The Stacks project

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

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


  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 ensure 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.

Comments (2)

Comment #7820 by Que on

minor typo: insure -> ensure

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07HJ. Beware of the difference between the letter 'O' and the digit '0'.