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%%%%%%% Cohen-Macaulay quotients of normal semigroup rings %%%%%%%
%%%%%%% via irreducible resolutions %%%%%%%
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%%%%%%% Ezra Miller %%%%%%%
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%\mbox{}\vspace{3ex}
\title[Cohen--Macaulay quotients via irreducible resolutions]
{Cohen--Macaulay quotients of normal semigroup rings via
irreducible resolutions}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\author[Ezra Miller]{Ezra Miller$^*$}%
%
% \footnote[$^*$]{The author was funded by the National Science
% Foundation.}
%}
\date{10 January 2002}
\begin{abstract}
\noindent
For a radical monomial ideal $I$\/ in a normal semigroup ring~$k[Q]$,
there is a unique minimal \bem{irreducible resolution} $0 \to k[Q]/I \to
\WW^0 \to \WW^1 \to \cdots$\ by modules $\WW^i$ of the form $\bigoplus_j
k[F_{ij}]$, where the $F_{ij}$ are (not necessarily distinct) faces
of~$Q$. That is, $\WW^i$ is a direct sum of quotients of $k[Q]$ by prime
ideals. This paper characterizes Cohen--Macaulay quotients $k[Q]/I$ as
those whose minimal irreducible resolutions are \bem{linear}, meaning
that $\WW^i$ is pure of dimension $\dim(k[Q]/I) - i$ for $i \geq 0$. The
proof exploits a graded ring-theoretic analogue of the Zeeman spectral
sequence~\cite{zeeIII}, thereby also providing a combinatorial
topological version involving no commutative algebra. The
characterization via linear irreducible resolutions reduces to the
Eagon--Reiner theorem \cite{ER} by Alexander duality
% \cite{Mil2,Rom}
\mbox{when $Q = \NN^d$.}
% , so that $k[Q]$ is a polynomial~ring.
\vskip 1ex
\noindent
{{\it 2000 AMS Classification:} 13C14, 14M05, 13D02, 55Txx (primary)
14M25, 13F55 (secondary)}
\end{abstract}
\maketitle
\footnotetext{$^*$The author was funded by the National Science
Foundation.}
\renewcommand{\thefootnote}{\arabic{footnote}}
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{}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%{\tableofcontents}
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\section{Introduction}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:intro}
Let $Q \subseteq \ZZ^d$ be an affine semigroup, which we require
to be saturated except in
% Sections~\ref{sec:zeeman} and~\ref{sec:cm}.
Section~\ref{sec:irr}. We assume everywhere for simplicity that~$Q$
generates~$\ZZ^d$ as a group, and that~$Q$ has trivial unit group. The
real cone $\RR_{\geq 0} Q$
% is the cone over some polytope $\ol Q$ (obtained by intersecting
% $\RR_{\geq 0} Q$ with a transversal hyperplane). Endow the polyhedral
% complex $\ol Q$ with an incidence function $\varepsilon$, and let
% $\Delta \subseteq \ol Q$ be a closed polyhedral subcomplex.
is a polyhedral cell complex; let $\Delta \subseteq \RR_{\geq 0} Q$ be a
closed polyhedral subcomplex. Corresponding to $\Delta$ is the ideal
$I_\Delta$ inside the semigroup ring~$k[Q]$, generated (as a $k$-vector
space) by all monomials in $k[Q]$ not lying on any face of~$\Delta$.
Thus $k[Q]/I_\Delta$ is spanned by monomials lying in~$\Delta$.
This paper has three goals, the latter two only for saturated affine
semigroups ~$Q$:
\begin{itemize}
\item
Define the notion of \bem{irreducible resolution} for $Q$-graded
$k[Q]$-modules.
\item
Introduce the \bem{Zeeman double complex} for $\Delta$.
\item
Characterize Cohen--Macaulay quotients $k[Q]/I_\Delta$ in terms of the
above items.
\end{itemize}
An irreducible resolution (Definition~\ref{d:irr}) of a $\ZZ^d$-graded
$k[Q]$-module is an injective-like resolution, in which the summands are
quotients of $k[Q]$ by irreducible monomial ideals rather than
indecomposable injectives. Minimal irreducible resolutions exist
uniquely up to isomorphism for all $Q$-graded modules~$M$
(Theorem~\ref{t:irr}). When $k[Q]$ is normal and $M = k[Q]/I_\Delta$,
every summand is isomorphic to a semigroup ring~$k[F]$, considered as a
quotient module of~$k[Q]$, for some face $F \in \Delta$
(Corollary~\ref{c:irr}).
The Zeeman double complex $D(\Delta)$ consists of $k[Q]$-modules that are
direct sums of semigroup rings $k[F]$ for faces $F \in \Delta$
(Definition~\ref{d:zeeman}). Its naturally defined differentials come
from the maps in the chain and cochain complexes of~$\Delta$.
Here is the idea behind the Cohen--Macaulay criterion,
Theorem~\ref{t:cm}. Although the total complex of the Zeeman double
complex $D(\Delta)$ is an example of an irreducible resolution
(Theorem~\ref{t:tot}), its large number of summands keeps it far from
being minimal. However, the cancellation afforded by the horizontal
differential of $D(\Delta)$ sometimes causes the resulting vertical
differential (on the horizontal cohomology) to be a minimal irreducible
resolution. This fortuitous cancellation occurs precisely when $\Delta$
is Cohen--Macaulay over~$k$, in which case the horizontal cohomology
occurs in exactly one column of $D(\Delta)$.
