $F_j \in \{0,1\}^n$ for all $j$, then $\CC^\spot_\TT$ is
the usual \v Cech complex
$$
0 \to S \to \bigoplus_{j=1}^r S[\xx^{-F_j}] \to \cdots \to
\bigoplus_{|X| = \ell} S[\xx^{-F_X}] \to \cdots \to
S[\xx^{-F_{\{1,\ldots,r\}}}] \to 0
$$
on the monomials $\xx^{F_1}, \ldots, \xx^{F_r}$ generating $I$.
\end{prop}
\begin{proof}
Each original summand $S(-F)$ in $\TT_\spot$ turns into a corresponding
summand $S(F-\1) = \hhom_S(S(-F),\omega_S)$ in
$\hhom_S(\TT_\spot,\omega_S)$. Lemma~\ref{lemma:cech} says
$\cech(S(F-\1)) \cong S[\xx^{-F}](F-\1)$, and this is isomorphic to
$S[\xx^{-F}](-\1)$ because multiplication by $\xx^F$ is a homogeneous
degree $F$ automorphism. Shifting by $\1$ yields the result.%
\end{proof}
The proof of Proposition~\ref{prop:taylor} says how to describe any
generalized \v Cech complex $\CC^\spot_\FF$ in more familiar terms: after
choosing bases in the complex $\FF^\spot = \hhom_S(\FF_\spot, \omega_S)$,
simply replace each summand $S(F-\1)$ by the localization
$S[\xx^{-F}](\1)$.
\begin{example} \rm \label{ex:maxcech}
The usual \v Cech complex $\CC^\spot_\mm = \CC^\spot(x_1,\ldots,x_n)$
% is the complex
% $$
% \CC^\spot_\mm:\ 0 \to S \to \bigoplus_{i=1}^n S[x_i\inv] \to \cdots \to
% \bigoplus_{|F| = \ell} S[\xx^{-F}] \to \cdots \to S[x_1\inv, \ldots,
% x_n\inv] \to 0\,,
% $$
% where $\xx^F = \prod_{i \in F} x_i$ for $F \subseteq \{1,\ldots,n\}$
% and $S[\xx^{-F}]$ is the localization of $S$ by the element $\xx^F$.
is the canonical \v Cech complex of the maximal $\ZZ^n$-graded ideal
$\mm$. In other words,
% since the Koszul complex is self-dual,
{\em the Cech hull of the Koszul complex is the \v Cech complex}, up to a
$\ZZ^n$-graded shift by $\1$. This example, or more generally
Proposition~\ref{prop:taylor}, is the reason for the term ``\v Cech
hull''.
\end{example}
\begin{remark} \rm
The canonical \v Cech complex of $I$ depends (up to isomorphism of
$\ZZ^n$-graded complexes) only on $I$, not on any system of
$\ZZ^n$-graded generators of $I$. The term ``canonical \v Cech complex''
refers both to this freedom from choices and the fact that it is, up to
$\ZZ^n$-graded and homological shift, the \v Cech hull of a free
resolution of the canonical module \label{osiref} $\omega_{S/I}$ when
$S/I$ is Cohen-Macaulay.
\end{remark}
\subsection{Local cohomology}\label{sub:localcoh}%%%%%%%%%%%%%%%%%%%%%%%%
If one is convinced that the minimal free resolution of $S/I$ is in any
sense ``better'' than the Taylor resolution (or any other free
resolution), then one should be equally convinced that the canonical \v
Cech complex $\CC^\spot_I$ is similarly ``better'' than the usual \v Cech
complex $\CC^\spot_\TT$ on the minimal generators of $I$, by
Proposition~\ref{prop:taylor}. Of course, the main use for the usual \v
Cech complex is in defining the {\em local cohomology modules}
$$
H_I^i(M) = H^i(M \otimes_S \CC^\spot_\TT),
$$
and one needs to be convinced that the canonical \v Cech complex is just
as good at this local cohomology computation. In fact, this holds for
any generalized \v Cech complex.
\begin{thm}[\cite{Mil2}] \label{thm:cech}
If $\FF_\spot$ is a free resolution of $S/I$ and $M$ is any $S$-module,
then
$$
H^i_I(M) = H^i(M \otimes_S \CC^\spot_\FF).
$$
\end{thm}
Although it is possible to give a concrete proof using
Lemma~\ref{lemma:frob} as in \cite[Theorem~6.2]{Mil2}, a new and more
conceptual proof is appropriate here, in anticipation of $\ZZ^n$-graded
Greenlees-May duality in Section~\ref{sec:gm}. The proof below pinpoints
the relation between the canonical \v Cech complex and the usual \v Cech
complex as being analogous to---and a consequence of---the relation
between the minimal free resolution of $S/I$ and the Taylor resolution.
It rests on Proposition~\ref{prop:cech->taylor}, which is the observation
that propels the rest of the paper: Theorem~\ref{thm:cech} and
Corollary~\ref{cor:L} depend on Proposition~\ref{prop:cech->taylor}, and
Proposition~\ref{prop:systems} is modelled upon it.
In what follows, a {\em homology isomorphism} of complexes (also known as
a quasi-isomorphism) is a map inducing an isomorphism on homology.
\begin{prop} \label{prop:cech->taylor}
If $\GG_\spot$ and $\FF_\spot$ are free resolutions of $S/I$, then there
is a homology isomorphism $\CC^\spot_{\GG} \to \CC^\spot_\FF$ between
their corresponding generalized \v Cech complexes.%
\end{prop}
\begin{proof}
If $\FF^I_\spot$ is a minimal free resolution of $S/I$, then there are
homology isomorphisms $\FF_\spot \to \FF^I_\spot$ and $\FF^I_\spot \to
\GG_\spot$ because $\FF^I_\spot$ is a split subcomplex of every free
resolution of $S/I$ \cite[Theorem~20.2]{Eis}. Composing these, we have a
homology isomorphism $\FF_\spot \to \GG_\spot$. Applying
$\hhom_S(-,\omega_S)$ yields a map $\GG^\spot \to \FF^\spot$ whose
induced map on cohomology is an isomorphism: both have cohomology
$\eext^\spot_S(S/I,\omega_S)$. The desired homology isomorphism is
therefore simply $(\cech \GG^\spot)(\1) \to (\cech \FF^\spot)(\1)$, since
the \v Cech hull and the shift by $\1$ are both exact.%
\end{proof}
It's worth stating explicitly the following lemma, even though it is
standard (so its proof, the K\"unneth spectral sequence, is omitted),
because it will be used so often. A {\em bounded below} complex is one
that has nonzero modules only in positive homological degrees; thus free
resolutions of modules are bounded below.
\begin{lemma} \label{lemma:quism}
If $\cL_\spot \to \cL'_\spot$ is a homology isomorphism of bounded below
complexes of flat modules and $M$ is any module, then $M \otimes
\cL_\spot \to M \otimes \cL'_\spot$ is a homology isomorphism.%
\end{lemma}
\begin{proofof}{Theorem~\ref{thm:cech}}
Proposition~\ref{prop:cech->taylor} produces a homology isomorphism
$\CC^\spot_{\TT(I)} \to \CC^\spot_\FF$. Since both of the complexes
$\CC^\spot_{\TT(I)}$ and $\CC^\spot_\FF$ are bounded below (and above),
Lemma~\ref{lemma:quism} says that $H^i_I(M) = H^i(M \otimes
\CC^\spot_{\TT(I)}) \to H^i(M \otimes \CC^\spot_\FF)$ is an isomorphism.
