Equivariant sheaf

In mathematics, given the action 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 -modules together with the isomorphism of -modules

 

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

.

On the stalk level, the cocycle condition says that the isomorphism is the same as the composition ; i.e., the associativity of the group action.

The unitarity of a group action, on the other hand, is a consequence: applying to both sides gives and so is the identity.

Note that 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 ) 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 -module F is the same as to give group homomorphisms for rings R over ,

.[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., 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 to that of so that the projection is equivariant.

(Locally free sheaves and vector bundles correspond contravariantly. Thus, if V is a vector bundle corresponding to F, then induces isomorphisms between fibers , 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 .

References

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