Local cohomology of Stanley-Reisner rings with supports in general monomial ideals

摘要

We study the local cohomology modules H-l Sigma(i)(k[Delta]) of the Stanley-Reisner ring k[A] of a simplicial complex A with support in the ideal I-Sigma subset of k[Delta] corresponding to a subcomplex Sigma subset of Delta. We give a combinatorial topological formula for the multigraded Hilbert series, and in the case where the ambient complex is Gorenstein. compare this with a second combinatorial formula that generalizes results of Mustata and Terai. The agreement between these two formulae is seen to be a disguised form of Alexander duality. Other results include a comparison of the local cohomology with certain Ext modules, results about when it is concentrated in a single homological degree, and combinatorial topological interpretations of some vanishing theorems. (C) 2001 Academic Press. [References: 22]