The Stacks project

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 ( 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 ( commutes with them. This formally implies part (3) as $\text{Cris}(C/A)$ is the fibre category of ( over $C$. $\square$

Comments (6)

Comment #6002 by Simon Paege on

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

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 is nilpotent in . For example, how about setting , , to be the set of positive integers, and with the obvious isomorphism , meaning ?

Comment #7053 by on

Yes, good catch. Your example shows that the category does not have infinite products. Finite products do exist. In your example you could also have used to be and with the usual divided powers. I will fix this the next time I go through all the comments. Thanks.

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