# Cohomology on Toric Varieties and Local Cohomology with Monomial Supports

@article{Eisenbud2000CohomologyOT, title={Cohomology on Toric Varieties and Local Cohomology with Monomial Supports}, author={David Eisenbud and Mircea Mustala and Michael Eugene Stillman}, journal={J. Symb. Comput.}, year={2000}, volume={29}, pages={583-600} }

We study the local cohomology modules H^i_B(R) for a reduced monomial ideal B in a polynomial ring R=k[X_1,...,X_n]. We consider a grading on R which is coarser than the Z^n-grading such that each component of H^i_B(R) is finite dimensional and we give an effective way to compute these components. Using Cox's description for sheaves on toric varieties, we apply these results to compute the cohomology of coherent sheaves on toric varieties. We give algorithms for this computation which have been…

## 81 Citations

Three lectures on local cohomology modules supported on monomial ideals

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- MathematicsCompositio Mathematica
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Let R be a d-dimensional regular local ring of characteristic p > 0 with maximal ideal $\mathfrak m$, let I be an ideal of R and let A = R/I. We describe some properties of the local cohomology…

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EULER CHARACTERISTIC OF COHERENT SHEAVES ON SIMPLICIAL TORICS VIA THE

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We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud-Mustaţǎ-Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for…

Klyachko Diagrams of Monomial Ideals

- MathematicsAlgebras and Representation Theory
- 2022

In this paper, we introduce the notion of a Klyachko diagram for a monomial ideal I in a certain multi-graded polynomial ring, namely the Cox ring R of a smooth complete toric variety, with…

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We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud–Mustaţǎ–Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for…

EULER CHARACTERISTIC OF LINE BUNDLES ON SIMPLICIAL TORICS VIA THE

- Mathematics
- 2010

We combine work of Cox on the homogeneous coordinate ring of a toric variety and results of Eisenbud-Mustaţǎ-Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for…

D-modules on Smooth Toric Varieties

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Abstract Let X be a smooth toric variety. Cox introduced the homogeneous coordinate ring S of X and its irrelevant ideal b . Let A denote the ring of differential operators on Spec(S). We show that…

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