Local Cohomology at Monomial Ideals

Abstract

We prove that if B ⊂ R = k [ X 1,⋯ , X n ] is a reduced monomial ideal, then H B i ( R) = ∪ d ≥ 1 Ext R i ( R / B [ d ] , R), where B [ d ] is the d th Frobenius power of B. We give two descriptions for H B i ( R) in each multidegree, as simplicial cohomology groups of certain simplicial complexes. As a first consequence, we derive a relation between Ext R ( R / B, R) and Tor R ( B ∨ , k), where B ∨ is the Alexander dual of B. As a further application, we give a filtration of Ext R i ( R / B, R) such that the quotients are suitable shifts of modules of the form R / ( X i 1 ,⋯ , X i r ). We conclude by giving a topological description of the associated primes of Ext R i ( R / B, R). In particular, we characterize the minimal associated primes of Ext R i ( R / B, R) using only the Betti numbers of B ∨ .