Equivariant sheaf

In mathematics, given an action σ:G×SX->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 𝒪X-modules together with the isomorphism of 𝒪G×SX-modules

ϕ:σ*F→≃p2*F

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

p23*ϕ∘(1G×σ)*ϕ=(m×1X)*ϕ.

## Contents

* 1 Notes on the definition
* 2 Linearized line bundles
* 3 Dual action on sections of equivariant sheaves
* 4 Equivariant vector bundle
* 5 Examples
* 6 See also
* 7 Notes
* 8 References
* 9 External links

## Notes on the definition

On the stalk level, the cocycle condition says that the isomorphism Fgh⋅x≃Fx is the same as the composition Fg⋅h⋅x≃Fh⋅x≃Fx; 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 to both sides to get (e×1)*ϕ∘(e×1)*ϕ=(e×1)*ϕ and so (e×1)*ϕ is the identity.

Note that ϕ 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≃p2*F automatically satisfies the cocycle condition.[3]

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 𝒪X-module F is the same as to give group homomorphisms for rings R over S,

G(R)->Aut⁡(X×SSpec⁡R,F⊗SR).[4]

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

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 Ln of L is linearizable.[5]

Also, if L is very ample and linearized, then there is a G-linear closed immersion from X to 𝐏N such that 𝒪𝐏N(1) is linearized and the linearization on L is induced by that of 𝒪𝐏N(1).[6]

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

Given an algebraic group G and a G-equivariant sheaf F on X over a field k, let V=Γ(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 for the group action, for each g in G and v in V, let

π(g)v=(φ∘σ*)(v)(g−1)

where σ*:V->Γ(G×X,σ*F) and φ:Γ(G×X,σ*F)->∼Γ(G×X,p2*F)=k[G]⊗kV is the isomorphism given by the equivariant-sheaf structure on F. The cocycle condition then ensures that π:G->GL(V) is a group homomorphism (i.e., π is a representation.)

Example: take X=G,F=𝒪G and σ= the action of G on itself. Then V=k[G], (φ∘σ*)(f)(g,h)=f(gh) and

(π(g)f)(h)=f(g−1h),

meaning π is the left regular representation of G.

The representation π defined above is a rational representation: for each vector v in V, there is a finite-dimensional G-submodule of V that contains v.[7]

## Equivariant vector bundle

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:Ex->Egx is a "linear" isomorphism of vector spaces.[8] In other words, an equivariant vector bundle is a pair consisting of a vector bundle and the lifting of the action G×X->X to that of G×E->E so that the projection E->X is equivariant.

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

## Examples

* 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

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

## Notes

1. ↑ MFK 1994, Ch 1. § 3. Definition 1.6.
2. ↑ Gaitsgory 2005, § 6.
3. ↑ MFK 1994, end of the proof of Ch. 1, § 3., Proposition 1.5.
4. ↑ Thomason 1987, § 1.2.
5. ↑ MFK 1994, Ch 1. § 3. Corollary 1.6.
6. ↑ MFK 1994, Ch 1. § 3. Proposition 1.7.
7. ↑ MFK 1994, Ch. 1. § 1. the lemma just after Definition 1.3.
8. ↑ If E is viewed as a sheaf, then g needs to be replaced by g−1.

## References

* J. Bernstein, V. Lunts, "Equivariant sheaves and functors," Springer Lecture Notes in Math. 1578 (1994).
* Mumford, David; Fogarty, John; Kirwan, Frances (1994) (in en). Geometric Invariant Theory. Berlin: Springer-Verlag. ISBN 978-3-540-56963-3. https://books.google.com/books?id=dFlv3zn_2-gC.
* Gaitsgory, D. (2005). "Geometric Representation theory, Math 267y, Fall 2005". http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf.
* Thomason, R.W. (1987). "Algebraic K-theory of group scheme actions". 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. 113. Princeton, N.J.: Princeton University Press. p. 539-563. ISBN 9780691084268.

## External links

* Equivariant sheaves

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