Equivariant sheaf

In mathematics, given an action ${\displaystyle \sigma :G\times _{S}X\to X}$ of a group scheme G on a scheme X over a base scheme S, an equivariant sheaf F on X is a sheaf of ${\displaystyle {\mathcal {O}}_{X}}$-modules together with the isomorphism of ${\displaystyle {\mathcal {O}}_{G\times _{S}X}}$-modules

${\displaystyle \phi :\sigma ^{*}F\xrightarrow {\simeq } p_{2}^{*}F}$  

that satisfies the cocycle condition:^[1][2] writing m for multiplication,

${\displaystyle p_{23}^{*}\phi \circ (1_{G}\times \sigma )^{*}\phi =(m\times 1_{X})^{*}\phi }$.

On the stalk level, the cocycle condition says that the isomorphism ${\displaystyle F_{gh\cdot x}\simeq F_{x}}$ is the same as the composition ${\displaystyle F_{g\cdot h\cdot x}\simeq F_{h\cdot x}\simeq F_{x}}$; 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 ${\displaystyle (e\times e\times 1)^{*},e:S\to G}$ to both sides to get ${\displaystyle (e\times 1)^{*}\phi \circ (e\times 1)^{*}\phi =(e\times 1)^{*}\phi }$ and so ${\displaystyle (e\times 1)^{*}\phi }$ is the identity.

Note that ${\displaystyle \phi }$ 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 ${\displaystyle \sigma ^{*}F\simeq p_{2}^{*}F}$ 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 ${\displaystyle {\mathcal {O}}_{X}}$-module F is the same as to give group homomorphisms for rings R over ${\displaystyle S}$,

${\displaystyle G(R)\to \operatorname {Aut} (X\times _{S}\operatorname {Spec} R,F\otimes _{S}R)}$.^[3]

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 ${\displaystyle L^{n}}$ of L is linearizable.^[4]

Also, if L is very ample and linearized, then there is a G-linear closed immersion from X to ${\displaystyle \mathbf {P} ^{N}}$ such that ${\displaystyle {\mathcal {O}}_{\mathbf {P} ^{N}}(1)}$ is linearized and the linearlization on L is induced by that of ${\displaystyle {\mathcal {O}}_{\mathbf {P} ^{N}}(1)}$.^[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 ${\displaystyle V=\Gamma (X,F)}$ 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 ${\displaystyle \sigma :G\times X\to X}$ for the group action, for each g in G and v in V, let

${\displaystyle \pi (g)v=(\varphi \circ \sigma ^{*})(v)(g^{-1})}$

where ${\displaystyle \sigma ^{*}:V\to \Gamma (G\times X,\sigma ^{*}F)}$ and ${\displaystyle \varphi :\Gamma (G\times X,\sigma ^{*}F){\overset {\sim }{\to }}\Gamma (G\times X,p_{2}^{*}F)=k[G]\otimes _{k}V}$ is the isomorphism given by the equivariant-sheaf structure on F. The cocycle condition then ensures that ${\displaystyle \pi :G\to GL(V)}$ is a group homomorphism (i.e., ${\displaystyle \pi }$ is a representation.)

Example: take ${\displaystyle X=G,F={\mathcal {O}}_{G}}$ and ${\displaystyle \sigma =}$ the action of G on itself. Then ${\displaystyle V=k[G]}$, ${\displaystyle (\varphi \circ \sigma ^{*})(f)(g,h)=f(gh)}$ and

${\displaystyle (\pi (g)f)(h)=f(g^{-1}h)}$,

meaning ${\displaystyle \pi }$ is the left regular representation of G.

The representation ${\displaystyle \pi }$ 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., ${\displaystyle g:E_{x}\to E_{gx}}$ 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 ${\displaystyle G\times X\to X}$ to that of ${\displaystyle G\times E\to E}$ so that the projection ${\displaystyle E\to X}$ is equivariant.

Just like in the non-equivariant setting, one can define an equivariant characteristic class of an equivariant vector bundle.

• 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(A) 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.^{[citation needed]}

• Equivariant algebraic K-theory
• Equivariant bundle
• Equivariant cohomology
• Quotient stack

• J. Bernstein, V. Lunts, "Equivariant sheaves and functors," Springer Lecture Notes in Math. 1578 (1994).
• Mumford, David; Fogarty, John; Kirwan, Frances (1994). Geometric Invariant Theory. Berlin: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906.
• Gaitsgory, D. (2005). "Geometric Representation theory, Math 267y, Fall 2005" (PDF). Archived from the original (PDF) on 22 January 2015.
• 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. ISBN 9780691084268.

• Equivariant sheaves