The Stacks project

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.


Comments (0)

There are also:

  • 5 comment(s) on Section 60.5: Affine crystalline site

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Your email address will not be published.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 07HM. Beware of the difference between the letter 'O' and the digit '0'.