In mathematics, given an action of a group schemeG on a scheme X over a base scheme S, an equivariant sheafF 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
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.
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 Wλ of B. Then GxWλ is a trivial vector bundle over G on which B acts. The quotient Lλ=GxBWλ 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(A) is an affine scheme, a Gm-action on X is the same thing as a Z grading on A. Similarly, a Gm equivariant quasicoherent sheaf on X is the same thing as a Z graded A module.[citation needed]
Thompson, R.W. (1987). "Algebraic K-theory of group scheme actions". In Browser, William (ed.). Algebraic topology and algebraic K-theory : proceedings of a conference, October 24-28, 1983 at Princeton University, dedicated to John C. Moore on his 60th birthday. Vol. 113. Princeton, N.J.: Princeton University Press. p. 539-563. ISBN9780691084268.