Local Cohomology at Monomial Ideals

We 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 dthFrobenius power of B. We give two descriptions for HBi(R) in each multidegree, as simplicialcohomology 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 .