Equivariant Primary Decomposition and Toric Sheaves

Understand

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.