Equivariant sheaf

In mathematics, given an action formula_1 of a group scheme "G" on a scheme "X" over a base scheme "S", an equivariant sheaf "F" on "X" is a sheaf of formula_2-modules together with the isomorphism of formula_3-modules
that satisfies the cocycle condition: writing "m" for multiplication,
Notes on the definition.
On the stalk level, the cocycle condition says that the isomorphism formula_6 is the same as the composition formula_7; i.e., the associativity of the group action. The unitarity of a group action is also a consequence: apply formula_8 to both sides to get formula_9 and so formula_10 is the identity.
Note that formula_11 is an additional data; it is "a lift" of the action of "G" on "X" to the sheaf "F". Moreover, when "G" is a connected algebraic group, "F" an invertible sheaf and "X" is reduced, the cocycle condition is automatic: any isomorphism formula_12 automatically satisfies the cocycle condition (this fact is noted at the end of the proof of Ch. 1, § 3., Proposition 1.5. of Mumford's "geometric invariant theory.")
If the action of "G" is free, then the notion of an equivariant sheaf simplifies to a sheaf on the quotient "X"/"G", because of the descent along torsors.
By Yoneda's lemma, to give the structure of an equivariant sheaf to an formula_2-module "F" is the same as to give group homomorphisms for rings "R" over formula_14,
There is also a definition of equivariant sheaves in terms of simplicial sheaves. Alternatively, one can define an equivariant sheaf to be an equivariant object in the category of, say, coherent sheaves.
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 formula_16 of "L" is linearizable.
Also, if "L" is very ample and linearized, then there is a "G"-linear closed immersion from "X" to formula_17 such that formula_18 is linearized and the linearlization on "L" is induced by that of formula_18.
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 formula_20 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 formula_21 for the group action, for each "g" in "G" and "v" in "V", let
where formula_23 and formula_24 is the isomorphism given by the equivariant-sheaf structure on "F". The cocycle condition then ensures that formula_25 is a group homomorphism (i.e., formula_26 is a representation.)
Example: take formula_27 and formula_28 the action of "G" on itself. Then formula_29, formula_30 and
meaning formula_26 is the left regular representation of "G".
The representation formula_26 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 (i.e., a variety corresponding to a locally free sheaf of constant rank). 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., formula_34 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 formula_35 to that of formula_36 so that the projection formula_37 is equivariant.
Just like in the non-equivariant setting, one can define an equivariant characteristic class of an equivariant vector bundle.