Equivariant sheaf

In mathematics, given an action σ : G × S X -> X {\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  O X {\displaystyle {\mathcal {O}}_{X}} -modules together with the isomorphism of  O G × S X {\displaystyle {\mathcal {O}}_{G\times _{S}X}} -modules

ϕ : σ ∗ F -> ≃ p 2 ∗ F {\displaystyle \phi :\sigma ^{*}F\xrightarrow {\simeq } p_{2}^{*}F}

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

p 23 ∗ ϕ ∘ ( 1 G × σ ) ∗ ϕ = ( m × 1 X ) ∗ ϕ {\displaystyle p_{23}^{*}\phi \circ (1_{G}\times \sigma )^{*}\phi =(m\times 1_{X})^{*}\phi } .

## Notes on the definition[edit]

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

Note that  ϕ {\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  σ ∗ F ≃ p 2 ∗ F {\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  O X {\displaystyle {\mathcal {O}}_{X}} -module F is the same as to give group homomorphisms for rings R over  S {\displaystyle S} ,

G ( R ) -> Aut ⁡ ( X × S Spec ⁡ R , F ⊗ S R ) {\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.

## Linearized line bundles[edit]

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  L n {\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  P N {\displaystyle \mathbf {P} ^{N}} such that  O P N ( 1 ) {\displaystyle {\mathcal {O}}_{\mathbf {P} ^{N}}(1)} is linearized and the linearlization on L is induced by that of  O P N ( 1 ) {\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.

## Dual action on sections of equivariant sheaves[edit]

Given an algebraic group G and a G-equivariant sheaf F on X over a field k, let  V = Γ ( X , F ) {\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  σ : G × X -> X {\displaystyle \sigma :G\times X\to X} for the group action, for each g in G and v in V, let

π ( g ) v = ( φ ∘ σ ∗ ) ( v ) ( g − 1 ) {\displaystyle \pi (g)v=(\varphi \circ \sigma ^{*})(v)(g^{-1})}

where  σ ∗ : V -> Γ ( G × X , σ ∗ F ) {\displaystyle \sigma ^{*}:V\to \Gamma (G\times X,\sigma ^{*}F)} and  φ : Γ ( G × X , σ ∗ F ) -> ∼ Γ ( G × X , p 2 ∗ F ) = k [ G ] ⊗ k V {\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  π : G -> G L ( V ) {\displaystyle \pi :G\to GL(V)} is a group homomorphism (i.e.,  π {\displaystyle \pi } is a representation.)

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

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

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

The representation  π {\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]

## Equivariant vector bundle[edit]

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.,  g : E x -> E g x {\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  G × X -> X {\displaystyle G\times X\to X} to that of  G × E -> E {\displaystyle G\times E\to E} so that the projection  E -> X {\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.

## Examples[edit]

* 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 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]

## See also[edit]

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

## Notes[edit]

1. ^ MFK 1994, Ch 1. § 3. Definition 1.6.
2. ^ Gaitsgory 2005, § 6.
3. ^ Thompson 1987, § 1.2.
4. ^ MFK 1994, Ch 1. § 3. Corollary 1.6.
5. ^ MFK 1994, Ch 1. § 3. Proposition 1.7.
6. ^ MFK 1994, Ch. 1. § 1. the lemma just after Definition 1.3.
7. ^ If E is viewed as a sheaf, then g needs to be replaced by  g − 1 {\displaystyle g^{-1}} .

## References[edit]

* 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.

## External links[edit]

* Equivariant sheaves