\end{proofof}
\begin{remark} \label{rk:Mod} \rm
If the module $M$ is $\ZZ^n$-graded, then the isomorphism in
Theorem~\ref{thm:cech} produces the natural $\ZZ^n$-grading on
$H^i_I(M)$, since the right-hand side is still $\ZZ^n$-graded.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The \v Cech hull and completion}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:completion}
\subsection{Graded completion}\label{sub:completion}%%%%%%%%%%%%%%%%%%%%%
Suppose for the moment that $I$ is an ideal in an ungraded commutative
ring $A$. Just as $\Gamma_I$ takes the direct limit of submodules
annihilated by powers of $I$, the {\em $I$-adic completion} functor
$\LambdaI$ takes the inverse limit of quotients annihilated by powers
of~$I$. More precisely, any $A$-module $M$ has a filtration by
submodules $I^t M$ for $1 \leq t \in \ZZ$, giving rise to an inverse
system $M/I^{t+1} M \to M/I^t M$. The $I$-adic completion $\LambdaI(M)$
is then defined as the inverse limit of this system, $\invlim t M/I^t M$.
In general, $I$-adic completion is neither left exact nor right exact
(intuition from finitely generated modules over noetherian rings fails
badly). However, one can still define the left derived functors
$L^I_\spot(M)$ of $\LambdaI$ by taking a free resolution of $M$ and
applying $\LambdaI$ to it. The catch is that the natural map from the
zeroth homology $L^I_0(M)$ of the resulting complex to $\LambdaI(M)$ need
not be an isomorphism, as it would be if $\LambdaI$ were right exact.
Everything in the last two paragraphs (and everything in the next
subsection, as well) makes complete~(!)\ sense, mutatis mutandis, in the
category of modules graded by some group or monoid over a similarly
graded algebra. This includes the case of the $\ZZ^n$-graded polynomial
$k$-algebra $S$ and $\ZZ^n$-graded modules, to which we return for
simplicity of exposition. The phrase ``mutatis mutandis'' here means
only that the inverse limits defining the $\ZZ^n$-graded completion
$\uLambdaI$ must be taken in the category of $\ZZ^n$-graded objects and
homogeneous maps {\em of degree zero}. Recall what this means for (say)
an inverse system $\{\FF^t\}$ of chain complexes of $\ZZ^n$-graded
$S$-modules: the $\ZZ^n$-graded inverse limit $^*\invlimt$ is defined by
\begin{equation} \label{eq:*invlim}
^*\Invlim t\, \FF^t = \bigoplus_{\bb \in \ZZ^n} \Invlim t\, \FF^t_\bb,
\end{equation}
where the ordinary inverse limits of the degree $\bb$ pieces $\FF^t_\bb$
on the right are in the category of chain complexes of $k$-vector spaces.
\begin{example} \label{ex:completion}
The polynomial ring $S$ is $\ZZ^n$-graded complete with respect to its
maximal ideal $\mm = \$, and hence with respect to
every $\ZZ^n$-graded ideal~$I$. Indeed, for any finitely generated
module $M$, ideal $I$, and fixed $\bb$, the inverse system
$(M/I^{t+1}M)_\bb \onto (M/I^t M)_\bb$ eventually stabilizes to become
$M_\bb \congto M_\bb$, so $\complete MI = M$. On the other hand,
infinitely generated modules may behave differently:
\begin{thmlist}
\vspace{-.2em}
\item
Completion of the injective hull $S^\vee$ of $k$ yields $\complete
{S^\vee}I = 0$, since $I^t S^\vee = S^\vee$ for all nonzero ideals $I^t$.
However, see Example~\ref{ex:L} for $\uL^I_\spot(S^\vee)$.
\vspace{-.1em}
\item
As in the ungraded case, $\complete MI = 0$ if $IM = M$. This can't
happen if $M \neq 0$ is finitely generated by Nakayama's Lemma, but can
occur for localizations ${M[\xx^{-F}]}$ (the notation is explained before
Lemma~\ref{lemma:cech}).
% Note that $\complete {M[\xx^{-F}]}I = \complete
% {M[\xx^{-F}]}{I[\xx^{-F}]}$ is either $M[\xx^{-F}]$ or zero when $M$~is
% finitely generated, since $S[\xx^{-F}]$ is complete with respect to
% $I[\xx^{-F}]$.
%
% Indeed, the inverse system whose $^*\invlimt$ yields $\complete
% {M[\xx^{-F}]}I$ is $\{(M[\xx^{-F}] \otimes_S S/I^t\} = \{M[\xx^{-F}]
% \otimes_{S[\xx^{-F}]} S[\xx^{-F}]/I^t[\xx^{-F}]\}$.
\end{thmlist}
Since graded modules behave so much like vector spaces, truly bad
behavior of completion requires some infinite dimensionality. With one
variable $x$, for instance, let $I = \ \subset k[x]$ and $M =
\bigoplus_{b \in \ZZ} k[x](-b)$. The completion $\complete MI$ has the
vector space ${\complete MI}_a = \prod_{b \leq a} k$ in $\ZZ$-degree $a$,
with multiplication by~$x$ being inclusion. Observe, however, that when
all of the direct summands are generated in the same $\ZZ$-degree, the
module $\bigoplus_{b \in \ZZ} k[x] = \complete {\bigoplus_{b \in \ZZ}
k[x]}I$ is fixed by completion.
For comparison, the ungraded completion $\LambdaI(\bigoplus_{b \in \ZZ}
k[[x]])$ is even more infinite than $\complete MI$. The former contains
any sequence $(x^{d_b})_{b \in \ZZ}$ in which, for each fixed~$d$, there
are finitely many $b$ satisfying $d_b \leq d$, whereas the latter
contains only semi-infinite sequences.%
\end{example}
\vspace{-.11em}
The goal of this section is the computation in Theorem~\ref{thm:L}
of $\uL^I_\spot$ in terms of generalized \v Cech complexes.
% (Definition~\ref{defn:cech}).
\subsection{Telescopes and microscopes}\label{sub:TelMic}%%%%%%%%%%%%%%%%
This section follows \cite{gm}, but with gradings. Given a homogeneous
map \hbox{$\phi : Z^\spot \to Y^\spot$} of cochain complexes with
differentials $\delta_z$ and $\delta_y$, form the associated double
complex $Y \oplus_\phi Z$ with $(Y^\spot,\delta_y)$ in the $0^\th$ row;
$(Z^\spot,-\delta_z)$ in row~$-1$; and vertical differential $\phi :
Z^\spot \to Y^\spot$. The {\em cone} or {\em cofiber} $\cone(\phi)$ is
the total complex $\tot(Y \oplus_\phi Z)$, which has $Y^i \oplus Z^{i+1}$
as its term in cohomological degree $i$ and differential
$\delta(y^i,z^{i+1}) = (\delta_y y^i + \phi z^{i+1}, -\delta_z z^{i+1})$.
