Section 60.7 (07I1): Divided power schemes

60.7 Divided power schemes

Some remarks on how to globalize the previous notions.

Definition 60.7.1. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$ . Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. A divided power structure $\gamma$ on $\mathcal{I}$ is a sequence of maps $\gamma _ n : \mathcal{I} \to \mathcal{I}$ , $n \geq 1$ such that for any object $U$ of $\mathcal{C}$ the triple

$(\mathcal{O}(U), \mathcal{I}(U), \gamma )$

is a divided power ring.

To be sure this applies in particular to sheaves of rings on topological spaces. But it's good to be a little bit more general as the structure sheaf of the crystalline site lives on a... site! A triple $(\mathcal{C}, \mathcal{I}, \gamma )$ as in the definition above is sometimes called a divided power topos in this chapter. Given a second $(\mathcal{C}', \mathcal{I}', \gamma ')$ and given a morphism of ringed topoi $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ we say that $(f, f^\sharp )$ induces a morphism of divided power topoi if $f^\sharp (f^{-1}\mathcal{I}') \subset \mathcal{I}$ and the diagrams

$\xymatrix{ f^{-1}\mathcal{I}' \ar[d]_{f^{-1}\gamma '_ n} \ar[r]_{f^\sharp } & \mathcal{I} \ar[d]^{\gamma _ n} \\ f^{-1}\mathcal{I}' \ar[r]^{f^\sharp } & \mathcal{I} }$

are commutative for all $n \geq 1$ . If $f$ comes from a morphism of sites induced by a functor $u : \mathcal{C}' \to \mathcal{C}$ then this just means that

$(\mathcal{O}'(U'), \mathcal{I}'(U'), \gamma ') \longrightarrow (\mathcal{O}(u(U')), \mathcal{I}(u(U')), \gamma )$

is a homomorphism of divided power rings for all $U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ .

In the case of schemes we require the divided power ideal to be quasi-coherent. But apart from this the definition is exactly the same as in the case of topoi. Here it is.

Definition 60.7.2. A divided power scheme is a triple $(S, \mathcal{I}, \gamma )$ where $S$ is a scheme, $\mathcal{I}$ is a quasi-coherent sheaf of ideals, and $\gamma$ is a divided power structure on $\mathcal{I}$ . A morphism of divided power schemes $(S, \mathcal{I}, \gamma ) \to (S', \mathcal{I}', \gamma ')$ is a morphism of schemes $f : S \to S'$ such that $f^{-1}\mathcal{I}'\mathcal{O}_ S \subset \mathcal{I}$ and such that

$(\mathcal{O}_{S'}(U'), \mathcal{I}'(U'), \gamma ') \longrightarrow (\mathcal{O}_ S(f^{-1}U'), \mathcal{I}(f^{-1}U'), \gamma )$

is a homomorphism of divided power rings for all $U' \subset S'$ open.

Recall that there is a 1-to-1 correspondence between quasi-coherent sheaves of ideals and closed immersions, see Morphisms, Section 29.2. Thus given a divided power scheme $(T, \mathcal{J}, \gamma )$ we get a canonical closed immersion $U \to T$ defined by $\mathcal{J}$ . Conversely, given a closed immersion $U \to T$ and a divided power structure $\gamma$ on the sheaf of ideals $\mathcal{J}$ associated to $U \to T$ we obtain a divided power scheme $(T, \mathcal{J}, \gamma )$ . In many situations we only want to consider such triples $(U, T, \gamma )$ when the morphism $U \to T$ is a thickening, see More on Morphisms, Definition 37.2.1.

Definition 60.7.3. A triple $(U, T, \gamma )$ as above is called a divided power thickening if $U \to T$ is a thickening.

Fibre products of divided power schemes exist when one of the three is a divided power thickening. Here is a formal statement.

Lemma 60.7.4. Let $(U', T', \delta ') \to (S'_0, S', \gamma ')$ and $(S_0, S, \gamma ) \to (S'_0, S', \gamma ')$ be morphisms of divided power schemes. If $(U', T', \delta ')$ is a divided power thickening, then there exists a divided power scheme $(T_0, T, \delta )$ and

$\xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' }$

which is a cartesian diagram in the category of divided power schemes.

Proof. Omitted. Hints: If $T$ exists, then $T_0 = S_0 \times _{S'_0} U'$ (argue as in Divided Power Algebra, Remark 23.3.5). Since $T'$ is a divided power thickening, we see that $T$ (if it exists) will be a divided power thickening too. Hence we can define $T$ as the scheme with underlying topological space the underlying topological space of $T_0 = S_0 \times _{S'_0} U'$ and as structure sheaf on affine pieces the ring given by Lemma 60.5.3. $\square$

We make the following observation. Suppose that $(U, T, \gamma )$ is triple as above. Assume that $T$ is a scheme over $\mathbf{Z}_{(p)}$ and that $p$ is locally nilpotent on $U$ . Then

$p$ locally nilpotent on $T \Leftrightarrow U \to T$ is a thickening (see Divided Power Algebra, Lemma 23.2.6), and
$p^ e\mathcal{O}_ T$ is locally on $T$ preserved by $\gamma$ for $e \gg 0$ (see Divided Power Algebra, Lemma 23.4.5).

This suggest that good results on divided power thickenings will be available under the following hypotheses.

Situation 60.7.5. Here $p$ is a prime number and $(S, \mathcal{I}, \gamma )$ is a divided power scheme over $\mathbf{Z}_{(p)}$ . We set $S_0 = V(\mathcal{I}) \subset S$ . Finally, $X \to S_0$ is a morphism of schemes such that $p$ is locally nilpotent on $X$ .

It is in this situation that we will define the big and small crystalline sites.

Comments (5)

Comment #1604 by Rakesh Pawar on August 18, 2015 at 05:09

In the definition 48.7.2 the expression of the morphism of divided power rings should be corrected. In the first triple S, I, \gamma should be replaced with S', I', \gamma' and in the next triple S' should be replaced with S.

Comment #1678 by Johan on October 08, 2015 at 17:06

Dear Rakesh, thanks for this and the other comments, they are now fixed here, here, here, here.

Comment #8721 by Eiki on November 05, 2023 at 11:20

In Definition 07I4, shouldn't it be $(T,U,\gamma)$ if we are identifying the closed subscheme $U$ as a quasicoherent sheaf of ideals?

Comment #9354 by Stacks project on June 04, 2024 at 10:34

Going to leave as is.

Comment #9783 by Frank on September 25, 2024 at 16:13

It doesn't seem like the proof sketch of Lemma 48.7.5 works in the generality that the lemma is stated. For instance, one could take $S' = (\mathbb{Z}, 0, \emptyset), T' = (\mathbb{Z}[x]/(x^2), (x), 0),$ and $S$ being the divided power polynomial algebra in one variable over $S'.$ Then, using the universal property of divided power polynomial algebras, $T$ should be the divided power algebra in one variable over $T'.$ But $T$ is not a thickening, since the ideal $\mathbb{Z}[x]/(x)^2\langle y \rangle_+$ is not locally nilpotent (which is counter to the statement given in the proof sketch).

Maybe I'm missing some hypotheses that were implicitly given in the chapter (but which weren't explicitly stated), or misunderstanding the definition of a thickening?

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60.7 Divided power schemes

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