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Cohomology and toric varieties and local cohomology with monomial supports. (English) Zbl 1044.14028
Consider local cohomology, with supports in a monomial ideal, of a finitely generated, graded module over a polynomial ring. The authors prove a criterion on the grading such that the corresponding homogeneous components of the local cohomology are finite-dimensional. Moreover, if this criterion holds, they give an algorithm for the computation of the components of local cohomology.
These results apply to toric varieties: coherent sheaves on a complete toric variety can be represented as graded modules over the associated homogeneous coordinate ring, where the grading is given by the group of invariant Weil divisor classes on the toric variety. The sheaf cohomology of the toric variety can be computed from the local cohomology of the graded module representing the sheaf, with supports in the irrelevant ideal of the homogeneous coordinate ring. Such gradings satisfy the criterion given in the article. As a result the authors obtain an explicit computation of sheaf cohomology of complete toric varieties. For the non-complete case they also provide other finiteness conditions.
Using these results, M. Mustaţǎ [Tôhoku Math. J. (2) 54, 451–470 (2002; Zbl 1092.14064)] has proved cohomology vanishing theorems on toric varieties that are valid in all characteristics, generalizing the well-known vanishing theorem of Kawamata and Viehweg.

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
13D45 Local cohomology and commutative rings
Software:
Macaulay2
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References:
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