The Stacks project

Remark 60.5.4. In Situation 60.5.1 we denote $\text{Cris}^\wedge (C/A)$ the category whose objects are pairs $(B \to C, \delta )$ such that

  1. $B$ is a $p$-adically complete $A$-algebra,

  2. $B \to C$ is a surjection of $A$-algebras,

  3. $\delta $ is a divided power structure on $\mathop{\mathrm{Ker}}(B \to C)$,

  4. $A \to B$ is a homomorphism of divided power rings.

Morphisms are defined as in Definition 60.5.2. Then $\text{Cris}(C/A) \subset \text{Cris}^\wedge (C/A)$ is the full subcategory consisting of those $B$ such that $p$ is nilpotent in $B$. Conversely, any object $(B \to C, \delta )$ of $\text{Cris}^\wedge (C/A)$ is equal to the limit

\[ (B \to C, \delta ) = \mathop{\mathrm{lim}}\nolimits _ e (B/p^ eB \to C, \delta ) \]

where for $e \gg 0$ the object $(B/p^ eB \to C, \delta )$ lies in $\text{Cris}(C/A)$, see Divided Power Algebra, Lemma 23.4.5. In particular, we see that $\text{Cris}^\wedge (C/A)$ is a full subcategory of the category of pro-objects of $\text{Cris}(C/A)$, see Categories, Remark 4.22.5.

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