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 that satisfies the cocycle condition:[1][2] writing m for multiplication, On the stalk level, the cocycle condition says that the isomorphism
; i.e., the associativity of the group action.
The condition that the unit of the group acts as the identity is also a consequence: apply
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
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
-module F is the same as to give group homomorphisms for rings R over
, 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.
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
[4] Also, if L is very ample and linearized, then there is a G-linear closed immersion from X to
is linearized and the linearlization on L is induced by that of
[5] 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 [1] for an example of a variety for which most line bundles are not linearizable.
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.
is the isomorphism given by the equivariant-sheaf structure on F. The cocycle condition then ensures that
defined above is a rational representation: for each vector v in V, there is a finite-dimensional G-submodule of V that contains v.[6] 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.,
is a "linear" isomorphism of vector spaces.
[7] In other words, an equivariant vector bundle is a pair consisting of a vector bundle and the lifting of the action
Just like in the non-equivariant setting, one can define an equivariant characteristic class of an equivariant vector bundle.