Processing math: 100%

The Stacks project

Remark 60.24.4 (Alternating 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{\mathrm{Spec}}(A) and S_0 = \mathop{\mathrm{Spec}}(A/I). Let X be a separated quasi-compact 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 a finite affine open covering X = \bigcup _{\lambda \in \Lambda } U_\lambda of X and a total ordering on \Lambda . Write U_\lambda = \mathop{\mathrm{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 an alternating Čech complex which computes R\Gamma (\text{Cris}(X/S), \mathcal{F}).

We are going to use the notation introduced in Remark 60.24.3. Denote \Omega _{\lambda _0 \ldots \lambda _ n} the p-adically completed module of differentials of D_{\lambda _0 \ldots \lambda _ n} over A compatible with the divided power structure. Let \nabla be the integrable connection on M_{\lambda _0 \ldots \lambda _ n} coming from Proposition 60.17.4. Consider the double complex M^{\bullet , \bullet } with terms

M^{n, m} = \bigoplus \nolimits _{\lambda _0 < \ldots < \lambda _ n} M_{\lambda _0 \ldots \lambda _ n} \otimes ^\wedge _{D_{\lambda _0 \ldots \lambda _ n}} \Omega ^ m_{D_{\lambda _0 \ldots \lambda _ n}}.

For the differential d_1 (increasing n) we use the usual Čech differential and for the differential d_2 we use the connection, i.e., the differential of the de Rham complex. We claim that

R\Gamma (\text{Cris}(X/S), \mathcal{F}) = \text{Tot}(M^{\bullet , \bullet })

Here \text{Tot}(-) denotes the total complex associated to a double complex, see Homology, Definition 12.18.3.

Hints: We have

R\Gamma (\text{Cris}(X/S), \mathcal{F}) = R\Gamma (\text{Cris}(X/S), \mathcal{F} \otimes _{\mathcal{O}_{X/S}} \Omega _{X/S}^\bullet )

by Proposition 60.23.1. The right hand side of the formula is simply the alternating Čech complex for the covering X = \bigcup _{\lambda \in \Lambda } U_\lambda (which induces an open covering of the final sheaf of \text{Cris}(X/S)) and the complex \mathcal{F} \otimes _{\mathcal{O}_{X/S}} \Omega _{X/S}^\bullet , see Proposition 60.21.3. Now the result follows from a general result in cohomology on sites, namely that the alternating Čech complex computes the cohomology provided it gives the correct answer on all the pieces (insert future reference here).


Comments (0)

There are also:

  • 2 comment(s) on Section 60.24: Some further results

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.