Cohomology of partially ordered sets and local cohomology of section rings

Details Cohomology of partially ordered sets and local cohomology of section rings Cohomology of partially ordered sets and local cohomology of section rings

2005-02-24

arxiv.org/abs/math/0502517v3

Adv. Math. 208, No. 1, 210-235 (2007)

Mathematics

Commutative Algebra

24 pages; To appear in Adv. Math. Revised version of the paper

Scientific paper

We study local cohomology of rings of global sections of sheafs on the Alexandrov space of a partially ordered set. We give a criterion for a splitting of the local cohomology groups into summands determined by the cohomology of the poset and the local cohomology of the stalks. The face ring of a rational pointed fan can be considered as the ring of global sections of a flasque sheaf on the face poset of the fan. Thus we obtain a decomposition of the local cohomology of such face rings. Since the Stanley-Reisner ring of a simplicial complex is the face ring of a rational pointed fan, our main result can be interpreted as a generalization of Hochster's decomposition of local cohomology of Stanley-Reisner rings.

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