Defining $Z[t]^\spot$ by $Z[t]^i = Z^{t+i}$ with differential
$(-1)^t\delta_z$, there is a short exact sequence $0 \to Y \to
\cone(\phi) \to Z[1] \to 0$ whose long exact cohomology sequence has
connecting homomorphism $\phi$.
Any cochain complex $Y^\spot$ is also a chain complex $Y_\spot$ with $Y^i
= Y_{-i}$. Define the {\em fiber} of $\psi : Y_\spot \to Z_\spot$ by
$\fiber(\psi) = \cone(-\psi)[-1]$. This is designed precisely so that if
$\phi : \tilde Z^\spot \to \tilde Y^\spot$ is a map of cochain complexes
of (say) $k$-vector spaces and $\psi : Y_\spot \to Z_\spot$ is its
transpose, so $\hom_k(\tilde Y^i,k) = Y_i$, then $\fiber(\psi)$ is the
transpose of $\cone(\phi)$.
Now suppose $\{\phi_t : \GG^\spot_t \to \GG^\spot_{t+1}\}_{t \geq 1}$ is
a sequence of cochain maps, homogeneous of degree zero, and define $\phi
: \bigoplus \GG^\spot_t \to \bigoplus \GG^\spot_t$ by $\phi(x) = x -
\phi_t(x)$ for $x \in \GG^\spot_t$. The {\em homotopy colimit} or {\em
telescope} of the sequence $\{\phi_t\}_{t \geq 1}$ is $\cone(\phi)$, and
denoted $\tel(\GG^\spot_t)$. Although the projection $\tel(\GG^\spot_t)
\to \bigoplus \GG^\spot_t$ to the target of $\phi$ (corresponding to
$Y^\spot$ above) is not a morphism of cochain complexes, it becomes so
after modding out by the image of~$\phi$. The resulting cochain map
$\tel(\GG^\spot_t) \to (\bigoplus \GG^\spot_t)/\im\phi = \dirlimt
\GG^\spot_t$ is a homology isomorphism, identifying
$H^i\tel(\GG^\spot_t)$ as $\dirlim t (H^i\GG^\spot_t)$.
Dual to the telescope is the {\em homotopy limit} or {\em microscope}
$\mic(\GG_\spot^t)$ of a sequence of homogeneous degree zero chain maps
$\{\psi^t : \GG^{t+1}_\spot \to \GG_\spot^t\}_{t \geq 1}$. It is defined
as $\fiber(\psi)$ for the map $\psi : \prod^* \GG_\spot^t \to \prod^*
\GG_\spot^t$ sending the element $(x^t)_{t \geq 1} \mapsto (x^t -
\psi^{t+1}(x^{t+1}))_{t \geq 1}$. The products here are in the category
of graded modules, and thus do not agree with the usual products. They
can be seen as special cases of $^*\invlimt$; more concretely, $\prod^*
\GG^t_\spot \subseteq \prod\GG^t_\spot$ is the submodule generated by
arbitrary products of elements of the same homological degree {\em and}
the same graded degree.
It is routine to verify the following relation between $\tel$ and $\mic$
(check that the maps $\phi$ and $\psi$ on the direct sum and product are
dual, before actually taking the cone and fiber to get $\tel$ and
$\mic$).
\begin{lemma} \label{lemma:TelMic}
If $\{\GG^\spot_t\}$ is a sequence of cochain complexes and $M$ is any
module, then $\mic\,\hhom_S(\GG^\spot_t, M) = \hhom_S(\tel
\GG^\spot_t,M)$. In particular, if $\GG^\spot_t = \FF^\spot_t(\1) =
\hhom_S(\FF_\spot^t,S)$ for a sequence of free resolutions $\FF_\spot^t$
of some ideals $I_t$, then
$$
\hhom_S(\tel \FF^\spot_t(\1),M) = \mic(\FF_\spot^t \otimes M).
$$
\end{lemma}
Note that $\mic$ depends on the grading while $\tel$ doesn't (hence the
underlining). The association of microscopes to inverse sequences
constitutes an exact functor on inverse sequences, since $\prod^*$ is
exact. Furthermore, setting $\GG_\spot^t = \hhom_S(\GG^\spot_t,S)$ for a
system of free complexes $\GG^\spot_t$, sending $M \mapsto
\mic(\GG_\spot^t \otimes M)$ is an exact functor of $M$.
The whole point of microscopes is that they compute $\uL^I_\spot$. To
get the precise statement, say that two chains $\{I_t\}$ and $\{I'_t\}$
of ideals are {\em cofinal} if each $I_t$ is contained in some $I'_{t'}$
and each $I'_{s'}$ is contained in some $I_s$. For instance, the
Frobenius powers $\{I^{[t]}\}$ of Definition~\ref{defn:frob} are cofinal
with the ordinary powers $\{I^t\}$.
\begin{lemma}[{\cite[Proposition~1.1]{gm}}] \label{lemma:gm}
Suppose $\{I_t\}$ is a chain of ideals cofinal with the powers $\{I^t\}$,
and let $\FF_\spot^t$ be an inverse sequence of free resolutions of
$S/I_t$, lifting the inverse sequence of surjections $S/I_{t+1} \onto
S/I_t$. If $\EE_\spot$ is a free resolution of some module $M$, then the
total complex of the double complex $\mic(\FF_\spot^t \otimes \EE_\spot)$
is homology isomorphic to both $\mic(\FF_\spot^t \otimes M)$ and
$\complete{\EE_\spot}I$. In particular,
$$
H_i \mic(\FF_\spot^t \otimes M) = \uL^I_i(M).
$$
\end{lemma}
\begin{proof}
For any module $J$, the short exact sequence for the fiber defining the
microscope runs $0 \to \prod^*(\FF_{1+\spot}^t \otimes J) \to
\mic(\FF_\spot^t \otimes J) \to \prod^*(\FF_\spot^t \otimes J) \to 0$.
The final four terms of the resulting long exact sequence are
$$
\hbox{$H_0 \mic(\FF_\spot^t \otimes J) \to \prod^* (S/I_t \otimes J)
\to \prod^*(S/I_t \otimes J) \to H_{-1}\mic(\FF_\spot^t \otimes J) \to
0$}.
$$
The $H_{-1}\mic$ term is zero because every map in the inverse system
$\{S/I_t \otimes J\}$ is surjective \cite[Lemma~3.5.3]{Wei}. If $J$ is
free, all of the higher terms vanish because each $\FF_\spot^t$ is a
resolution, and the $H_0\mic$ term is $\complete JI$ by the cofinality
assumption. Setting $J = \EE_\spot$ this shows that the homology of the
double complex $\mic(\FF_\spot^t \otimes \EE_\spot)$ in the $\FF$
direction is $\complete {\EE_\spot}I$.
