Equivariant sheaf

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

[math]\displaystyle{ \phi: \sigma^* F \xrightarrow{\simeq} p_2^*F }[/math]

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

[math]\displaystyle{ p_{23}^* \phi \circ (1_G \times \sigma)^* \phi = (m \times 1_X)^* \phi }[/math].

## 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 [math]\displaystyle{ F_{gh \cdot x} \simeq F_x }[/math] is the same as the composition [math]\displaystyle{ F_{g \cdot h \cdot x} \simeq F_{h \cdot x} \simeq F_x }[/math]; 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 [math]\displaystyle{ (e \times e \times 1)^*, e: S \to G }[/math] to both sides to get [math]\displaystyle{ (e \times 1)^* \phi \circ (e \times 1)^* \phi = (e \times 1)^* \phi }[/math] and so [math]\displaystyle{ (e \times 1)^* \phi }[/math] is the identity.

Note that [math]\displaystyle{ \phi }[/math] 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 [math]\displaystyle{ \sigma^* F \simeq p_2^* F }[/math] 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 [math]\displaystyle{ \mathcal{O}_X }[/math]-module F is the same as to give group homomorphisms for rings R over [math]\displaystyle{ S }[/math],

[math]\displaystyle{ G(R) \to \operatorname{Aut}(X \times_S \operatorname{Spec}R, F \otimes_S R) }[/math].[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

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

Also, if L is very ample and linearized, then there is a G-linear closed immersion from X to [math]\displaystyle{ \mathbf{P}^N }[/math] such that [math]\displaystyle{ \mathcal{O}_{\mathbf{P}^N}(1) }[/math] is linearized and the linearlization on L is induced by that of [math]\displaystyle{ \mathcal{O}_{\mathbf{P}^N}(1) }[/math].[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

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

[math]\displaystyle{ \pi(g)v = (\varphi \circ \sigma^*)(v)(g^{-1}) }[/math]

where [math]\displaystyle{ \sigma^*: V \to \Gamma(G \times X, \sigma^* F) }[/math] and [math]\displaystyle{ \varphi: \Gamma(G \times X, \sigma^* F) \overset{\sim}\to \Gamma(G \times X, p_2^* F) = k[G] \otimes_k V }[/math] is the isomorphism given by the equivariant-sheaf structure on F. The cocycle condition then ensures that [math]\displaystyle{ \pi: G \to GL(V) }[/math] is a group homomorphism (i.e., [math]\displaystyle{ \pi }[/math] is a representation.)

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

[math]\displaystyle{ (\pi(g)f)(h) = f(g^{-1} h) }[/math],

meaning [math]\displaystyle{ \pi }[/math] is the left regular representation of G.

The representation [math]\displaystyle{ \pi }[/math] 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

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., [math]\displaystyle{ g: E_x \to E_{gx} }[/math] 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 [math]\displaystyle{ G \times X \to X }[/math] to that of [math]\displaystyle{ G \times E \to E }[/math] so that the projection [math]\displaystyle{ E \to X }[/math] 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. ↑ 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 [math]\displaystyle{ g^{-1} }[/math].

## 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.
* Thompson, 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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