Cohomología local con soporte un ideal monomial (D-módulos y combinatoria)
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hdl:2117/1030
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Data publicació2000
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Abstract
We study, by using the theory of algebraic $\cD$-modules, the local
cohomology modules supported on a monomial ideal $I$ of the
polynomial ring $R=k[x_1,\dots,x_n]$, where $k$ is a field of
characteristic zero. We compute the characteristic cycle of
$H_I^r(R)$ and $H_{{\p}}^p(H_I^r(R))$, where ${\p}$ is an
homogeneous prime ideal of $R$. By using these results we can
describe the support of these modules, in particular we can decide
when the local cohomology module $H_I^r(R)$ vanishes in terms of the
minimal primary decomposition of the monomial ideal $I$, compute the
Bass numbers of $H_I^r(R)$ and describe its associated primes. The
characteristic cycles also give some invariants of the ring $R/I$.
We use these invariants to compute the Hilbert function of $R/I$,
the minimal free resolutions of squarefree monomial ideals and the
cohomology groups of the complement of an arrangement of linear
varieties given by the monomial ideal $I$. Finally, we determine the
local cohomology modules by using the category
introduced by Galligo-Granger-Maisonobe \cite{GGM85b} and compute
its Hilbert function.
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