Part~\ref{ordinary} of Theorem~\ref{t:cm}, which characterizes the
Cohen--Macaulay property by collapsing at $E^1$ of the ordinary Zeeman
spectral sequence for $\Delta$ (Definition~\ref{d:sequence}), may be of
interest to algebraic or combinatorial topologists. Its statement as
well as its proof
% of Theorem~\ref{t:cm}.\ref{ordinary}
are independent of the surrounding commutative algebra.
The methods involving Zeeman double complexes and irreducible resolutions
should have applications beyond those investigated here; see
Section~\ref{sec:further} for possibilities.
\subsection*{Notational conventions}
The saturation of $Q$~is the intersection with $\ZZ^d$ of the positive
half-spaces defined by primitive integer-valued functionals $\tau_1,
\ldots, \tau_n$ on~$\ZZ^d$. The $i^\th$ \bem{facet} of~$Q$ (for $i =
1,\ldots,n$) is the subset $F_i \subseteq Q$ on which $\tau_i$ vanishes.
More generally, an arbitrary \bem{face} of~$Q$ is defined by the
vanishing of a linear functional on $\ZZ^d$ that is nonnegative on~$Q$
(it is not required that $Q$ be saturated for this to make sense). The
(Laurent) monomial in~$k[\ZZ^d]$ with exponent~$\alpha$ is denoted by
$\xx^\alpha$, although sets of monomials in $k[\ZZ^d]$ are frequently
identified with their exponent sets in~$\ZZ^d$.
All cellular homology and cohomology groups are taken with coefficients
in the field~$k$ unless otherwise stated.
% As usual, we consider the empty set~$\nothing$ as a face of a cell
% complex only in the context of reduced (co)homology.
We work here always with nonreduced (co)homology of the usually unbounded
polyhedral complex~$\Delta$, which corresponds to the reduced
(co)homology of an always bounded transverse hyperplane section
of~$\Delta$, homologically shifted by~$1$.
All modules in this paper, including injective modules, are
$\ZZ^d$-graded unless otherwise stated. Elementary facts regarding the
category of $\ZZ^d$-graded $k[Q]$-modules, especially $\ZZ^d$-graded
injective modules, hulls, and resolutions, can be found in~\cite{GWii},
although the necessary facts will be reviewed as necessary.
%\end{section}{Introduction}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Irreducible resolutions}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:irr}
The definitions and results in this section hold for unsaturated as well
as saturated affine semigroups~$Q$, with the same proofs.
Recall that an ideal $W$ inside of $k[Q]$ is \bem{irreducible} if $W$ can
not be expressed as an intersection of two ideals properly containing~it.
\begin{defn} \label{d:irr}
An \bem{irreducible resolution} $\WW^\spot$ of a $k[Q]$-module~$M$ is an
exact sequence
$$
0 \to M \to \WW^0 \to \WW^1 \to \cdots \qquad\quad \WW^i = \bigoplus_{j
= 1}^{\mu_i} k[Q]/W^{ij}
$$
in which each $W^{ij}$ is an irreducible monomial ideal of~$k[Q]$. The
irreducible resolution is called \bem{minimal} if all the numbers $\mu_i$
are simultaneously minimized (among irreducible resolutions of~$M$), and
\bem{linear} if $\WW^i$ is pure of Krull dimension $\dim(M) - i$ for
all~$i$. (By convention, modules of negative dimension are zero.)%
\end{defn}
The fundamental properties of quotients $\WW := k[Q]/W$ by irreducible
monomial ideals $W$ are inherited from the corresponding properties of
indecomposable injective modules. Recall that each such
\bem{indecomposable injective module} is a vector space $k\{\alpha +
E_F\}$ spanned by the monomials in $\alpha + E_F$, where
\begin{eqnarray} \label{eq:EF}
E_F &=& \{f-a \mid f \in F \hbox{ and } a \in Q\}
\end{eqnarray}
is the ``negative tangent cone'' along the face $F$ of~$Q$. The vector
space $k\{\alpha+E_F\}$ carries an obvious structure of $k[Q]$-module.
In what follows, the $\ZZ^d$-graded injective hull of a $\ZZ^d$-graded
module~$M$ is denoted by~$E(M)$, so that, in particular, $E(k[F]) =
k\{E_F\}$. Elementary behavior of $\ZZ^d$-graded injective hulls can be
found in \cite{GWii}, the main facts required here being:
\begin{periodlist}
\item
$M$ has a minimal injective resolution, unique up to noncanonical
isomorphism.
\item
Any injective resolution of~$M$ is (noncanonically) the direct sum of a
minimal injective resolution and a split exact injective resolution
of\/~$0$.
\item
The minimal injective resolution of $M$ has finitely many indecomposable
summands in each cohomological degree, if $M$ is finitely generated.
\end{periodlist}
Define the \bem{$Q$-graded part} of~$M$ to be the submodule $\bigoplus_{a
\in Q} M_a$ generated by elements whose degrees lie in~$Q$.
\begin{lemma} \label{l:irrideal}
A monomial ideal $W$ is irreducible if and only if $\WW := k[Q]/W$ is the
$Q$-graded part of some indecomposable injective module.
