Equivariant sheaf

In mathematics, given the action ${\displaystyle \sigma :G\times _{S}X\to X}$ 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 ${\displaystyle {\mathcal {O}}_{X}}$-modules together with the isomorphism of ${\displaystyle {\mathcal {O}}_{G\times _{S}X}}$-modules

...
Wikipedia