Taking homology in the $\EE$ direction yields $\mic(\FF_\spot^t \otimes
M)$ by exactness of the microscope construction. The standard spectral
sequence argument now applies.
\end{proof}
\subsection{Derived functors of completion}\label{sub:derived}%%%%%%%%%%%
The main result, Theorem~\ref{thm:L}, says that for many modules $M$, the
left derived functors of $\ZZ^n$-graded completion can be calculated
using the flat complex $\CC^\spot_\FF$ that is approximated by the
telescope $\tel\hhom_S(\FFlspotT,S)$. (Recall that $\FFlspotT$ is the
Frobenius power of a free resolution of $S/I$ as in
Definition~\ref{defn:frob}). This theorem is interesting for three
reasons. First, $\CC^\spot_\FF$ is never projective, so it doesn't seem
agile enough to detect derived functors. Second, its finitely many
indecomposable summands make $\CC^\spot_\FF$ much smaller than either a
microscope or a telescope: $\CC^\spot_\FF$ is {\em $\ZZ^n$-finite} when
$\FF_\spot$ is, meanining that its degree $\bb$ pieces are finite
$k$-vector spaces for all $\bb \in \ZZ^n$ (note that~$\cech$ preserves
$\ZZ^n$-finiteness). Finally, it allows the use of {\em any} free
resolution to make the generalized \v Cech complex. This last point is
crucial for Corollary~\ref{cor:L}.
\begin{thm} \label{thm:L}
If $\FF_\spot$ is a free resolution of $S/I$ and $M = J^\vee$ for some
$J$ (this includes all $\ZZ^n$-finite modules $M$), then
$$
\uL^I_i(M) = H_i \hhom_S(\CC^\spot_\FF,M).
$$
More precisely, $\hhom_S(\CC^\spot_\FF,M)$ is homology isomorphic to
$\mic(\FFlspotT \otimes M)$.
\end{thm}
Noteworthy is the case when $\FF_\spot = \TT_\spot$ is the Taylor
resolution (Lemma~\ref{lemma:[tay]}), because $\CC^\spot_\TT$ is then the
usual \v Cech complex on squarefree generators for $I$, by
Proposition~\ref{prop:taylor}.
\begin{proof}
Let $\FFhspotT = \hhom_S(\FFlspotT,\omega_S)$, as usual. Assuming for
the time being that $\FF_\spot$ is minimal, consider the following
diagram:
$$
\begin{array}{c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c}
\hhom_S(\CC^\spot_I,J^\vee)
& = &(\dirlim t \FFhspotT(\1) \otimes J)^\vee \\
& & \downarrow \\
& &(\tel\FFhspotT(\1)\otimes J)^\vee
& = &\hhom_S(\tel \FFhspotT(\1), J^\vee).
\end{array}
$$
Both ``$=$'' symbols use Lemma~\ref{lemma:vee}, while the first uses also
Lemma~\ref{lemma:frob} and Definition~\ref{defn:cech}. The downward map
is obtained by tensoring the homology isomorphism $\tel \FFhspotT(\1) \to
\dirlim t \FFhspotT(\1)$ with $J$, and then taking Matlis duals. That
the downward map is itself a homology isomorphism is an application of
Lemma~\ref{lemma:quism}. The diagram therefore represents a homology
isomorphism $\hhom_S(\CC^\spot_I,J^\vee) \to \mic(\FFlspotT \otimes M)$
by Lemma~\ref{lemma:TelMic}, and since the homology is $\uL^I_\spot(M)$
by Lemma~\ref{lemma:gm}, the result is proved when $\FF_\spot$ is
minimal.
When $\FF_\spot$ isn't minimal, the top end of the downward map is still
homology isomorphic to $(\CC^\spot_I \otimes J)^\vee$ by
Lemma~\ref{lemma:frob}, Proposition~\ref{prop:cech->taylor}, and
Lemma~\ref{lemma:quism}.%
\end{proof}
\begin{remark} \label{rk:finite}
The assumption $M = J^\vee$ for some $J$ in Theorem~\ref{thm:L} is
annoying, but it's unclear how to get rid of it. Anyway, even the
restriction of $\ZZ^n$-finiteness isn't so bad. It allows for a
significant number of infinitely generated modules, including \v Cech
hulls of finitely generated modules and their localizations.
\end{remark}
\begin{example} \label{ex:L}
Using Theorem~\ref{thm:L}, we compute $\uL^I_\spot(S^\vee) =
(H^i\CC^\spot_I)^\vee$. Since $H^i\CC^\spot_I =
(\cech\eext^i(S/I,\omega_S))(\1)$, we conclude that
$$
\uL^I_i(S^\vee) = (\cech\eext^i_S(S/I,\omega_S))^\vee(-\1).
$$
By definition, $\uL^I_i(S^\vee)$ is therefore the \v Cech hull of the
{\em Alexander dual} of $\eext^i(S/I,\omega_S)$ as defined in
\cite{Mil2,Rom}, and more cleanly denoted by
$\cech\eext^i_S(S/I,\omega_S)^\1$. No matter the notation, the vanishing
and nonvanishing of $\uL^I_i(S^\vee)$ follows the same pattern as
$\eext^i_S(S/I,\omega_S)$, since the latter can be recovered from the
former. In particular, $\uL^I_i(S^\vee)$ is nonzero precisely for $d =
\codim(I)$ if and only if $S/I$ is Cohen-Macaulay, in which case
$$
\uL^I_d(S^\vee) = \cech\omega_{S/I}^\1
$$
is the \v Cech hull of the Alexander dual of the canonical module.
Notice that neither an injective hull nor a localization can substitute
for the \v Cech hull here in relating $\uL^I_i(S^\vee)$ to the finitely
generated module $\eext^i_S(S/I,\omega_S)^\1$.
\end{example}
\subsection{Local homology}\label{sub:localhom}%%%%%%%%%%%%%%%%%%%%%%%%%%
Closely related to the left derived functors of completion $\uL^I_\spot$
is another collection of functors $\uH^I_\spot$. In the definition to
come, $\KK^\spot_t$ is the Koszul cochain complex on $m_1^t, \ldots,
m_r^t$. To be precise about homological and $\ZZ^n$-grading,
$\KK^\spot_t$ is the tensor product $\bigotimes_{j=1}^r (S \to S(\deg
m_j^t))$, with $S$ in cohomological degree $0$ and $S(\deg m_j^t)$, which
is generated in $\ZZ^n$-degree $-\!\deg(m_j^t)$, in cohomological
degree~$1$.
\begin{defnlabeled}{{\cite[Definition~2.4]{gm}}} \label{defn:localhom}
Let $I$ be generated by $m_1, \ldots, m_r$. The {\em local homology}
of $M$ at $I$ is
$$
\uH^I_i(M) = H_i\hhom_S(\tel \KK^\spot_t, M).
$$
\end{defnlabeled}
Of course, this definition works for any finitely generated graded ideal
in a commutative ring with any grading when the $m_j$ are arbitrary
homogeneous elements. For instance, in the ungraded setting, Greenlees
and May proved that $H^I_\spot = L^I_\spot$ under mild assumptions on $I$
and the ambient ring. In the present $\ZZ^n$-graded context over~$S$,
this isomorphism of functors is an easy corollary to Theorem~\ref{thm:L}.
\begin{cor} \label{cor:L}
If $M = J^\vee$ for some $J$, then $\uL^I_\spot(M) \cong \uH^I_\spot(M)$.