\end{lemma}
\begin{proof}
($\Leftarrow$) The module $k\{\alpha + E_F\}_Q$ is clearly isomorphic
to~$\WW$ for some ideal~$W$. Supposing that $W \neq k[Q]$, we may as
well assume $\alpha \in Q$ by adding an element far inside~$F$, so that
$\xx^\alpha \in \WW$ generates an essential submodule $k\{\alpha + F\}$;
that is, every nonzero submodule of $\WW$ intersects $k\{\alpha + F\}$
nontrivially. Suppose $W = W_1 \cap W_2$. The copy of $k\{\alpha+F\}$
inside $\WW$ must include into $\WW_{\!j}$ for $j = 1$ or $2$. Indeed,
if both induced maps $k\{\alpha+F\} \to \WW_{\!j}$ have nonzero kernels,
then they intersect in a nonzero submodule of $k\{\alpha+F\}$ because
$k[F]$ is a domain. The essentiality of $k\{\alpha+F\} \subseteq \WW$
then forces $\WW \to \WW_{\!j}$ to be an inclusion for some~$j$. We
conclude that $W$ contains this $W_j$, so $W = W_j$ is irreducible.
($\implies$) Let $W$ be an irreducible ideal and $\WW = k[Q]/W$.
Considering the injective hull~$E(\WW) = J_1 \oplus \cdots \oplus J_r$,
the composite $k[Q] \to \WW \to E(\WW)$ has kernel $W = W_1 \cap \cdots
\cap W_r$, where $\WW_{\!j} = (J_j)_Q$. Since $W$ is irreducible, we
must have $W_j = W$ for some~$j$. We conclude that $E(\WW) = J_j$, and
$W = W_j$.
\end{proof}
\begin{lemma} \label{l:isom}
For any finitely generated module~$M$, there exists $\beta \in \ZZ^d$
such that $M_\beta \neq 0$, and for all $\gamma \in \beta + Q$, the
inclusion $M_\gamma \into E(M)_\gamma$ is an isomorphism.
\end{lemma}
\begin{proof}
Suppose that $E(M) = \bigoplus_{\alpha,F} k\{\alpha +
E_F\}^{\mu(\alpha,F)}$, where we assume that $\alpha + E_F \neq \alpha' +
E_{F'}$ whenever $(\alpha,F) \neq (\alpha',F')$. Now fix a pair
$(\alpha,F)$ so that $\alpha + E_F$ is maximal inside~$\ZZ^d$ among all
such subsets appearing in the direct sum. Clearly we may assume $F$ is
maximal among faces of~$Q$ appearing in the direct sum. Pick an element
$f$\/ that lies in the relative interior of~$F$.
By~(\ref{eq:EF}) and the maximality of~$F$, some choice of $r \in \NN$
pushes the $\ZZ^d$-degree $\alpha + r\cdot f \in \alpha + E_F$ outside of
$\alpha' + E_{F'}$ for all $(\alpha',F')$ satisfying $F' \neq F$.
Moreover, the maximality of $(\alpha,F)$ implies that $\alpha + r\cdot f
\not\in \alpha' + E_F$ whenever $\alpha' \neq \alpha$.
The prime ideal $P_F$ satisfying $k[Q]/P_F = k[F]$ is minimal over the
annihilator of~$M$. Therefore, if $M' = (0:_M P_F)$ is the submodule
of~$M$ annihilated by~$P_F$, the composite injection $M' \into M \into
E(M)$ becomes an isomorphism onto its image after homogeneous
localization at~$P_F$---that is, after inverting the monomial $\xx^f$.
It follows that choosing $r \in \NN$ large enough forces isomorphisms
\begin{equation} \label{eq:isom}
M_{\alpha + r\cdot f}\ \: \congto\ \: E(M)_{\alpha + r\cdot f} \ \:
\congto\ \: \Bigl(k\{\alpha+E_F\}^{\mu(\alpha,F)}\Bigr){}_{\alpha +
r\cdot f}\ \: \cong \ \: k^{\mu(\alpha,F)}.
\end{equation}
Setting $\beta = \alpha + r\cdot f$, the multiplication map
$\xx^{\gamma-\beta} : E(M)_\beta \to E(M)_\gamma$ for $\gamma \in \beta +
Q$ is either zero or an isomorphism, because $E(M)$ agrees with
$k\{\alpha+E_F\}^{\mu(\alpha,F)}$ in degrees $\beta$ and $\gamma$ by
construction. The previous sentence holds with $M$ in place of $E(M)$
by~(\ref{eq:isom}), because $M$ is a submodule of $E(M)$.
\end{proof}
Not every module has an irreducible resolution, because being $Q$-graded
is a prerequisite. However, $Q$-gradedness is the only restriction, as
the next theorem shows.
\begin{thm} \label{t:irr}
Let $M = M_Q$ be a finitely generated $Q$-graded module. Then:
\begin{periodlist}
\item \label{unique}
$M$ has a minimal irreducible resolution, unique up to noncanonical
isomorphism.
\item \label{split}
Any irreducible resolution of~$M$ is (noncanonically) the direct sum of a
minimal irreducible resolution and a split exact irreducible resolution
of\/ $0$.
\item \label{finite}
The minimal irreducible resolution of $M$ has finitely many irreducible
summands in each cohomological degree.
\item \label{length}
The minimal irreducible resolution of $M$ has finite length; that is, it
vanishes in all sufficiently high cohomological degrees.
\item \label{Q-graded}
The $Q$-graded part of any injective resolution of~$M$ is an irreducible
resolution.
\item \label{injres}
Every irreducible resolution of~$M$ is the $Q$-graded part of an
injective resolution.
\end{periodlist}
\end{thm}
\begin{proof}
Lemma~\ref{l:irrideal} implies part~\ref{Q-graded}. Parts%
% \ref{unique}, \ref{split}, and~\ref{finite}
~\ref{unique}--\ref{finite} therefore follow from part~\ref{injres} and
the corresponding facts about $\ZZ^d$-graded injective resolutions before
Lemma~\ref{l:irrideal}. Part~\ref{length}, on the other hand, is false
for injective resolutions whenever $k[Q]$ is not isomorphic to a
polynomial ring, so we prove it separately at the end.
