Equivariant sheaf


In mathematics, given an action of a group scheme G on a scheme X over a base scheme S, an equivariant sheaf F on X is a sheaf of -modules together with the isomorphism of -modules
that satisfies the cocycle condition: writing m for multiplication,

Linearized line bundles

A structure of an equivariant sheaf on an invertible sheaf or a line bundle is also called a linearization.
Let X be a complete variety over an algebraically closed field acted by a connected reductive group G and L an invertible sheaf on it. If X is normal, then some tensor power of L is linearizable.
Also, if L is very ample and linearized, then there is a G-linear closed immersion from X to such that is linearized and the linearlization on L is induced by that of.
Tensor products and the inverses of linearized invertible sheaves are again linearized in the natural way. Thus, the isomorphism classes of the linearized invertible sheaves on a scheme X form a subgroup of the Picard group of X.
See Example 2.16 of for an example of a variety for which most line bundles are not linearizable.

Dual action on sections of equivariant sheaves

Given an algebraic group G and a G-equivariant sheaf F on X over a field k, let be the space of global sections. It then admits the structure of a G-module; i.e., V is a linear representation of G as follows. Writing for the group action, for each g in G and v in V, let
where and is the isomorphism given by the equivariant-sheaf structure on F. The cocycle condition then ensures that is a group homomorphism
Example: take and the action of G on itself. Then, and
meaning is the left regular representation of G.
The representation defined above is a rational representation: for each vector v in V, there is a finite-dimensional G-submodule of V that contains v.

Equivariant vector bundle

A definition is simpler for a vector bundle. We say a vector bundle E on an algebraic variety X acted by an algebraic group G is equivariant if G acts fiberwise: i.e., is a "linear" isomorphism of vector spaces. In other words, an equivariant vector bundle is a pair consisting of a vector bundle and the lifting of the action to that of so that the projection is equivariant.
Just like in the non-equivariant setting, one can define an equivariant characteristic class of an equivariant vector bundle.

Examples