\end{cor}
\begin{proof}
Start with the homology isomorphism $\tel \KK^\spot_t \to \dirlimt
\KK^\spot_t = \CC^\spot_\TT$ to the \v Cech complex, tensor with $J$, and
take Matlis duals to get a homology isomorphism $(\tel \KK^\spot_t
\otimes J)^\vee \from (\CC^\spot_\TT \otimes J)^\vee$, by
Lemma~\ref{lemma:quism}. Now apply Lemma~\ref{lemma:vee} and
Theorem~\ref{thm:L} with $\FF_\spot = \TT_\spot$ being the Taylor
resolution.
\end{proof}
The convenient feature of Theorem~\ref{thm:L} for this corollary is that
the microscopes there already compute $\uL^I_\spot$, rather than
$\uH^I_\spot$ as in \cite{gm}. Thus, to get $\uH^I_\spot \cong
\uL^I_\spot$, it suffices to compare telescopes, which is easier than
comparing microscopes because direct limits behave better than inverse
limits. This point will resurface in Section~\ref{sec:graded}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$\ZZ^n$-graded Greenlees-May duality}%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:gm}
\subsection{The $\ZZ^n$-graded derived category}\label{sub:DD}%%%%%%%%%%%
The theorem of Greenlees and May is a remarkable adjointness between
local cohomology and the left derived functors of completion. It greatly
generalizes the local duality theorem of Grothendieck, and unifies a
number of other dualities after it has been appropriately sheafified; see
\cite{AJL} for a detailed explanation of these assertions. As with the
ungraded versions in \cite{AJL}, the $\ZZ^n$-graded version
(Theorem~\ref{thm:gm}) is most naturally stated in terms of the {\em
derived category $\DD$} of $\ZZ^n$graded $S$-modules. However, since
hardly any of the machinery is used, all of the pertinent definitions and
facts concerning $\DD$ can be presented from scratch, so this is done in
the next few lines (without actually~defining~$\DD$).
If $\GG$ is a bounded complex of $\ZZ^n$-graded $S$ modules---so $\GG$ is
nonzero in only finitely many (co)homological degrees and has homogeneous
maps of degree $\0$---then $\GG$ represents an object in $\DD$. Every
homogeneous degree $\bb$ homomorphism $\phi$ of such complexes is a
morphism of degree $\bb$ in $\DD$. Moreover, if $\phi$ is a homology
isomorphism then $\phi$ is an {\em isomorphism} (of degree $\bb$) in
$\DD$. The usual functors $\eext$, $H_I^\spot$, and $\uL^I_\spot$ are
replaced by their derived categorical versions $\RR\hhom$, $\RR\Gamma_I$,
and $\LL\uLambdaI$ as follows.
Suppose $\GG$ is a fixed complex. The right derived functor
$\RR\hhom(\GG,-)$ is calculated on a complex $\EE$ by applying
$\hhom(\GG,-)$ to an injective resolution of $\EE$. By definition, such
an injective resolution is a complex $\JJ$ of injectives along with a
homology isomorphism $\EE \to \JJ$. The result of taking
$\hhom(\GG,\JJ)$ is a double complex whose total complex is defined to be
$\RR\hhom(\GG,\EE)$. It is a fact that $\RR\hhom(\GG,\EE)$ doesn't
depend (up to isomorphism in $\DD$) on the choice of injective resolution
$\JJ$.
Alternatively, as with $\eext$, one also gets $\RR\hhom(\GG,\EE)$ by
taking a free resolution of $\GG$ and applying $\hhom(-,\EE)$ to it.
Dually to an injective resolution, a free resolution of $\GG$ is a
homology isomorphism $\FF \to \GG$ from a complex of free modules to
$\GG$. Again, $\hhom(\FF,\EE)$ yields a double complex, whose total
complex $\tot\hhom(\FF,\EE)$ is isomorphic in $\DD$ to
$\RR\hhom(\GG,\EE)$, independent of the free resolution $\FF \to \GG$.
Observe that if either $\GG$ is already a complex of free modules or
$\EE$ is already a complex of injectives, then $\hhom(\GG,\EE)$
represents the right derived functor, and the $\RR$ may be left off. The
relation between $\RR\hhom(\GG,\EE)$ and $\eext$ modules is seen when
$\GG = G$ and $\EE = E$ are both modules: the usual notions of free and
injective resolutions of modules may be regarded as homology isomorphisms
as above, and $\eext^\spot(G,E)$ is the cohomology of $\RR\hhom(G,E)$
(calculated either way).
The discussion above works just as well with $\RR\Gamma_I$ in place of
$\RR\hhom(\GG,-)$ (injective resolution of ``$-$'' required) and with
$\LL\uLambdaI$ in place of $\RR\hhom(-,\EE)$ (free resolution of ``$-$''
required), except that no double complexes appear, so there's no need to
take any total complexes.
\begin{remark} \rm \label{rk:gm}
Proposition~\ref{prop:cech->taylor} says that $\CC^\spot_\FF$ and
$\CC^\spot_{\TT(I)}$ are isomorphic in $\DD$. Lemma~\ref{lemma:gm} says
that $\mic(\FF_\spot^t,M) \cong_\DD \LL\uLambdaI(M)$; in fact, $M$ there
can be replaced by a bounded complex $\EE$, since Lemma~\ref{lemma:quism}
works with bounded $\EE$ in place of $M$. All resolutions of bounded
complexes over $S$ can be chosen bounded, so we restrict to bounded
complexes to avoid technical issues.
\end{remark}
As in the previous remark, the next lemma says that certain objects are
isomorphic in $\DD$. Its proof uses the standard spectral sequence
arguments, and is omitted. The symbols $\hisom$ and $\isomh$ denote
homology isomorphisms, so as to keep track of their directions (and thus
avoid getting too steeped in $\DD$, where $\cong_\DD$ would suffice
for~both).
\begin{lemma} \label{lemma:quisms}
Suppose $\FF \hisom \GG$ and $\EE \hisom \JJ$. Then:
\begin{thmlist}
\item \label{FF}
If $\FF$ is free then $\hhom(\FF,\EE) \hisom \hhom(\FF,\JJ)$.
\item \label{JJ}
If $\JJ$ is injective then $\hhom(\FF,\JJ) \isomh \hhom(\GG,\JJ)$.
\end{thmlist}
\end{lemma}
\subsection{The duality theorem}\label{sub:gm}%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm} \label{thm:gm}
If $\GG$ and $\EE$ are any bounded complexes of $\ZZ^n$-graded
$S$-modules, then
$$
\RR\hhom(\RR\Gamma_I(\GG),\EE) \cong_\DD
\RR\hhom(\GG,\LL\uLambdaI(\EE)).
$$
\end{thm}
\begin{proof}
Let $\EE \to \JJ$ be an injective resolution, $\TT_\spot^{[t]}$ the
Frobenius power of a Taylor resolution for $I$ on squarefree generators,
and $\TT^\spot_{[t]} = \hhom(\TT_\spot^{[t]},\omega_S)$, as usual.