Focusing now on part~\ref{injres}, let $\WW^\spot$ be an irreducible
resolution of~$M$, and set $J^0 = E(\WW^0)$.
% Would use $\cech J^\spot$, if I didn't have to introduce things
The inclusion $M \into J^0$ has $Q$-graded part $M \into \WW^0$ by
Lemma~\ref{l:irrideal}. Making use of the defining property of injective
modules, extend the composite inclusion $\WW^0/M \into \WW^1 \into
E(\WW^1)$ to a map $J^0/M \to E(\WW^1)$, and let $K^0$ be the kernel.
Then $K^0$ has zero $Q$-graded part because $\WW^0/M \into \WW^1$ is a
monomorphism. The injective hull $K^0 \into E(K^0)$ therefore has zero
$Q$-graded part. Extending $K^0 \into E(K^0)$ to a map $J^0/M \to
E(K^0)$ yields an injection $J^0/M \to J^1 := E(K^0) \oplus E(\WW^1)$
whose $Q$-graded part is $\WW^0/M \into \WW^1$. Replacing~$M$,
$0$~and~$1$ by ${\rm image}(\WW^{i-1} \to \WW^i)$, $i$~and~$i+1$ in this
discussion produces the desired injective resolution by induction.%
Finally, for the length-finiteness in part~\ref{length}, consider the set
$V(M)$ of degrees $a \in Q$ such that $M_b$ vanishes for all $b \in a+Q$.
The vector space $k\{V(M)\}$ is naturally an ideal in~$k[Q]$.
Lemma~\ref{l:isom} implies that $V(M) \subsetneq V(\WW/M)$ whenever $\WW$
is the $Q$-graded part of an injective hull of~$M$ and $M \neq 0$ (that
is, $V(M) \neq Q$). The noetherianity of $k[Q]$ plus this strict
containment force the sequence of ideals
$$
k\{V(M)\}\ \subseteq\ k\{V(\WW^0/M)\}\ \subseteq\ k\{V(\WW^1/{\rm
image}(\WW^0))\}\ \subseteq\ \cdots
$$
to stabilize at the unit ideal of~$k[Q]$ after finitely many steps.%
\end{proof}
\begin{remark} \label{rk:yan1}
{}From another perspective, Theorem~\ref{t:irr} says that the category of
$Q$-graded modules has enough injectives, with the resulting ``minimal
injective resolutions'' being unique and finite. Lemma~\ref{l:irrideal}
says that the indecomposable injectives in the category of $Q$-graded
modules have the form $k[Q]/W$ for irreducible ideals $W$.%
\end{remark}
Examples of irreducible resolutions include
% Proposition~\ref{p:tot},
Theorem~\ref{t:tot}, below, as well as the proof of Lemma~\ref{l:vert},
which contains the irreducible resolution of the canonical module of
$k[F]$ in~(\ref{eq:F}). In general, any example of an injective
resolution of any $\ZZ^d$-graded module yields an irreducible resolution
of its $Q$-graded part, although the indecomposable injective summands
with zero $Q$-graded part get erased. In particular, the ``cellular
injective resolutions'' of \cite{Mil2} become what should be called
``cellular irreducible resolutions'' here.
%\end{section}{Irreducible resolutions}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Zeeman double complex}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:zeeman}
This section introduces the Zeeman double complex and its resulting
spectral sequences for saturated affine semigroups~$Q$. The total
complex of the Zeeman double complex in
% Proposition~\ref{p:tot}
Theorem~\ref{t:tot} provides a natural but generally nonminimal
irreducible resolution for $k[Q]/I_\Delta$.
Assume for this section that~$Q$ is saturated. For each face $G \in
\Delta$, let $k[G]$ be the affine semigroup ring for~$G$, considered as a
quotient of~$k[Q]$, and denote by~$e_G$ the canonical generator of $k[G]$
in $\ZZ^d$-graded degree~$\0$. Also, for each face $F \in \Delta$, let
$k F$ be a $1$-dimensional $k$-vector space spanned by~$F$ in
$\ZZ^d$-graded degree~$\0$.
\begin{defn} \label{d:zeeman}
Consider the $k[Q]$-module $D(\Delta) = \bigoplus_{F \supseteq G} kF
\otimes_k k[G]$ generated by
$$
\{F \otimes_k e_G \mid F,G \in \Delta \hbox{ and }F \supseteq G\},
$$
with $k[Q]$ acting only on the right hand factors~$k[G]$. Doubly index
the generators so that $D(\Delta)_{pq}$ is generated by
$$
\{F \otimes e_G \mid p = \dim F \hbox{ and } -q = \dim G\},
$$
and hence $\{\0\} \otimes e_{\{\0\}} \in D(\Delta)_{00}$, with the rest
of the double complex in the fourth quadrant. Now define the \bem{Zeeman
double complex} of $\ZZ^d$-graded $k[Q]$-modules to be $D(\Delta)$, with
vertical differential $\partial$ and horizontal differential $\delta$ as
in the diagram:
$$
\partial e_G\ =\hspace{-1ex} \sum_\twoline{G' \subset G}{\text{is a
facet}} \hspace{-1ex}\varepsilon(G',G) e_{G'}
\qquad\!
\begin{array}{@{}ccc@{}}
F \otimes \makebox[0pt][l]{$\partial e_G$}\phantom{e_G}
\\[3pt] \partial\uparrow\phantom{F}
\\[-5pt]F\otimes e_G & \stackrel{\textstyle \delta}{\fillrightmap}
& \delta F \otimes e_G
\end{array}
\quad\
(-1)^q \delta F\ =\hspace{-1ex} \sum_\twoline{F \subset F'}{\text{is a
facet}} \hspace{-1ex}\varepsilon(F,F') F',
$$
where the signs $\varepsilon(G',G)$ and $\varepsilon(F,F')$ come from a
fixed incidence function on~$\Delta$ (defined by relative orientations of
the polyhedral faces, say).
