## 60.5 Affine crystalline site

In this section we discuss the algebraic variant of the crystalline site. Our basic situation in which we discuss this material will be as follows.

Situation 60.5.1. Here $p$ is a prime number, $(A, I, \gamma )$ is a divided power ring such that $A$ is a $\mathbf{Z}_{(p)}$-algebra, and $A \to C$ is a ring map such that $IC = 0$ and such that $p$ is nilpotent in $C$.

Usually the prime number $p$ will be contained in the divided power ideal $I$.

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.

Note that for a divided power thickening $(B, J, \delta )$ as above the ideal $J$ is locally nilpotent, see Divided Power Algebra, Lemma 23.2.6. There is a canonical functor

60.5.2.1
$$\label{crystalline-equation-forget-affine} \text{CRIS}(C/A) \longrightarrow C\text{-algebras},\quad (B, J, \delta ) \longmapsto B/J$$

This category does not have equalizers or fibre products in general. It also doesn't have an initial object ($=$ empty colimit) in general.

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 $C$. $\square$

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.

Lemma 60.5.5. In Situation 60.5.1. Let $P \to C$ be a surjection of $A$-algebras with kernel $J$. Write $D_{P, \gamma }(J) = (D, \bar J, \bar\gamma )$. Let $(D^\wedge , J^\wedge , \bar\gamma ^\wedge )$ be the $p$-adic completion of $D$, see Divided Power Algebra, Lemma 23.4.5. For every $e \geq 1$ set $P_ e = P/p^ eP$ and $J_ e \subset P_ e$ the image of $J$ and write $D_{P_ e, \gamma }(J_ e) = (D_ e, \bar J_ e, \bar\gamma )$. Then for all $e$ large enough we have

1. $p^ eD \subset \bar J$ and $p^ eD^\wedge \subset \bar J^\wedge$ are preserved by divided powers,

2. $D^\wedge /p^ eD^\wedge = D/p^ eD = D_ e$ as divided power rings,

3. $(D_ e, \bar J_ e, \bar\gamma )$ is an object of $\text{Cris}(C/A)$,

4. $(D^\wedge , \bar J^\wedge , \bar\gamma ^\wedge )$ is equal to $\mathop{\mathrm{lim}}\nolimits _ e (D_ e, \bar J_ e, \bar\gamma )$, and

5. $(D^\wedge , \bar J^\wedge , \bar\gamma ^\wedge )$ is an object of $\text{Cris}^\wedge (C/A)$.

Proof. Part (1) follows from Divided Power Algebra, Lemma 23.4.5. It is a general property of $p$-adic completion that $D/p^ eD = D^\wedge /p^ eD^\wedge$. Since $D/p^ eD$ is a divided power ring and since $P \to D/p^ eD$ factors through $P_ e$, the universal property of $D_ e$ produces a map $D_ e \to D/p^ eD$. Conversely, the universal property of $D$ produces a map $D \to D_ e$ which factors through $D/p^ eD$. We omit the verification that these maps are mutually inverse. This proves (2). If $e$ is large enough, then $p^ eC = 0$, hence we see (3) holds. Part (4) follows from Divided Power Algebra, Lemma 23.4.5. Part (5) is clear from the definitions. $\square$

Lemma 60.5.6. In Situation 60.5.1. Let $P$ be a polynomial algebra over $A$ and let $P \to C$ be a surjection of $A$-algebras with kernel $J$. With $(D_ e, \bar J_ e, \bar\gamma )$ as in Lemma 60.5.5: for every object $(B, J_ B, \delta )$ of $\text{CRIS}(C/A)$ there exists an $e$ and a morphism $D_ e \to B$ of $\text{CRIS}(C/A)$.

Proof. We can find an $A$-algebra homomorphism $P \to B$ lifting the map $C \to B/J_ B$. By our definition of $\text{CRIS}(C/A)$ we see that $p^ eB = 0$ for some $e$ hence $P \to B$ factors as $P \to P_ e \to B$. By the universal property of the divided power envelope we conclude that $P_ e \to B$ factors through $D_ e$. $\square$

Lemma 60.5.7. In Situation 60.5.1. Let $P$ be a polynomial algebra over $A$ and let $P \to C$ be a surjection of $A$-algebras with kernel $J$. Let $(D, \bar J, \bar\gamma )$ be the $p$-adic completion of $D_{P, \gamma }(J)$. For every object $(B \to C, \delta )$ of $\text{Cris}^\wedge (C/A)$ there exists a morphism $D \to B$ of $\text{Cris}^\wedge (C/A)$.

Proof. We can find an $A$-algebra homomorphism $P \to B$ compatible with maps to $C$. By our definition of $\text{Cris}^\wedge (C/A)$ we see that $P \to B$ factors as $P \to D_{P, \gamma }(J) \to B$. As $B$ is $p$-adically complete we can factor this map through $D$. $\square$

Comment #1464 by sdf on

How should one interpret the commutative diagram appearing in Definition 47.5.2? The language seems to be strongly suggesting something viz. the use of "thickening" in "P.D. thickening" ?

Comment #1465 by on

When I interpret your question as "what is the relationship between 'divided power thickenings' and 'PD-thickening'", the answer is that PD is a French abbreviation for "puissances divisÃ©es", which translates to "divided power" in English. But maybe this is not what you are asking.

Comment #1466 by sdf on

No sorry, my question was badly written. I was wondering of the use of the word "thickening" - what object should we view as being thickened when we have this structure? It is like we should view C being "thickened" by the homomorphism, because we have this sort of compatible divided power structure on A->B? I am just interested in knowing why the choice was made to use the word "thickening" since it is very suggestive.

Comment #1486 by on

Maybe the terminology could be better... The idea is that if $(B, J, \delta)$ is a divided power algebra, then we think of $\Spec(B)$ as a thickening of $\Spec(B/J)$ (see Definition 37.2.1). This isn't entirely correct, because the ideal $J$ does not have to be locally nilpotent, although it is if $p$ is nilpotent in $B$, see Lemma 23.2.6. So a "divided power thickening" isn't literally a thickening in this setting. Thus in the situation of the Definition 60.5.2 we view $\Spec(B/J)$ as a scheme over $\Spec(C)$ which comes with a divided power thickening over the base $(A, I, \gamma)$.

Comment #1500 by sdf on

Ah, I wasn't aware of the notion of thickening in the sense of Definition 04EX. That clears everything up, thanks.

Comment #8696 by Xiangru Zeng on

In the proof of Lemma 07KI, I believe the $\text{Cris}(C/A)$ should be $\text{Cris}^{\wedge}(C/A)$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).