Equivariant sheaf
In mathematics, given an action σ : G × S X → X {\displaystyle \sigma :G\times _{S}X\to X} of a group scheme G on a scheme X over a base scheme S, an equivariant sheaf F on X is a sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules together with the isomorphism of O G × S X {\displaystyle {\mathcal {O}}_{G\times _{S}X}} -modules ϕ : σ ∗ F → ≃ p 2 ∗ F {\displaystyle \phi :\sigma ^{*}F\xrightarrow {\simeq } p_{2}^{*}F} that satisfies the cocycle condition: writing m for multiplication, p 23 ∗ ϕ ∘ ( 1 G × σ ) ∗ ϕ = ( m × 1 X ) ∗ ϕ {\displaystyle p_{23}^{*}\phi \circ (1_{G}\times \sigma )^{*}\phi =(m\times 1_{X})^{*}\phi } . == Notes on the definition == On the stalk level, the cocycle condition says that the isomorphism F g h ⋅ x ≃ F x {\displaystyle F_{gh\cdot x}\simeq F_{x}} is the same as the composition F g ⋅ h ⋅ x ≃ F h ⋅ x ≃ F x {\displaystyle F_{g\cdot h\cdot x}\simeq F_{h\cdot x}\simeq F_{x}} ; i.e., the associativity of the group action.