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: 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 $S$,

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

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. 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.

See also

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

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. MR1304906 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