Local Cohomology of Stanley–Reisner Rings with Supports in General Monomial Ideals

We study the local cohomology modules HiIΣ(k[Δ]) of the Stanley–Reisner ring k[Δ] of a simplicial complex Δ with support in the ideal IΣ k[Δ] corresponding to a subcomplex Σ Δ. 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.