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 ∘ ∗ ϕ = ∗ ϕ {\displaystyle p_{23}^{*}\phi \circ ^{*}\phi =^{*}\phi }. Wikipedia