zbMATH — the first resource for mathematics

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.

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
13D45 Local cohomology and commutative rings
Full Text: DOI arXiv
[1] D. Bayer, M. Stillman
[2] Brodmann, M.; Sharp, R., Local cohomology: an algebraic introduction with geometric applications, (1998), Cambridge University Press · Zbl 0903.13006
[3] Cox, D., The homogeneous coordinate ring of a toric variety, J.algebr. geom., 4, 17-50, (1995) · Zbl 0846.14032
[4] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, (1995), Springer New York · Zbl 0819.13001
[5] Fulton, W., Introduction to toric varieties, volume 131 of annals of mathematical studies, (1993), Princeton University Press Princeton, NJ
[6] Godement, R., Topologie algebrique et theorie des faisceax, (1958), Hermann Paris
[7] D. Grayson, M. Stillman
[8] Grothendieck, A., Local cohomology, volume 41 of Springer lecture notes in mathematics, (1967), Springer-Verlag Heidelberg · Zbl 0185.49202
[9] Huneke, C.; Lyubeznik, G., On the vanishing of local cohomology modules, Invent.math., 102, 73-93, (1990) · Zbl 0717.13011
[10] Mustaţa, M., Local cohomology at monomial ideals, J. symb. comput., 29, 704-720, (2000) · Zbl 0966.13010
[11] Smith, G., Computing global extension modules for coherent sheaves on a projective scheme, J. symb. comput., 29, 729-746, (2000) · Zbl 0978.13008
[12] Vasconcelos, W., Computational methods in commutative algebra and algebraic geometry, volume 2 of algorithms and computation in mathematics, (1998), Springer-Verlag
[13] Weibel, C., An introduction to homological algebra, volume 38 of Cambridge studies in advanced mathematics, (1994), Cambridge University Press
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.