The Stacks project

Definition 60.7.2. A divided power scheme is a triple $(S, \mathcal{I}, \gamma )$ where $S$ is a scheme, $\mathcal{I}$ is a quasi-coherent sheaf of ideals, and $\gamma$ is a divided power structure on $\mathcal{I}$. A morphism of divided power schemes $(S, \mathcal{I}, \gamma ) \to (S', \mathcal{I}', \gamma ')$ is a morphism of schemes $f : S \to S'$ such that $f^{-1}\mathcal{I}'\mathcal{O}_ S \subset \mathcal{I}$ and such that

is a homomorphism of divided power rings for all $U' \subset S'$ open.