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59.7 Divided power schemes

Some remarks on how to globalize the previous notions.

Definition 59.7.1. 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 59.7.2. 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 59.7.3. 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 59.7.4. 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 59.5.3. $\square$

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

  1. $p$ locally nilpotent on $T \Leftrightarrow U \to T$ is a thickening (see Divided Power Algebra, Lemma 23.2.6), and

  2. $p^ e\mathcal{O}_ T$ 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 59.7.5. 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 (2)

Comment #1604 by Rakesh Pawar on

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.


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