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60.4 Compatibility

This section isn't required reading; it explains how our discussion fits with that of [Berthelot]. Consider the following technical notion.

Definition 60.4.1. Let $(A, I, \gamma )$ and $(B, J, \delta )$ be divided power rings. Let $A \to B$ be a ring map. We say $\delta $ is compatible with $\gamma $ if there exists a divided power structure $\bar\gamma $ on $J + IB$ such that

\[ (A, I, \gamma ) \to (B, J + IB, \bar\gamma )\quad \text{and}\quad (B, J, \delta ) \to (B, J + IB, \bar\gamma ) \]

are homomorphisms of divided power rings.

Let $p$ be a prime number. Let $(A, I, \gamma )$ be a divided power ring. Let $A \to C$ be a ring map with $p$ nilpotent in $C$. Assume that $\gamma $ extends to $IC$ (see Divided Power Algebra, Lemma 23.4.2). In this situation, the (big affine) crystalline site of $\mathop{\mathrm{Spec}}(C)$ over $\mathop{\mathrm{Spec}}(A)$ as defined in [Berthelot] is the opposite of the category of systems

\[ (B, J, \delta , A \to B, C \to B/J) \]

where

  1. $(B, J, \delta )$ is a divided power ring with $p$ nilpotent in $B$,

  2. $\delta $ is compatible with $\gamma $, and

  3. the diagram

    \[ \xymatrix{ B \ar[r] & B/J \\ A \ar[u] \ar[r] & C \ar[u] } \]

    is commutative.

The conditions “$\gamma $ extends to $C$ and $\delta $ compatible with $\gamma $” are used in [Berthelot] to insure that the crystalline cohomology of $\mathop{\mathrm{Spec}}(C)$ is the same as the crystalline cohomology of $\mathop{\mathrm{Spec}}(C/IC)$. We will avoid this issue by working exclusively with $C$ such that $IC = 0$1. In this case, for a system $(B, J, \delta , A \to B, C \to B/J)$ as above, the commutativity of the displayed diagram above implies $IB \subset J$ and compatibility is equivalent to the condition that $(A, I, \gamma ) \to (B, J, \delta )$ is a homomorphism of divided power rings.

[1] Of course there will be a price to pay.

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