Local Cohomology at Monomial Ideals

AbstractWe prove that if B⊂R=k [ X1,⋯ , Xn] is a reduced monomial ideal, then HBi(R) = ∪d≥1ExtRi(R/B[ d ], R), where B[ d ]is the d th Frobenius power of B. We give two descriptions for HBi(R) in each multidegree, as simplicial cohomology groups of certain simplicial complexes. As a first consequence, we derive a relation between ExtR(R/B, R) and TorR(B∨, k), where B∨is the Alexander dual of B. As a further application, we give a filtration of ExtRi(R/B, R) such that the quotients are suitable shifts of modules of the form R/(Xi1,⋯ , Xir). We conclude by giving a topological description of the associated primes of ExtRi(R/B, R). In particular, we characterize the minimal associated primes of ExtRi(R/B, R) using only the Betti numbers of B∨