Definition 60.5.2. In Situation 60.5.1.

1. A divided power thickening of $C$ over $(A, I, \gamma )$ is a homomorphism of divided power algebras $(A, I, \gamma ) \to (B, J, \delta )$ such that $p$ is nilpotent in $B$ and a ring map $C \to B/J$ such that

$\xymatrix{ B \ar[r] & B/J \\ & C \ar[u] \\ A \ar[uu] \ar[r] & A/I \ar[u] }$

is commutative.

2. A homomorphism of divided power thickenings

$(B, J, \delta , C \to B/J) \longrightarrow (B', J', \delta ', C \to B'/J')$

is a homomorphism $\varphi : B \to B'$ of divided power $A$-algebras such that $C \to B/J \to B'/J'$ is the given map $C \to B'/J'$.

3. We denote $\text{CRIS}(C/A, I, \gamma )$ or simply $\text{CRIS}(C/A)$ the category of divided power thickenings of $C$ over $(A, I, \gamma )$.

4. We denote $\text{Cris}(C/A, I, \gamma )$ or simply $\text{Cris}(C/A)$ the full subcategory consisting of $(B, J, \delta , C \to B/J)$ such that $C \to B/J$ is an isomorphism. We often denote such an object $(B \to C, \delta )$ with $J = \mathop{\mathrm{Ker}}(B \to C)$ being understood.

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