Local cohomology modules of Stanley-Reisner rings and Alexander duality

Kohji Yanagawa

Osaka University, Japan

March 8,
refreshments at 3:45pm


Stanley-Reisner rings and affine semigroup rings are central concepts of combinatorial commutative algebra. For these studies, explicit descriptions of the local cohomology modules with supports in maximal monomial ideals are fundamental. In the Stanley-Reisner ring case, Hochster's formula gives a topological description. But recently, for affine semigroup rings, several authors investigated the case where the support ideal is not maximal. In this case, the local cohomology modules are neither noetherian nor artinian, but still have nice graded structure.

This talk will concern the local cohomologies of a Stanley-Reisner ring with supports in a general monomial ideal. A monomial ideal of a Stanley-Reisner ring associated to a subcomplex of an (abstract) simplicial complex. I will 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 formula that generalizes results of Mustata and Terai. The agreement between these two formulae is seen to be a disguised form of Alexander duality.

This is a joint work with V. Reiner and V. Welker.

Speaker's Contact Info: yanagawa(at-sign)math.sci.osaka-u.ac.jp

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