% This $\delta$ is nothing more than the coboundary map on the simplex
% $F$, homogenized via the variables $z_i$ to be compatible with the
% $\ZZ^n$-grading: $\deg z_i e_{G \cup i} = \deg e_G = -G$.
\end{defn}
For each fixed $G$, the elements $F \otimes e_G$ generate a summand of
$D(\Delta)$ closed under the horizontal differential $\delta$. Taking
the sum over all~$G$ yields the horizontal complex
$$
(D(\Delta), \delta) = \bigoplus_{G \in \Delta} C^\spot(\Delta_G)
\otimes k[G], \quad {\rm where} \quad \Delta_G = \{F \in \Delta \mid F
\supseteq G\}
$$
is the \bem{part of $\Delta$ above}~$G$. It is straightforward to verify
that $C^\spot(\Delta_G)$ is isomorphic to the reduced cochain complex
$\wt C^\spot \link(G,\Delta)$ of the link of~$G$ in~$\Delta$ (also known
as the vertex figure of~$G$ in~$\Delta$), but with $\nothing$ in
homological degree~$\dim G$ instead of~$-1$:
\begin{eqnarray*}
C^{i+1+\dim G}(\Delta_G) &\cong& \wt C^{i} \link(G,\Delta).
\end{eqnarray*}
The cohomology $H^i(C^\spot(\Delta_G))$ is also called the \bem{local
cohomology~$H^i_G(\Delta)$} of $\Delta$ near~$G$. Since the complex
$C^\spot(\Delta_G)$ is naturally a subcomplex of $C^\spot(\Delta_{G'})$
whenever $G' \subseteq G$, the natural restriction maps $H^i_G(\Delta)
\to H^i_{G'}(\Delta)$ make local cohomology into a sheaf on $\Delta$.
The following is immediate from the above discussion.
\begin{lemma} \label{l:hor}
In column~$p$, the vertical complex $(H_\delta D(\Delta), \partial)$ of
$k[Q]$-modules has $\bigoplus_{\dim G = q} H^p_G(\Delta) \otimes_k k[G]$
in cohomological degree~$-q$. The vertical differential $\partial$ is
comprised of the natural maps $H^p_G(\Delta) \otimes e_G \to
H^p_{G'}(\Delta) \otimes \varepsilon(G',G) e_{G'}$ for \mbox{facets $G'$
of~$G$.}
\end{lemma}
We'll need to know the vertical cohomology $H_\partial D(\Delta)$ of
$D(\Delta)$, too.
\begin{lemma} \label{l:vert}
$H_\partial D(\Delta) = \bigoplus_{F \in \Delta} \omega_{k[F]}$, where
$\omega_{k[F]}$ is the canonical module of $k[F]$, and each summand
$\omega_{k[F]}$ sits along the diagonal in bidegree $(p,q) = (\dim F,
-\dim F)$.
\end{lemma}
\begin{proof}
Collecting the terms with fixed~$F$ yields the tensor product of $kF$
with
\begin{equation} \label{eq:F}
0 \to k[F] \to \bigoplus_{\facets\ F'\ {\rm of}\ F} k[F'] \to \cdots
\to \bigoplus_{\rays\ v \in F} k[v] \to k \to 0.
\end{equation}
The $\ZZ^d$-graded degree~$a$ part of this complex is zero unless $a \in
F$, in which case we get the local homology complex $C_\spot(F_{F''})$ of
$F$ near the face $F''$ containing~$a$ in its relative interior. Such
local homology is zero unless $F'' = F$. Therefore, the only homology of
(\ref{eq:F}) is the canonical module $\omega_{k[F]}$, being the kernel of
the first map.%
\end{proof}
\begin{thm} \label{t:tot} %\label{p:tot}
The total complex $\tot D(\Delta)$ of the Zeeman double complex is an
irreducible resolution of $k[Q]/I_\Delta$.
\end{thm}
\begin{proof}
The spectral sequence obtained by first taking vertical cohomology of
$D(\Delta)$ has $H_\partial D(\Delta) = E^1 = E^\infty$ by
Lemma~\ref{l:vert}. The same lemma implies that the cohomology of $\tot
D(\Delta)$ is zero except in degree $p + q = 0$, and that the nonzero
cohomology has a filtration whose associated graded module is
$\bigoplus_{F \in \Delta} \omega_{k[F]}$. The Hilbert series of this
cohomology module equals that of~$k[Q]/I_\Delta$.
