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%Intended for Kohji, \qquad \qquad \qquad cc: David Helm
\begin{center}\bf Local cohomology has finite Bass numbers over
Stanley--Reisner rings
\end{center}
\medskip
\begin{center}
David Helm and Ezra Miller
\end{center}
\bigskip
\medskip
\noindent
Here are some thoughts about finiteness of Bass numbers of local
cohomology modules over Stanley-Reisner rings with support in
$\NN^n$-graded ideals. Our \v Cech hull machinery, for which we prove a
change of rings statement, is directly and easily applicable to
% , and we (David and I) proved
the following theorem. (To get the basic idea of the proof, read
Proposition~\ref{prop:finite} first.) All modules and functors in what
follows are $\ZZ^n$-graded.
\begin{thm} \label{thm:bass}
Let $R = S/J$ be the quotient of the polynomial ring $S$ by the monomial
ideal $J$ (not necessarily squarefree). If $I \subset R$ is an ideal and
$M$ is a finitely generated $R$-module, then the Bass numbers (over $R$)
of $H^i_I(M)$ are finite.
\end{thm}
\proof The local cohomology modules $H^i_I(M)$ are modules over both $R$
and $S$, since they can be calculated using the \v Cech complex. It
follows from Corollary~5.3 in [HM] along with Lemma~\ref{lemma:fixed},
below, that $H^i_I(M)(-\alpha)$ is fixed by $\cech_S$ for some $\alpha
\in \ZZ^n$. Proposition~5.4 in [HM] implies that this
$H^i_I(M)(-\alpha)$ has finitely generated $\NN^n$-graded part. The
theorem has been recast as a special case of
Proposition~\ref{prop:finite}. \endproof
\medskip
\noindent
Alternatively, one can simply prove, using the methods for semigroup
rings in [HM], that $H^i_I(M)(-\alpha)$ is $R$-straight for some
$\alpha$, and proceed from there.
\medskip
Let $Q$ be a semigroup and $\cech$ the \v Cech hull functor over some
$Q^{\rm gp}$-graded ring~[HM]. For a module $M$ over this ring, let $M_Q
= \bigoplus_{q \in Q}M_q$ be the $Q$-graded part of~$M$.
\begin{lemma} \label{lemma:fixed}
Suppose $A$ is fixed by $\cech$. Then $A(-a)$ is fixed by $\cech$ for
all $a \in Q$. Equivalently, $(\cech A')(-a) = \cech((\cech A')(-a))$ for
all $a \in Q$ and all modules $A'$.
\end{lemma}
\proof The two claims are equivalent because a module is fixed by $\cech$
if and only if it is the \v Cech hull of something. We prove the second,
using uniqueness of adjoints, for which it suffices to verify that
$(\cech(\dash))(-a)$ and $\cech(\cech(\dash)(-a))$ are right adjoints to
isomorphic functors. But their adjoints are, respectively,
$((\dash)(a))_Q$ and $((\dash)_Q(a))_Q$, which are obviously
isomorphic as functors. \endproof
\medskip
% We will need to use the Cech hull functor over several rings,
Let $\cech_S$ be the \v Cech hull over the polynomial ring $S$ and
$\cech_R$ the \v Cech hull over $R = S/J$.
%, and simply $\cech$ when working in a general context. Also, we
Use $\EE_S$ and $\EE_R$ to denote $\ZZ^n$-graded injective hulls over $R$
and~$S$.
\begin{lemma} \label{lemma:overR}
If $A$ is any $R$-module then $\cech_R(A) = \hhom_S(R,\cech_S(A))$. In
particular, $\hhom_S(R,\EE_S(R/\pp)) = \EE_R(R/\pp)$ for all primes $\pp$
of $R$.
\end{lemma}
\proof $\cech_R$ is the right adjoint to taking $\NN^n$-graded parts,
denoted $(\dash)_{\succeq 0}$. In other words, $\hom_R(B_{\succeq 0},A)
= \hom_R(B, \cech_R(A))$, where $\hom(\dash,\dash)$ denotes homogeneous
homomorphsims {\em of degree 0}. We have for all $R$-modules $A$ and~$B$,
$$
\begin{array}{@{}rcl@{\ }l@{}}
\hom_R(B,\cech_RA)
&=& \hom_R(B_{\succeq 0}, A)
\\&=& \hom_S(B_{\succeq 0}, A) \ \ \hbox{since the $S$-action factors
through $R$}
\\&=& \hom_S(B, \cech_SA)
\\&=& \hom_R(B, \hhom_S(R,\cech_SA)) \ \ \hbox{since the image of $B$ is
killed by $J$}
\end{array}
$$
This says that $\cech_R(\dash)$ and $\hhom_S(R,\cech_S(\dash))$ are both
right adjoint to $(\dash)_{\succeq 0}$. Therefore, these two functors
are isomorphic as functors on $R$-modules. The ``in particular'' follows
from the fact that $\EE_S(R/\pp) = \cech_S(R/\pp)$ and $\EE_R(R/\pp) =
\cech_R(R/\pp)$. \endproof
\medskip
\noindent
More generally, the proof can be appropriately souped up to show
$\hhom_S(R, \cech_S M) = \cech_{R\,}\hhom_S(R,M)$.
