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: [1] [2] writing m for multiplication,
On the stalk level, the cocycle condition says that the isomorphism is the same as the composition ; 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 to both sides to get and so is the identity.
Note that 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 of L is linearizable. [4]
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 . [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. 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 (i.e., is a representation.)
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. [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 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.
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: [1] [2] writing m for multiplication,
On the stalk level, the cocycle condition says that the isomorphism is the same as the composition ; 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 to both sides to get and so is the identity.
Note that 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 of L is linearizable. [4]
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 . [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. 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 (i.e., is a representation.)
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. [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 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.