Resolutions and Cohomologies of Toric Sheaves. The affine case

TL;DR

This paper investigates the structure of toric sheaves on affine toric varieties, providing explicit resolutions and cohomology calculations, and constructing new Cohen-Macaulay modules with higher rank in both smooth and non-smooth cases.

Contribution

It develops machinery for minimal free resolutions and local cohomology of toric sheaves, and constructs new indecomposable Cohen-Macaulay modules in higher dimensions.

Findings

Explicit combinatorial formulas for Betti numbers and local cohomology of reflexive modules.

Construction of indecomposable maximal Cohen-Macaulay modules of rank d-1 in non-smooth cases.

Resolution techniques applicable to both smooth and non-smooth affine toric varieties.

Abstract

We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of decomposable maximal Cohen-Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a certain family of filtrations. Within this setting, we develop machinery which facilitates the construction of minimal free resolutions for the smooth case as well as resolutions which are acyclic with respect to local cohomology functors for the general case. We give two main applications. First, over the polynomial ring, we determine in explicit combinatorial terms the Z^n-graded Betti numbers and local cohomology of reflexive modules whose associated filtrations form a hyperplane arrangement. Second, for the non-smooth, simplicial case in dimension d >= 3, we construct new examples of indecomposable maximal Cohen-Macaulay modules of rank d - 1.