Calculate:
\begin{eqnarray}
\LL\uLambdaI(\EE) \nonumber\label{cong1}
&\cong &\mic(\TT_\spot^{[t]} \otimes \EE) \\\nonumber\label{hisom}
&\hisom&\mic(\TT_\spot^{[t]} \otimes \JJ) \\\nonumber\label{cong2}
&\cong &\hhom(\tel \TT^\spot_{[t]}(\1),\JJ) \\\label{eq:isomh}
&\isomh&\hhom(\CC^\spot_\TT,\JJ).
\end{eqnarray}
The first $\cong$
% Eq.~(\ref{cong1})
is by Lemma~\ref{lemma:gm}, where Remark~\ref{rk:gm} has been used to
justify replacing $M$ by a complex $\EE$; the $\hisom$
% Eq.~(\ref{hisom})
uses the exactness of formation of microscopes when the inverse system is
flat; the second $\cong$
% Eq.~(\ref{cong2})
uses Lemma~\ref{lemma:TelMic}; and Eq.~(\ref{eq:isomh}) is by
Lemma~\ref{lemma:quisms}.\ref{JJ}.
Observe that Eq.~(\ref{eq:isomh}) is a complex of injectives (proof: the
functor of ``$-$'' given by $\hhom(-,\hhom(\hbox{flat},
\hbox{injective})) = \hhom(- \otimes \hbox{flat},\hbox{injective})$ is
exact). Therefore
\begin{eqnarray*}
\RR\hhom(\GG,\LL\uLambdaI(\EE))
&\cong_\DD&\hhom(\GG,\hhom(\CC^\spot_\TT,\JJ))\\
&\cong\ \;&\hhom(\GG\otimes\CC^\spot_\TT,\JJ)\\
&\cong_\DD&\RR\hhom(\GG\otimes\CC^\spot_\TT,\EE),
\end{eqnarray*}
and the result is a consequence of the standard isomorphism $\GG \otimes
\CC^\spot_\TT \hisom \RR\Gamma_I(\GG)$ (which is proved by applying
$\Gamma_I$ to an injective resolution of $\GG$).
\end{proof}
\begin{remark}
The restrictions on $M$ that appear in Theorem~\ref{thm:L} and
Corollary~\ref{cor:L} don't appear in Theorem~\ref{thm:gm} because we
allowed ourselves to replace $M$ by an injective resolution $\JJ$: the
homology isomorphism in Eq.~(\ref{eq:isomh}) follows without the
dualization argument used for Theorem~\ref{thm:L}. The fact that
$\CC^\spot_\TT$ isn't free means we can't \textsl{a~priori} replace $\JJ$
by $\EE$ as in Lemma~\ref{lemma:quisms}.\ref{FF} (or $M$ as in
Theorem~\ref{thm:L}).
\end{remark}
$\ZZ^n$-graded local duality is a special case of Theorem~\ref{thm:gm}.
\begin{cor}[Local duality with monomial support \cite{Mil2}]
\label{cor:duality}
Suppose $\FF_\spot$ is a minimal free resolution of $S/I$, and let
$\CC^\spot_\FF$ be the generalized \v Cech complex determined
by~$\FF_\spot$. For arbitrary $\ZZ^n$-graded modules $M$,
$$
H^i_I(M)^\vee \cong H_i\hhom_S(M,(\CC^\spot_\FF)^\vee).
$$
In particular, if $S/I$ is Cohen-Macaulay of codimension $d$ and
$\omega_{S/I}^\1$ is the Alexander dual of the canonical module
$\omega_{S/I}$ (Example~\ref{ex:L}), then
$$
H_I^i(M)^\vee \cong \eext^{d-i}_S(M,\cech\omega_{S/I}^\1).
$$
\end{cor}
\begin{proof}
Setting $\EE = S^\vee$ and $\GG = M$ in Theorem~\ref{thm:gm} yields the
first displayed equation by Example~\ref{ex:L} (note that $S^\vee$ is
$\ZZ^n$-finite). By Lemma~\ref{lemma:flinj} $(\CC^\spot_\FF)^\vee$ is a
complex of injectives (decreasing in homological degree). Under the
Cohen-Macaulay hypothesis, Example~\ref{ex:L} also says the homology
of $(\CC^\spot_\FF)^\vee$ is $\cech\omega_{S/I}^\1$, in homological
degree~$d$.%
\end{proof}
The usual $\ZZ^n$-graded Grothendieck-Serre local duality theorem
(support taken on $\mm = \$) is a special case of
Corollary~\ref{cor:duality}, by Example~\ref{ex:dualizing}. Another
consequence is the theorem in combinatorial commutative algebra relating
local cohomology of $S$ with support on $I$ to local cohomology of $S/I$
with maximal support. The combinatorial interpretation is as an equality
of $\ZZ^n$-graded Hilbert series whose coefficients are Betti numbers of
certain simplicial complexes related to the {\em Stanley-Reisner
simplicial complex} of $I$.
\begin{cor}[\cite{Mus1,TerLocalCoh,Mil2}] \label{cor:gendual}
For a squarefree monomial ideal $I$,
$$
H^i_I(\omega_S) \cong \cech(H^{n-i}_\mm(S/I)^\vee) \cong
\cech\eext^i_S(S/I,\omega_S).
$$
\end{cor}
\begin{proof}
The second isomorphism is by usual local duality, and the first is the
Matlis dual of Corollary~\ref{cor:duality} with $M = \omega_S = S(-\1)$,
using Example~\ref{ex:L}.
\end{proof}
Judging from the relation between the usual local duality theorem and
Serre duality for projective schemes, it seems clear, in view of the
connections made in \cite{Cox, EMS, Mus2} between local cohomology with
monomial support in polynomial rings and sheaf cohomology on toric
varieties, that graded Greenlees-May duality has something to do with
Serre duality on toric varieties. It would be interesting to see exactly
how the details work out. In particular, what will be the role played by
the {\em cellular flat complexes} related to toric varieties in
\cite[Example~6.6]{Mil2} and \cite[Proposition~5.4, Examples~6.3
and~8.14]{MP}?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Graded noetherian rings}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:graded}
Let $I = \<\alpha_1, \ldots, \alpha_r\>$ be a finitely generated graded
ideal in a commutative ring~$A$ graded by a commutative monoid.
% It is assumed henceforth that there are enough graded injectives (all
% gradings this author has ever seen, including all finitely generated
% submonoids of $\ZZ^n$, admit enough injectives).
The $\ZZ^n$-graded methods of this paper suggest a transparent proof of
GM duality in this general graded case, at least when $A$ is noetherian.
This is not to say that the graded case doesn't follow with care from
known proofs for proregular sequences in arbitrary commutative rings
\cite{gm,AJL}. Rather, the innovation here is the simplicity of the
proof; the consideration of a grading is done ``for the record'', because
it requires no extra effort at this point.
