Equivariant sheaf

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

\phi: \sigma^* F \simeq p_2^*F

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

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 $F_{gh \cdot x} \simeq F_x$ is the same as the composition $F_{g \cdot h \cdot x} \simeq F_{h \cdot x} \simeq F_x$ ; i.e., the associativity of the group action.

The unitarity of a group action, on the other hand, is a consequence: applying $(e \times e \times 1)^*, e: S \to G$ to both sides gives $(e \times 1)^* \circ (e \times 1)^* \phi = (e \times 1)^* \phi$ and so $(e \times 1)^* \phi$ is the identity.

Note that $\phi$ is an additional data; it is "a lift" of the action of G on X to the sheaf F. A structure of an equivariant sheaf on a sheaf (namely $\phi$ ) is also called a linearization. In practice, one typically imposes further conditions; e.g., F is quasi-coherent, G is smooth and affine.

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 ${\mathcal {O}}_{X}$ -module F is the same as to give group homomorphisms for rings R over $S$ ,

G(R) \to \operatorname{Aut}(X \times_S \operatorname{Spec}R, F \otimes_S R)

.^[2]

Remark: There is also a definition of equivariant sheaves in terms of simplicial sheaves.

One example of an equivariant sheaf is a linearlized line bundle in geometric invariant theory. Another example is the sheaf of equivariant differential forms.

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., $g: E_x \to E_{gx}$ is a "linear" isomorphism of vector spaces.^[3] In other words, an equivariant vector bundle is a pair consisting of a vector bundle and the lifting of the action $G \times X \to X$ to that of $G \times E \to E$ so that the projection $E \to X$ is equivariant.

(Locally free sheaves and vector bundles correspond contravariantly. Thus, if V is a vector bundle corresponding to F, then $\phi$ induces isomorphisms between fibers $V_x \overset{\simeq}\to V_{gx}$ , which are linear maps.)

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.

Notes

↑ MFK 1994, Ch 1. § 3. Definition 1.6.
↑ Thomason 1987, § 1.2.
↑ 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, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. MR 1304906 ISBN 3-540-56963-4
Thomason, R.W.:Algebraic K-theory of group scheme actions. In: Browder, W. (ed.) Algebraic topology and algebraic K-theory. (Ann. Math. Stud., vol. 113, pp. 539–563) Princeton: Princeton University Press 1987

External links

http://math.stackexchange.com/questions/264796/equivariant-sheaves

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