Lemma 60.5.3. In Situation 60.5.1.

1. $\text{CRIS}(C/A)$ has finite products (but not infinite ones),

2. $\text{CRIS}(C/A)$ has all finite nonempty colimits and (60.5.2.1) commutes with these, and

3. $\text{Cris}(C/A)$ has all finite nonempty colimits and $\text{Cris}(C/A) \to \text{CRIS}(C/A)$ commutes with them.

Proof. The empty product, i.e., the final object in the category of divided power thickenings of $C$ over $(A, I, \gamma )$, is the zero ring viewed as an $A$-algebra endowed with the zero ideal and the unique divided powers on the zero ideal and finally endowed with the unique homomorphism of $C$ to the zero ring. If $(B_ t, J_ t, \delta _ t)_{t \in T}$ is a family of objects of $\text{CRIS}(C/A)$ then we can form the product $(\prod _ t B_ t, \prod _ t J_ t, \prod _ t \delta _ t)$ as in Divided Power Algebra, Lemma 23.3.2. The map $C \to \prod B_ t/\prod J_ t = \prod B_ t/J_ t$ is clear. However, we are only guaranteed that $p$ is nilpotent in $\prod _ t B_ t$ if $T$ is finite.

Given two objects $(B, J, \gamma )$ and $(B', J', \gamma ')$ of $\text{CRIS}(C/A)$ we can form a cocartesian diagram

$\xymatrix{ (B, J, \delta ) \ar[r] & (B'', J'', \delta '') \\ (A, I, \gamma ) \ar[r] \ar[u] & (B', J', \delta ') \ar[u] }$

in the category of divided power rings. Then we see that we have

$B''/J'' = B/J \otimes _{A/I} B'/J' \longleftarrow C \otimes _{A/I} C$

see Divided Power Algebra, Remark 23.3.5. Denote $J'' \subset K \subset B''$ the ideal such that

$\xymatrix{ B''/J'' \ar[r] & B''/K \\ C \otimes _{A/I} C \ar[r] \ar[u] & C \ar[u] }$

is a pushout, i.e., $B''/K \cong B/J \otimes _ C B'/J'$. Let $D_{B''}(K) = (D, \bar K, \bar\delta )$ be the divided power envelope of $K$ in $B''$ relative to $(B'', J'', \delta '')$. Then it is easily verified that $(D, \bar K, \bar\delta )$ is a coproduct of $(B, J, \delta )$ and $(B', J', \delta ')$ in $\text{CRIS}(C/A)$.

Next, we come to coequalizers. Let $\alpha , \beta : (B, J, \delta ) \to (B', J', \delta ')$ be morphisms of $\text{CRIS}(C/A)$. Consider $B'' = B'/ (\alpha (b) - \beta (b))$. Let $J'' \subset B''$ be the image of $J'$. Let $D_{B''}(J'') = (D, \bar J, \bar\delta )$ be the divided power envelope of $J''$ in $B''$ relative to $(B', J', \delta ')$. Then it is easily verified that $(D, \bar J, \bar\delta )$ is the coequalizer of $(B, J, \delta )$ and $(B', J', \delta ')$ in $\text{CRIS}(C/A)$.

By Categories, Lemma 4.18.6 we have all finite nonempty colimits in $\text{CRIS}(C/A)$. The constructions above shows that (60.5.2.1) commutes with them. This formally implies part (3) as $\text{Cris}(C/A)$ is the fibre category of (60.5.2.1) over $C$. $\square$

Comment #6002 by Simon Paege on

The empty product of thickenings is not $C$ but the zero ring, right? In general there shouldn't be a canonical map $B \to C$ for a thickening $B$.

Comment #7048 by nkym on

I guess that in the proof of (1), 07GV should be cited instead of 07GX.

Comment #7052 by nkym on

In fact, I am now wondering if the usual product construction in the proof of (1) is legitimate since it does not seem to assure that $p$ is nilpotent in $\prod B_t$. For example, how about setting $(A, I, \gamma) = (\mathbb{Z}_{(p)}, 0, \emptyset)$, $C = \mathbb{Z}/p\mathbb{Z}$, $T$ to be the set of positive integers, and $(B_t, J_t, \delta_t) = (\mathbb{Z}/p^t\mathbb{Z}, p\mathbb{Z}/p^t\mathbb{Z}, (px\mapsto \frac{p^n}{n!}x^n)_{n>0})$ with the obvious isomorphism $C\to B_t/J_t$, $\emptyset$ meaning $(0\to 0)_{n>0}$?

Comment #7053 by on

Yes, good catch. Your example shows that the category $\text{CRIS}(C/A)$ does not have infinite products. Finite products do exist. In your example you could also have used $(A, I, \gamma)$ to be $A = \mathbf{Z}_{(p)}$ and $I = (p)$ with the usual divided powers. I will fix this the next time I go through all the comments. Thanks.

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