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 linearization 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 an abelian group. There is a homomorphism to the Picard group of X which forgets the linearization; this homomorphism is neither injective nor surjective in general, and its kernel can be identified with the isomorphism classes of linearizations of the trivial line bundle.
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

The tangent bundle of a manifold or a smooth variety is an equivariant vector bundle.
The sheaf of equivariant differential forms.
Let G be a semisimple algebraic group, and λ:H→C a character on a maximal torus H''. It extends to a Borel subgroup λ:B→C, giving a one dimensional representation W_λ of B''. Then GxW_λ is a trivial vector bundle over G on which B acts. The quotient L_λ=Gx_BW_λ by the action of B is a line bundle over the flag variety G/B. Note that G→G/B is a B bundle, so this is just an example of the associated bundle construction. The Borel–Weil–Bott theorem says that all representations of G arise as the cohomologies of such line bundles.
If X=Spec is an affine scheme, a G_m-action on X is the same thing as a Z grading on A. Similarly, a G_m equivariant quasicoherent sheaf on X is the same thing as a Z graded A module.