Equivariant sheaf

In mathematics, given an action ${\displaystyle \sigma :G\times _{S}X\to X}$ of a group scheme G on a scheme X over a base scheme S, an equivariant sheaf F on X is a sheaf of ${\displaystyle {\mathcal {O}}_{X}}$-modules together with the isomorphism of ${\displaystyle {\mathcal {O}}_{G\times _{S}X}}$-modules

${\displaystyle \phi :\sigma ^{*}F\xrightarrow {\simeq } p_{2}^{*}F}$  

that satisfies the cocycle condition:^[1][2] writing m for multiplication,

${\displaystyle p_{23}^{*}\phi \circ (1_{G}\times \sigma )^{*}\phi =(m\times 1_{X})^{*}\phi }$.