On the other hand, the map $\phi: k[Q] \to D(\Delta)$ sending $1 \mapsto
\sum_{F \in \Delta} \epsilon_F F \otimes e_F$ has kernel~$I_\Delta$, for
any choice of signs $\epsilon_F = \pm 1$. The image of~$\phi$ is thus
isomorphic to $k[Q]/I_\Delta$. Choosing the signs
$$
\epsilon_F = (-1)^{\dim F(\dim F + 1)/2} =
\left\{
\begin{array}{@{}r@{\ \ {\rm if}\ }l@{}}
-1 & \dim F \equiv 1,2 \mod 4\\
1 & \dim F \equiv 0,3 \mod 4
\end{array}
\right.
$$
forces $(\delta + \partial)(\sum_{F \in \Delta} \epsilon_F F \otimes e_F)
= 0$, thanks to the factor $(-1)^q$ in the definition~of~$\delta$. By
Hilbert series considerations, the image of~$\phi$ equals the kernel of
$\delta + \partial$.%
\end{proof}
\begin{cor} \label{c:irr}
Every summand in the minimal irreducible resolution of the quotient
$k[Q]/I_\Delta$ by a radical monomial ideal $I_\Delta$ is isomorphic to
$k[F]$ for some face $F$ of~$Q$.
\end{cor}
\begin{proof}
Every summand in the total Zeeman complex of
% Proposition~\ref{p:tot}
Theorem~\ref{t:tot} has the desired form. Now apply
Theorem~\ref{t:irr}.\ref{split}.
\end{proof}
The spectral sequence in the proof of
% Proposition~\ref{p:tot}
Theorem~\ref{t:tot} always converges rather early, at~$E^1$. The other
spectral sequence, however, obtained by first taking the horizontal
cohomology~$H_\delta$, may be highly nontrivial.
\begin{defn} \label{d:sequence}
The \bem{$\ZZ^d$-graded Zeeman spectral sequence} for the polyhedral
complex $\Delta$ is the spectral sequence $\ZZ E^\spot_{pq}(\Delta)$ on
the double complex $D(\Delta)$ obtained by taking horizontal homology
first, so $\ZZ E^2_{pq}(\Delta) = H_\partial H_\delta D(\Delta)$.
% (The $E^1$ term is given in Lemma~\ref{l:hor}),
The \bem{ordinary Zeeman spectral sequence} for $\Delta$ is the
$\ZZ^d$-graded degree~$\0$ piece $\mathit{ZE}^\spot_{pq}(\Delta) = \ZZ
E^\spot_{pq}(\Delta)_\0$.
\end{defn}
%\end{section}{Zeeman double complex}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Characterization of Cohen--Macaulay quotients}%%%%%%%%%%%%%%%%%%
\label{sec:cm}
This section contains a characterization of Cohen--Macaulayness in terms
of irreducible resolutions coming from the Zeeman double complex
$D(\Delta)$. As in the previous section, assume that $Q$ is saturated.
\begin{defn} \label{d:cm}
The polyhedral complex $\Delta$ is \bem{Cohen--Macaulay over $k$} if the
local cohomology over $k$ of $\Delta$ near every face $G \in \Delta$
satisfies $H^i_G(\Delta) = 0$ for $i < \dim \Delta$.
\end{defn}
\begin{thm} \label{t:cm}
Let $I = I_\Delta$ be a radical monomial ideal. The following are
equivalent.
\begin{periodlist}
\item \label{cm/k}
$\Delta$ is Cohen--Macaulay over $k$.
\item \label{ordinary}
The only nonzero vector spaces $\mathit{ZE}^1_{pq}(\Delta)$ lie in
column~$p = \dim(\Delta)$.
\item \label{E1}
The complex $\ZZ E^1(\Delta)$ is a minimal linear irreducible resolution
of~$k[Q]/I$.
\item \label{lin}
$k[Q]/I$ has a linear irreducible resolution.
\item \label{cm}
$k[Q]/I$ is a Cohen--Macaulay ring.
\end{periodlist}
\end{thm}
\begin{proof}
\ref{cm/k} $\Leftrightarrow$ \ref{ordinary}: The $\ZZ^d$-degree $\0$ part
of Lemma~\ref{l:hor} says that $\mathit{ZE}^1$ has $\bigoplus_{\dim G =
q} H^p_G(\Delta)$ in cohomological degree~$-q$. The equivalence is now
immediate from Definition~\ref{d:cm}.
\ref{cm/k} $\implies$ \ref{E1}: The $E^1$ term in question is the complex
$H_\delta D(\Delta)$, with the differential $\partial$ in
Lemma~\ref{l:hor}. That lemma together with Definition~\ref{d:cm}
implies that the horizontal cohomology $H_\delta D(\Delta)$ has one
column (indexed by $\dim \Delta$), that must therefore be a resolution of
something having the same Hilbert series as $k[Q]/I_\Delta$ by
% Proposition~\ref{p:tot}.
Theorem~\ref{t:tot}. Set $n = \dim \Delta$. Since $H^n_F(\Delta) = kF$
for facets $F \in \Delta$, it is enough to check that the diagonal
embedding $k[Q]/I_\Delta \into \bigoplus_{\facets\ F \in \Delta} k[F]$ is
contained in the kernel of the first map of~$(H_\delta D(\Delta),
\partial)$. If $\dim G = n-1$ for some face $G \in \Delta$, then
\begin{eqnarray*}
H^n_G(\Delta) &=& ({\textstyle\bigoplus kF)/\<\textstyle \sum
\varepsilon(G,F) F\>},
\end{eqnarray*}
both sums being over all facets $F \in \Delta$ containing~$G$. Now
calculate
$$
\partial\Bigl(\sum_{\dim F = n\,} F \otimes e_F\Bigr)\ =\!\sum_{\dim F
= n} \sum_{F \supset G} F \otimes \varepsilon(G,F) e_G\ =\!\!\!\!
\sum_{\dim G = n-1} \Bigl(\sum_{F \supset G} \varepsilon(G,F) F\Bigr)
\otimes e_G\ =\ 0.
$$
\ref{E1} $\implies$ \ref{lin}: Trivial.
