Samiksha Jaiswal (Editor)

Equivariant sheaf

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In mathematics, given the action σ : G × S X → 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 O X -modules together with the isomorphism of O G × S X -modules

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ϕ : σ ∗ F ≃ p 2 ∗ F  

that satisfies the cocycle condition: writing m for multiplication,

p 23 ∗ ϕ ∘ ( 1 G × σ ) ∗ ϕ = ( m × 1 X ) ∗ ϕ .

On the stalk level, the cocycle condition says that the isomorphism F g h ⋅ x ≃ F x is the same as the composition F g ⋅ h ⋅ x ≃ F h ⋅ x ≃ 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 × e × 1 ) ∗ , e : S → G to both sides gives ( e × 1 ) ∗ ∘ ( e × 1 ) ∗ ϕ = ( e × 1 ) ∗ ϕ and so ( e × 1 ) ∗ ϕ is the identity.

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

G ( R ) → Aut ⁡ ( X × S Spec ⁡ R , F ⊗ 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 → E g x 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 × X → X to that of G × E → E so that the projection E → X is equivariant.

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

    • References

    Equivariant sheaf Wikipedia
    (Text) CC BY-SA