\begin{defn} \sl
The {\em $\ZZ^n$-graded regularity} $\reg(M)$ of a finitely generated
$S$-module $M$ is the join of the Betti degrees of $M$.
\end{defn}
\noindent
Equivalently, if $M \from \FF_\spot$ is a minimal free resolution and
$\FF^\spot = \hhom_S(\FF_\spot,S)$, then $\reg(M)$ is the smallest vector
$\alpha$ in $\ZZ^n$ such that $\FF^\spot(-\alpha)$ is $\NN^n$-graded.
\begin{prop} \label{prop:reg}
If $A$ is an $S$-module that is fixed by $\cech_S$, then
$\hhom_S(R,A)(-\alpha)$ is fixed by $\cech_S$ for all $\alpha \succeq
\reg(R)$.
% In particular, the indecomposable injectives $\EE_R(R/\pp)(-\alpha)$
% are fixed by $\cech_S$ when $\alpha \succeq \reg(R)$.
\end{prop}
\proof By Lemma~\ref{lemma:fixed}, below,
% and the exactness of $\cech_S$,
we may assume $\alpha = \reg(R)$. Now use the fact that $\hhom_S(R,A) =
H^0\hhom_S(\FF_\spot,A) = H^0(\FF^\spot \otimes_S A)$, where $\FF_\spot$
is a minimal free resolution of $R$ over $S$ and $\FF^\spot =
\hhom_S(\FF_\spot,S)$. By definition of $\reg(R)$, the complex
$\FF^\spot \otimes A$ is, as a module, a direct sum of {\em nonnegative}
shifts of $A$. Thus Lemma~\ref{lemma:fixed} says that this complex is
fixed by $\cech_S$. Exactness of $\cech_S$ completes the proof.
\endproof
\begin{prop} \label{prop:finite}
If $H$ is an $R$-module fixed by $\cech_S$ and having $H_{\succeq 0}$
finitely generated, then the Bass numbers of $H$ over $R$ are finite.
\end{prop}
\proof The injective hull $\EE_S(H)$ is fixed by $\cech_S$, because
applying $\cech_S$ to the inclusion $H \hookrightarrow \EE_S(H)$ yields
an inclusion $H \hookrightarrow \cech_S\EE_S(H)$ in which any purported
indecomposable summands of $\EE_S(H)$ not fixed by $\cech_S$ are erased
[Lemma~3.2, HM]. Since $H$ is an $R$-module, the inclusion $H
\hookrightarrow \EE_S(H)$ factors through $\hhom_S(R,\EE_R(H))$, which
equals $\EE_R(H)$ by Lemma~\ref{lemma:overR}. It follows that the zeroth
Bass numbers of $H$ are finite.
Proposition~\ref{prop:reg} implies that $\EE_R(H)(-\reg(R))$ is fixed by
$\cech_S$, while Proposition~5.4 of [HM] implies that its $\NN^n$-graded
part is finitely generated. By Lemma~\ref{lemma:fixed} and the same
proposition in [HM], the same holds for $H(-\reg(R))$. Therefore
$(\EE_R(H)/H)(-\reg(R))$ is fixed by $\cech_S$ and has finitely generated
$\NN^n$-graded part, as well. The proof is complete by induction on the
cohomological degree. \endproof
\bigskip
We believe our proof can be made (without much trouble) to work with
``polynomial ring'' replaced by ``simplicial semigroup ring''. Observe,
for instance, that both Lemmas are completely general for the \v Cech
hull, and both Propositions have reformulations in terms of straight
modules.
\begin{thebibliography}{HM00}
\bibitem[HM00]{HelM}
David Helm and Ezra Miller, \emph{Bass numbers of semigroup-graded local
cohomology}, Preprint ({\sf math.AG/0010003}), 2000.
\end{thebibliography}
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