\ref{lin} $\implies$ \ref{cm}: If $M$ is any module having a linear
irreducible resolution $\WW^\spot$ in which each summand of $\WW^i$ is
isomorphic to $k[F]$ for some face~$F$, such as $M = k[Q]/I$ as in
Corollary~\ref{c:irr}, then $M$ is Cohen--Macaulay. This can be seen by
induction on $\dim(M)$ via the long exact sequence for local cohomology
$H^i_\mm$, where $\mm$ is the graded maximal ideal. The induction
requires the modules $\WW^i$ to be Cohen--Macaulay themselves, which
holds
% by Hochster's theorem
because $Q$ and hence all of its faces~$F$ are saturated.
\ref{cm} $\implies$ \ref{cm/k}: $k[Q]/I$ being Cohen--Macaulay implies
that $\eext^i_{k[Q]}(M,\omega_{k[Q]})$ is zero for $i \neq d - n$, where
$n = \dim \Delta$. In particular, if $\Omega^\spot$ is the $Q$-graded
part of the minimal injective resolution of $\omega_{k[Q]}$, then the
$i^\th$ cohomology of $\hhom(k[Q]/I,\Omega^\spot)$ is zero unless $i = d
- n$. The complex $\Omega^\spot$ is the linear irreducible resolution of
$\omega_{k[Q]}$ in which each quotient $k[F]$ for $F \in \RR_{\geq 0} Q$
appears precisely once; see~(\ref{eq:F}).
% in the proof of Lemma~\ref{l:vert}).
Since $\hom(k[Q]/I,k[F]) = k[F]$ if $F \in \Delta$ and zero otherwise,
$\hhom(k[Q]/I,\Omega^\spot)$ is
$$
0 \to \bigoplus_\twoline{F \in \Delta}{\dim F = n} k[F] \to \cdots \to
\bigoplus_\twoline{F \in \Delta}{\dim F = \ell} k[F] \to \cdots \to k
\to 0.
$$
If $a \in Q$ is in the relative interior of $G \in \Delta$, then the
$\ZZ^d$-graded degree~$a$ component of this complex is the homological
shift of $C^\spot(\Delta_G)$ whose $i^\th$ cohomology is
$H^{d-i}_G(\Delta)$.%
\end{proof}
\begin{remark}
Note the interaction of Theorem~\ref{t:cm} with the characteristic of
$k$: the horizontal cohomology of the Zeeman double complex $D(\Delta)$
can depend on $\char(k)$, just as the other parts of the theorem can.%
\end{remark}
When the semigroup $Q$ is $\NN^d$, so that $k[Q]$ is just the polynomial
ring in $d$~variables $z_1,\ldots,z_d$ over~$k$, the polyhedral complex
$\Delta$ becomes a simplicial complex. Thinking of $\Delta$ as an order
ideal in the lattice $2^{[d]}$ of subsets of $[d] := \{1,\ldots,d\}$, the
\bem{Alexander dual} simplicial complex $\dv$ is the complement
of $\Delta$ in $2^{[d]}$, but with the partial order reversed. Another
way to say this is that $\dv = \{[d] \minus F \mid F \not\in
\Delta\}$.
Theorem~\ref{t:cm} can be thought of as the extension to arbitrary normal
semigroup rings of the Eagon--Reiner theorem \cite{ER}, which concerns
the case $Q = \NN^d$, via the Alexander duality functors defined in
\cite{Mil2,Rom}. To see how, recall that a $\ZZ$-graded
$k[\NN^d]$-module is said to have \bem{linear free resolution} if its
minimal $\ZZ$-graded free resolution over $k[z_1, \ldots, z_d]$ can be
written using matrices filled with linear forms.
\begin{cor}[Eagon--Reiner] \label{c:ER}
If $\Delta$ is a simplicial complex on $\{1, \ldots, d\}$, then $\Delta$
is Cohen--Macaulay if and only if $I_{\dv}$ has linear free
resolution.
\end{cor}
\begin{proof}
The minimal free resolution of $I_{\dv}$ is the functorial Alexander dual
(see \cite[Definition~1.9]{Rom} or \cite[Theorem~2.6]{Mil2} with
${\mathbf a} = \1$) of the minimal irreducible resolution of
$k[\NN^d]/I_\Delta$ guaranteed by Theorem~\ref{t:cm}. Linearity of the
irreducible resolution translates directly into linearity of the free
resolution of~$I_\dv$.
\end{proof}
\begin{remark} \label{rk:yan2}
If $M$ is a `squarefree module' in the sense of Yanagawa \cite{YanPoset},
then a minimal irreducible resolution of~$M$ is a minimal injective
resolution of~$M$ in the category of squarefree modules. The equivalence
of parts \ref{lin} and~\ref{cm} in Theorem~\ref{t:cm} therefore holds
with an arbitrary squarefree module in place of $k[Q]/I$, by
\cite[Corollary~4.17]{YanPoset}. More generally, the analogue of Terai's
theorem~\cite{TerER}, which measures the difference between depth and
dimension, holds for squarefree modules by
\cite[Theorem~4.15]{YanPoset}.%
\end{remark}
\begin{remark}
Is there a generalization of Theorem~\ref{t:cm} to the sequential
Cohen--Macaulay case that works for arbitrary saturated semigroups,
analogous and Alexander dual to the generalization \cite{HRW} of
Corollary~\ref{c:ER}? Probably; and if so, it will likely say that the
ordinary and $\ZZ^d$-graded Zeeman spectral sequences collapse at $E^2$
(i.e.\ all differentials in $E^{\geq 3}$ vanish).
