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Bass Numbers of Semigroup-Graded Local Cohomology

David Helm·Ezra Miller
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Abstract

Given a module M over a ring R which has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology of M at any Q-graded ideal I in terms of Ext modules. This method is used to obtain finiteness results for the local cohomology of graded modules over semigroup rings; in particular we prove that for a semigroup Q whose saturation is simplicial, the Bass numbers of such local cohomology modules are finite. Conversely, if the saturation of Q is not simplicial, one can find a graded ideal I and a graded R-module M whose local cohomology at I in some degree has an infinite-dimensional socle. We introduce and exploit the combinatorially defined essential set of a semigroup.

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David Helm, Ezra Miller. 2000-10-17. Bass Numbers of Semigroup-Graded Local Cohomology. https://arxiv.org/abs/math/0010003

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