The interesting part about the proofs in Sections~\ref{sec:completion}
and~\ref{sec:gm} is that they work with the direct limits associated to
telescopes, whereas previous methods in \cite{gm,AJL} worked with the
inverse limits associated to microscopes. The direct limit technique is
applied below; it has the advantage that the colimits are exact, and the
adjunction between $\RR\Gamma_I$ and $\LL\LambdaI$ gets reduced directly
to the adjunction between $\hhom$ and $\otimes$.
In general, the adjointness of $H_I^\spot$ and $\uH^I_\spot$ is
essentially by definition, while the identification of the latter with
$\uL^I_\spot$ requires hypotheses. The proof here relies on two facts
about noetherian graded rings. The first, whose standard proof is
omitted, is that the \v Cech complex $C^\spot(\alpha_1,\ldots,\alpha_r)$
is isomorphic in the derived category to $\RR\Gamma_I A$ (apply $-
\otimes C^\spot(\alpha_1,\ldots,\alpha_r)$ to a graded injective
resolution).
% and use
% the method of \cite[p.~130]{BH}).
% the fact that $C^\spot(\alpha_1,\ldots,\alpha_r) \otimes\,$injective is
% acyclic).
The second is contained in the next subsection.
\subsection{General analogue of \v Cech hull colimits}\label{sub:analogue}
The main feature of the \v Cech hull that made it useful earlier was its
expression as a direct limit, resulting in the homology isomorphisms of
Proposition~\ref{prop:cech->taylor}. Although the combinatorial
construction is lost with general gradings, the direct limit still makes
sense, and the homology isomorphism survives, thanks to the next
proposition. Let
$$
\KK_\spot^t = \KK_\spot(\alpha_1^t, \ldots, \alpha_r^t) =
\bigotimes_{j=1}^r \bigl(A(-\deg\alpha_j^t) \stackrel{\alpha_j^t}\too
A\bigr)
$$
be the Koszul chain complex whose tensor factors are in homological
degrees~$1$ and~$0$.
\begin{prop} \label{prop:systems}
Suppose that $A$ is arbitrary, but for every $t \geq 1$ the ideal
$I^{[t]} = \<\alpha_1^t, \ldots, \alpha_r^t\>$ has a resolution by
finite-rank free $A$-modules. Then there is a morphism $\{\rho^t\} :
\{\KK_\spot^t\} \to \{\FF_\spot^t\}$ of inverse systems of complexes
(indexed by $t \geq 1$) in which
\begin{thmlist}
\item \label{1}
$\FF_\spot^t$ is a resolution of $A/I^{[t]}$ by finite rank free modules
for all $t \geq 1$;
\item \label{2}
the maps $\phi^t : \FF_\spot^{t+1} \to \FF_\spot^t$ lift the surjections
$A/I^{[t+1]} \onto A/I^{[t]}$;
\item
the maps on tensor factors given by the identity in homological
degree~$0$ and multiplication by $\alpha_j$ in homological degree~$1$
determine the maps $\kappa^t : \KK_\spot^{t+1} \to \KK_\spot^t$; and
\item \label{4}
each map $\rho^t : \KK_\spot^t \to \FF_\spot^t$ induces an isomorphism on
homology in degree~$0$.
\end{thmlist}
Under these conditions, the transpose direct system $\{\rho_t\} :
\{\FF^\spot_t\} \to \{\KK^\spot_t\}$ obtained by applying $\hhom_A(-,A)$
to $\{\rho^t\}$ induces a homology isomorphism
$$
\Dirlim t\FF^\spot_t \hisom \Dirlim t \KK^\spot_t.
$$
\end{prop}
\begin{proof}
All complexes of free modules appearing in this proof will be assumed to
have finite rank in each homological degree, by the hypothesis on the
ideals~$I^{[t]}$. Conditions~\ref{1} and~\ref{2} can be forced upon any
list $\{\tilde\FF_\spot^t\}$ of resolutions for the quotients
$A/I^{[t]}$. Moreover, the acyclicity of $\tilde\FF_\spot^t$ and the
freeness of $\KK_\spot^t$ imply that maps $\tilde\rho^t : \KK_\spot^t \to
\tilde\FF_\spot^t$ as in condition~\ref{4} exist and are unique up to
homotopy \cite[Porism~2.2.7]{Wei}. Although $\{\tilde\rho^t\}$ may not
\textsl{a~priori} constitute a morphism of inverse systems, the
uniqueness up to homotopy can be used to remedy this, by constructing
$\FF_\spot^t$, $\phi^t$, and $\rho^t$ inductively, starting with a
resolution $\FF_\spot^1$ of $A/I$ and a chain map $\rho^1$ as in
condition~\ref{4}.
Having defined $\FF_\spot^t$ and $\rho^t$, choose a resolution
$\tilde\FF_\spot^{t+1}$ of $A/I^{[t+1]}$ and a map $\tilde\rho^{t+1} :
\KK_\spot^{t+1} \to \tilde\FF_\spot^{t+1}$ as in condition~\ref{4}, and
let $\tilde\phi^t : \tilde\FF_\spot^{t+1} \to \FF_\spot^t$ be any lift of
the surjection $A/I^{[t+1]} \to A/I^{[t]}$. Then
$\tilde\phi^{t+1}\tilde\rho^{t+1}$ is homotopic to $\rho^t\kappa^t$.
Letting a lowered ``$t$'' index denote transpose, so (for instance)
$\rho_t : \FF^\spot_t \to \KK^\spot_t$ is obtained by applying
$\hhom_A(-,A)$ to $\rho^t$, it follows that
$\tilde\rho_{t+1}\tilde\phi_{t+1}$ is homotopic to $\kappa_t\rho_t :
\FF^\spot_t \to \KK^\spot_{t+1}$.
Any choice of homotopy induces a map $\rho_{t+1} : \cyl(\tilde\phi_t) \to
\KK^\spot_{t+1}$ from the {\em mapping cylinder} of $\tilde\phi_t$ to
$\KK^\spot_{t+1}$. (See \cite[Section~1.5]{Wei} for definitions and
generalities concerning mapping cylinders; only the properties of
$\cyl(\tilde\phi_t)$ required here are presented below.) The map
$\rho_t$ is produced essentially by \cite[Exercise~1.5.3]{Wei}, and
satisfies:
\begin{textlist}
\item \label{i}
There is an inclusion $\phi_t : \FF^\spot_t \into \cyl(\tilde\phi_t)$ of
complexes, and $\rho_{t+1}\phi_t = \kappa_t\rho_t$.
\item
$\tilde\FF^\spot_{t+1}$ injects into $\cyl(\tilde\phi_t)$, and the
composite $\tilde\FF^\spot_{t+1} \to \cyl(\tilde\phi_t)
\stackrel{\rho_{t+1}}\too \KK^\spot_{t+1}$ is $\tilde\rho_{t+1}$.
\item \label{iii}
The inclusion $\tilde\FF^\spot_{t+1} \into \cyl(\tilde\phi_t)$ is a homotopy
equivalence, and the composite $\FF^\spot_t \to \cyl(\tilde\phi^t) \to
\tilde\FF^\spot_{t+1}$ is just $\tilde\phi_t$.
