Equivariant Primary Decomposition and Toric Sheaves

TL;DR

This paper investigates equivariant primary decompositions of sheaves on schemes with algebraic group actions, especially focusing on toric varieties and their homogeneous coordinate rings, establishing existence and equivalence conditions.

Contribution

It proves the existence of equivariant primary decompositions for connected groups and relates these to graded module decompositions in toric varieties.

Findings

Equivariant primary decompositions exist for connected algebraic groups.

Primary decompositions of sheaves are equivalent to graded module decompositions under free group actions.

Provides explicit examples for toric varieties.

Abstract

We study global primary decompositions in the category of sheaves on a scheme which are equivariant under the action of an algebraic group. We show that equivariant primary decompositions exist if the group is connected. As main application we consider the case of varieties which are quotients of a quasi-affine variety by the action of a diagonalizable group and thus admit a homogeneous coordinate ring, such as toric varieties. Comparing these decompositions with primary decompositions of graded modules over the homogeneous coordinate ring, we show that these are equivalent if the action of the diagonalizable group is free. We give some specific examples for the case of toric varieties.