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

See also

Notes

  1. MFK 1994, Ch 1. § 3. Definition 1.6.
  2. Thomason 1987, § 1.2.
  3. If E is viewed as a sheaf, then g needs to be replaced by g^{-1}.

References

External links

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