\end{textlist}
The transpose $\rho^{t+1} : \KK_\spot^{t+1} \to
\hhom_A(\cyl(\tilde\phi_t),A) =: \FF_\spot^{t+1}$ satisfies the required
conditions, with $\phi^t : \FF_\spot^{t+1} \to \FF_\spot^t$ being the
transpose of the map $\phi_t$ in~(\ref{i}) and~(\ref{iii}).%
The induced map on direct limits is a homology isomorphism because
$$
\dirlimt \eext^i_A(A/I^{[t]},A) = \dirlimt H^i(\FF^\spot_t) =
H^i\dirlimt(\FF^\spot_t) \to H^i\dirlimt(\KK^\spot_t) = H_I^i(A)
$$
is the canonical isomorphism on the local cohomology module $H_I^i(A)$;
note that the graded module $\eext^i_A(A/I^{[t]},A)$ is naturally
isomorphic to the usual ungraded module $\ext^i_A(A/I^{[t]},A)$ because
$A/I^{[t]}$ is finitely presented.
\end{proof}
\begin{remark} \label{rk:systems}
Like the generalized \v Cech complexes of Definition~\ref{defn:cech}, the
object $\dirlimt \FF^\spot_t$ of Proposition~\ref{prop:systems} is a
complex of flat modules. This is because the colimits are taken over
directed systems of complexes of free modules, so complexes of flat
modules result by a theorem of Govorov and Lazard
\cite[Theorem~A6.6]{Eis}.
\end{remark}
It is unclear whether the methods of Proposition~\ref{prop:systems} can
be made to apply when the only assumption is that $(\alpha_1, \ldots,
\alpha_r) \subset A$ is a proregular sequence \cite{gm,AJL}. In general,
under what conditions will $\<\alpha_1^t, \ldots, \alpha_r^t\>$ have
finite Betti numbers as an $A$-module for infinitely many $t \geq 1$?
Perhaps characteristic~$p > 0$ criteria are possible.
\subsection{Graded noetherian Greenlees-May duality}\label{sub:noeth}%%%%
In the following theorem, the derived category statements concern
complexes $\GG$ and $\EE$ that are bounded, for simplicity. The proof is
nearly the same as that of Theorem~\ref{thm:gm}, but is repeated in full
to be self-contained, and for notation's sake.
\begin{thm}[\cite{gm, AJL}] \label{thm:graded}
Let $A$ be a noetherian ring graded by a monoid and $I \subset A$ a
graded ideal. Then $\uH^I_\spot(M) \cong \uL^I_\spot(M)$ for any graded
$A$-module~$M$. If $\GG$ and $\EE$ are bounded complexes of graded
$A$-modules, then
$$
\RR\hhom(\RR\Gamma_I(\GG),\EE) \cong \RR\hhom(\GG,\LL\uLambdaI(\EE)).
$$
\end{thm}
\begin{proof}
Let the ideals $I^{[t]}$, the free resolutions $\FF_\spot^t$, their
transposes $\FF^\spot_t$, the Koszul chain complexes $\KK_\spot^t$, and
their transposes $\KK^\spot_t$ be as in Proposition~\ref{prop:systems}.
If $\EE \to \JJ$ is an injective resolution, then
\begin{eqnarray}
\LL\uLambdaI(\EE) \label{gm3}
&\cong &\mic(\FF_\spot^t \otimes \EE) \\\label{gm4}
&\hisom&\mic(\FF_\spot^t \otimes \JJ) \\\label{gm5}
&\cong &\hhom(\tel \FF^\spot_t ,\JJ) \\\label{gm6}
&\isomh&\hhom(\dirlimt\FF^\spot_t,\JJ) \\\label{eq:systems}
&\isomh&\hhom(\dirlimt\KK^\spot_t,\JJ) \\\label{gm8}
&\hisom&\hhom(\tel\KK^\spot_t ,\JJ) \\\label{gm9}
&\isomh&\hhom(\tel\KK^\spot_t ,\EE).
\end{eqnarray}
Eq.~(\ref{gm3}) follows by replacing $M$ and its free resolution
$\EE_\spot$ in Lemma~\ref{lemma:gm} with a complex $\EE$ and a free
resolution of it (the proof is the same). Eq.~(\ref{gm4}) uses the
exactness of formation of microscopes when the inverse system is flat.
Eq.~(\ref{gm5}) is by Lemma~\ref{lemma:TelMic}. Eq.~(\ref{gm6}) is
because $\JJ$ is injective and $\tel \FF^\spot_t \hisom \dirlimt
\FF^\spot_t$, and similarly for Eq.~(\ref{gm8}).
Proposition~\ref{prop:systems} implies Eq.~(\ref{eq:systems}) (this is
the key step!). Finally, Eq.~(\ref{gm9}) uses the fact that $\tel
\KK^\spot_t$ is free.
When $\EE = M$ is a module, the conclusion of Eq.~(\ref{gm9}) implies
that $\uL^I_\spot(M) \cong \uH^I_\spot(M)$, by definition. To get the
derived categorical statement, observe first that Eq.~(\ref{eq:systems})
is a complex of injectives because the functor given by
$\hhom(-,\hhom(\hbox{flat}, \hbox{injective})) = \hhom(- \otimes
\hbox{flat},\hbox{injective})$ is exact. Therefore
\begin{eqnarray*}
\RR\hhom(\GG,\LL\uLambdaI(\EE))
&\cong&\hhom(\GG , \hhom(\dirlimt\KK^\spot_t , \JJ))\\
&\cong&\hhom(\GG\otimes\dirlimt\KK^\spot_t,\JJ)\\
&\cong&\RR\hhom(\GG\otimes\dirlimt\KK^\spot_t,\EE)\\
&\cong&\RR\hhom(\RR\Gamma_I\GG,\EE),
\end{eqnarray*}
the second isomorphism being at the level of complexes while the rest are
in the derived category. The last isomorphism is the standard
isomorphism for noetherian graded~rings, mentioned at the beginning of
Section~\ref{sec:graded}.%
\end{proof}
The above proof essentially requires $\FF_\spot^t =
\hhom(\hhom(\FF_\spot^t,A),A)$ to be its own double dual (e.g., in
Eq.~(\ref{gm5})), thus using again the finite rank conditions.
\begin{remark}
Looking back at Theorem~\ref{thm:L}, under what circumstances can the
injective resolution $\JJ$ in Eq.~(\ref{gm5}) be replaced by a module
$M$? I.e., when will $\uL^I_i(M)$ be isomorphic to $H_i\hhom_A(\dirlim t
\FF_t^\spot,M)$ (Remark~\ref{rk:systems}) or $\hhom_A(\dirlim t
\KK_t^\spot,M)$, rather than having to replace the colimits by projective
approximations in the form of telescopes?
\end{remark}
%\vspace{-.5ex}
%\end{section}{Introduction}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\footnotesize
%\bibliographystyle{amsalpha}
%\bibliography{biblio}
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