The Stacks Project


Tag 07MM

Chapter 54: Crystalline Cohomology > Section 54.24: Some further results

Remark 54.24.3 (Čech complex). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $X$ be a separated1 scheme over $S_0$ such that $p$ is locally nilpotent on $X$. Let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.

Choose an affine open covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$. Write $U_\lambda = \mathop{\rm Spec}(C_\lambda)$. Choose a polynomial algebra $P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$. Having fixed these choices we can construct a Čech complex which computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.

Given $n \geq 0$ and $\lambda_0, \ldots, \lambda_n \in \Lambda$ write $U_{\lambda_0 \ldots \lambda_n} = U_{\lambda_0} \cap \ldots \cap U_{\lambda_n}$. This is an affine scheme by assumption. Write $U_{\lambda_0 \ldots \lambda_n} = \mathop{\rm Spec}(C_{\lambda_0 \ldots \lambda_n})$. Set $$ P_{\lambda_0 \ldots \lambda_n} = P_{\lambda_0} \otimes_A \ldots \otimes_A P_{\lambda_n} $$ which comes with a canonical surjection onto $C_{\lambda_0 \ldots \lambda_n}$. Denote the kernel $J_{\lambda_0 \ldots \lambda_n}$ and set $D_{\lambda_0 \ldots \lambda_n}$ the $p$-adically completed divided power envelope of $J_{\lambda_0 \ldots \lambda_n}$ in $P_{\lambda_0 \ldots \lambda_n}$ relative to $\gamma$. Let $M_{\lambda_0 \ldots \lambda_n}$ be the $P_{\lambda_0 \ldots \lambda_n}$-module corresponding to the restriction of $\mathcal{F}$ to $\text{Cris}(U_{\lambda_0 \ldots \lambda_n}/S)$ via Proposition 54.17.4. By construction we obtain a cosimplicial divided power ring $D(*)$ having in degree $n$ the ring $$ D(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n} D_{\lambda_0 \ldots \lambda_n} $$ (use that divided power envelopes are functorial and the trivial cosimplicial structure on the ring $P(*)$ defined similarly). Since $M_{\lambda_0 \ldots \lambda_n}$ is the ''value'' of $\mathcal{F}$ on the objects $\mathop{\rm Spec}(D_{\lambda_0 \ldots \lambda_n})$ we see that $M(*)$ defined by the rule $$ M(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n} M_{\lambda_0 \ldots \lambda_n} $$ forms a cosimplicial $D(*)$-module. Now we claim that we have $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = s(M(*)) $$ Here $s(-)$ denotes the cochain complex associated to a cosimplicial module (see Simplicial, Section 14.25).

Hints: The proof of this is similar to the proof of Proposition 54.21.1 (in particular the result holds for any module satisfying the assumptions of that proposition).

  1. This assumption is not strictly necessary, as using hypercoverings the construction of the remark can be extended to the general case.

The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4472–4539 (see updates for more information).

\begin{remark}[{\v C}ech complex]
\label{remark-cech-complex}
Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \Spec(A)$ and
$S_0 = \Spec(A/I)$. Let $X$ be a separated\footnote{This assumption is
not strictly necessary, as using hypercoverings the construction of the
remark can be extended to the general case.} scheme over
$S_0$ such that $p$ is locally nilpotent on $X$. Let $\mathcal{F}$ be a
crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.

\medskip\noindent
Choose an affine open covering
$X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$.
Write $U_\lambda = \Spec(C_\lambda)$. Choose a polynomial algebra
$P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$.
Having fixed these choices we can construct a {\v C}ech complex which
computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.

\medskip\noindent
Given $n \geq 0$ and $\lambda_0, \ldots, \lambda_n \in \Lambda$
write $U_{\lambda_0 \ldots \lambda_n} = U_{\lambda_0} \cap \ldots
\cap U_{\lambda_n}$. This is an affine scheme by assumption. Write
$U_{\lambda_0 \ldots \lambda_n} = \Spec(C_{\lambda_0 \ldots \lambda_n})$.
Set
$$
P_{\lambda_0 \ldots \lambda_n} =
P_{\lambda_0} \otimes_A \ldots \otimes_A P_{\lambda_n}
$$
which comes with a canonical surjection onto $C_{\lambda_0 \ldots \lambda_n}$.
Denote the kernel $J_{\lambda_0 \ldots \lambda_n}$ and set
$D_{\lambda_0 \ldots \lambda_n}$
the $p$-adically completed divided power envelope of
$J_{\lambda_0 \ldots \lambda_n}$ in $P_{\lambda_0 \ldots \lambda_n}$
relative to $\gamma$. Let $M_{\lambda_0 \ldots \lambda_n}$ be the
$P_{\lambda_0 \ldots \lambda_n}$-module corresponding
to the restriction of $\mathcal{F}$ to
$\text{Cris}(U_{\lambda_0 \ldots \lambda_n}/S)$ via
Proposition \ref{proposition-crystals-on-affine}.
By construction we obtain a cosimplicial divided power ring $D(*)$
having in degree $n$ the ring
$$
D(n) =
\prod\nolimits_{\lambda_0 \ldots \lambda_n}
D_{\lambda_0 \ldots \lambda_n}
$$
(use that divided power envelopes are functorial and the trivial
cosimplicial structure on the ring $P(*)$ defined similarly).
Since $M_{\lambda_0 \ldots \lambda_n}$ is the ``value'' of $\mathcal{F}$
on the objects $\Spec(D_{\lambda_0 \ldots \lambda_n})$ we see that
$M(*)$ defined by the rule
$$
M(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n}
M_{\lambda_0 \ldots \lambda_n}
$$
forms a cosimplicial $D(*)$-module. Now we claim that we have
$$
R\Gamma(\text{Cris}(X/S), \mathcal{F}) = s(M(*))
$$
Here $s(-)$ denotes the cochain complex associated to a cosimplicial
module (see
Simplicial, Section \ref{simplicial-section-dold-kan-cosimplicial}).

\medskip\noindent
Hints: The proof of this is similar to the proof of
Proposition \ref{proposition-compute-cohomology} (in particular
the result holds for any module satisfying the assumptions of
that proposition).
\end{remark}

Comments (0)

There are no comments yet for this tag.

There are also 2 comments on Section 54.24: Crystalline Cohomology.

Add a comment on tag 07MM

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 lower-right corner).

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 box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

This captcha seems more appropriate than the usual illegible gibberish, right?