We study the local cohomology modules HIΣi(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.
Bibliographical noteFunding Information:
The authors thank Professor Gennady Lyubeznik for valuable comments. The first author is partially supported by the NSF. This research was began when the third author was visiting Mathematical Science Research Institute and University of California Berkeley, he is grateful to these institutes for their warm hospitality.
- Alexander duality
- Gorenstein complex
- Lichtenbaum-Hartshorne vanishing theorem
- Local cohomology modules
- Stanley-Reisner rings