\end{remark}
% \begin{cor} \label{cor:hilb}
% Use Theorem~\ref{t:cm} to give a formula for the Hilbert series of a
% reduced Cohen--Macaulay quotient. Does it agree with any known
% formulas?
% \end{cor}
\begin{remark}
The Alexander dual of the complex $\ZZ E^1(\Delta) = (H_\delta D(\Delta),
\partial)$, which provides a linear free resolution of $I_\dv$ in the
Cohen--Macaulay case, also provides the ``linear part'' of the free
resolution of $I_\dv$ when $\Delta$ is arbitrary \cite{RW}. It is
possible to give an apropos proof of this fact using the Alexander dual
of the Zeeman spectral sequence for a Stanley--Reisner ring along with an
argument due to J.~Eagon \cite{Eag} concerning how to make spectral
sequences into minimal free resolutions.%
\end{remark}
%\end{section}{Characterization of Cohen--Macaulay quotients}%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Remarks and further directions}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:further}
Zeeman's original spectral sequence appears verbatim as the ordinary
Zeeman spectral sequence $\mathit{ZE}^\spot_{pq}$ in
Definition~\ref{d:sequence}, with $Q = \NN^d$. Zeeman used his double
complex and spectral sequence to provide an extension of Poincar\'e
duality for singular triangulated topological spaces
\cite{zeeI,zeeII,zeeIII}. When the topological space is a manifold, of
course, usual Poincar\'e duality results. In the present context,
Zeeman's version of the Poincar\'e duality isomorphism should glue two
complexes of irreducible quotients of $k[\NN^d]$ together to form the
minimal irreducible resolution for the Stanley--Reisner ring of any
Buchsbaum simplicial complex---these simplicial complexes behave much
like manifolds. This gluing procedure should work also for the more
general Buchsbaum polyhedral complexes~$\Delta$ obtained by considering
arbitrary saturated affine semigroups~$Q$.
% We believe that Zeeman's work on simplicial homology can have further
% significant applications to resolutions of monomial ideals.
Theorem~\ref{t:cm} is likely capable of providing a combinatorial
construction of the ``canonical \v Cech complex'' for $I_\dv$
\cite{Mil2,YanMonSup} when $\Delta$ is Cohen--Macaulay, or even Buchsbaum
(if the previous paragraph works).
% This complex is the ``minimal'' complex of localizations of
% $k[Q^\star]$ whose cohomology is $H^i_{I_\dv}(k[Q^\star])$.
Although $\dv$ has only been defined a~priori for simplicial complexes
$\Delta$, when $Q = \NN^d$, the definition of functorial squarefree
Alexander duality extends easily to the case of arbitrary saturated
semigroups \cite[Remark~4.18]{YanPoset}. The catch is that $I_\dv$ is
not an ideal in~$k[Q]$, but rather an ideal in the semigroup ring
$k[Q^\star]$ for the cone $Q^\star$ dual to~$Q$. Combinatorially
speaking, the face poset of~$Q$ is not usually self-dual, as it is when
$Q = \NN^d$, so the process of ``reversing the partial order''
geometrically forces the switch to $Q^\star$. The functorial part of
Alexander duality follows the same pattern as the case $Q = \NN^d$:
quotients $k[F]$ of $k[Q]$ are dual to prime ideals $P_{F^\star}$ inside
$k[Q^\star]$, where $F^\star$ is the face of $Q^\star$ dual to~$F$. See
also \cite[Section~6]{YanMonSup}.
In general, irreducible resolutions---and perhaps other resolutions
% of this sort
by structure sheaves of subschemes---can be useful for computing the
$K$-homology classes of reduced subschemes that are unions of
transverally intersecting components. When the ambient scheme is
regular, this method is an alternative to calculating free resolutions,
which produce $K$-\emph{co}homology classes. In particular, this holds
for subspace arrangements in projective spaces. This philosophy
underlies the application of irreducible resolutions in
\cite[Appendix~A.3]{grobGeom} to the definition of ``multidegrees''.
Note that when $Q \not\cong \NN^d$, irreducible resolutions are the only
finite resolutions to be had: free and injective resolutions of finitely
generated modules rarely terminate. In particular, an understanding of
the Hilbert series of irreducible quotients of $k[Q]$---a polyhedral
problem---would give rise to formulae for Hilbert series of $Q$-graded
modules. Similarly, algorithmic computations with irreducible
resolutions can allow explicit computation of injective resolutions,
local cohomology, and perhaps other homological invariants in the
$\ZZ^d$-graded setting over semigroup rings.
% These connections between irreducible resolutions, Buchsbaum cell
% complexes, and local cohomology will be pursued elsewhere.
\subsection*{Acknowledgements.}
The author is indebted to John Greenlees for directing him to Zeeman's
work, and Vic Reiner
% The idea to work out a linearity result for resolutions over semigroup
% rings arose in discussions with Vic Reiner.
for inspiring discussions. Kohji Yanagawa kindly pointed out
Remarks~\ref{rk:yan1} and~\ref{rk:yan2}.
%\end{section}{Further directions}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\footnotesize%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibliographystyle{amsalpha}
%\bibliography{biblio}
\def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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\end{thebibliography}
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\vbox{\footnotesize \baselineskip 10pt
\bigskip\noindent
% Add your address here.
%
% e-mail: {\tt reiner@math.umn.edu}
%
% \smallskip\noindent
Applied Mathematics, 2-363A, Massachusetts Institute of Technology, 77
Massachusetts Avenue,
Cambridge, MA 02139 \qquad email: {\tt ezra@math.mit.edu}}
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