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{\sc Abstract: Sequentially Cohen-Macaulay modules and local cohomomology}
{Juergen Herzog}
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In a combinatorial context Stanley introduced the notion of sequentially Cohen-Macaulay %%@
modules, and calls a simplicial complex sequentially Cohen-Macaulay over a field $K$ if the %%@
Stanley Reisner ring $K[\Delta]$ is sequentially Cohen-Macaulay. This is implied over any %%@
field $K$ by the hypothesis that the Alexander dual of $\Delta$ is non-pure shellable in the %%@
sense of Bj\"orner and Wachs.
For all $k\geq 0$, the simplicial complex $\Delta^{(k)}$, which is generated by all %%@
$k$-dimensional faces of $\Delta$ is called the pure $k$-skeleton of $\Delta$. It has been %%@
shown by Duval that $\Delta$ is sequentially Cohen-Macaulay over $K$ if and only if the pure %%@
$k$-skeleton of $\Delta$ is Cohen-Macaulay over $K$ for all $k\geq 0$.
Peskine gave a nice homological characterization of sequentially Co\-hen-Macaulay modules. He %%@
showed that over a local (or standard graded) Cohen-Macaulay ring with canonical module %%@
$\omega_R$, a finitely generated (graded) module is sequentially Cohen-Macaulay if for any %%@
$0\leq i\leq \dim R$, the module $\Ext^i(M,\omega_R)$ is either $0$ or Cohen-Macaulay of %%@
dimension $\dim R-i$.
In this lecture a joint result with Sbarra will be presented in which we give another %%@
surprising characterization of sequentially Cohen-Macaulay modules. The result, in the case of %%@
graded rings, says the following: Let $K$ be a field, let $I\subset R$ be a graded ideal in %%@
the polynomial ring $R=K[x_1,\ldots,x_n]$, and let $\Gin(I)$ denote the generic initial ideal %%@
of $I$ with respect to the reverse lexicographical order. Then the local cohomology modules %%@
of $R/I$ and $R/\Gin(I)$ have the same Hilbert functions, if and only if $R/I$ is %%@
sequentially Cohen-Macaulay. Notice that for arbitrary ideals the Hilbert functions of the %%@
local cohomology modules of $R/I$ are only bounded above by those of $R/\Gin(I)$.
A similar result can be shown for simplicial complexes. Kalai introduced the symmetric %%@
algebraic shifted complex $\Delta^s$ of $\Delta$. It is known that $\Delta$ and $\Delta^s$ %%@
share many basic properties, while on the other hand $\Delta^s$ has a much simpler %%@
combinatorial structure than $\Delta$. We show that $\Delta$ is sequentially Cohen-Macaulay %%@
over $K$ if and only if the local cohomology modules of $K[\Delta]$ and $K[\Delta^s]$ have the %%@
same Hilbert function.
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