nLab
Witt cohomology

Contents

Idea

Serre decided to try taking coefficients in the Witt vectors as an early attempt at a Weil cohomology theory. Ultimately, it wasn’t successful for this purpose, but has been generalized in several ways for other purposes with great success.

Sheaf of Witt vectors

Let XX be a scheme over a perfect field kk of positive characteristic pp. Let WW and W nW_n be the functors of Witt vectors and truncated Witt vectors respectively. The functorial nature allows us to define a sheaf of Witt vectors 𝒲\mathcal{W} and 𝒲 n\mathcal{W}_n just by taking Witt vectors of the rings of sections of 𝒪 X\mathcal{O}_X.

Note that as a sheaf of sets 𝒲 n\mathcal{W}_n is just 𝒪 X n\mathcal{O}_X^n. The ring structure is just the addition and multiplication of the Witt vectors. The operations on the Witt vectors sheafify as well. When nmn\geq m we have the exact sequence 0𝒲 mV𝒲 nR𝒲 nm00\to \mathcal{W}_m\stackrel{V}{\to} \mathcal{W}_n\stackrel{R}{\to}\mathcal{W}_{n-m}\to 0. If we take m=1m=1, then we get the sequence 0𝒪 X𝒲 n𝒲 n100\to \mathcal{O}_X\to \mathcal{W}_n\to \mathcal{W}_{n-1}\to 0

Definition

The sheaf of Witt vectors is an abelian sheaf, so we just define cohomology H q(X,𝒲 n)H^q(X, \mathcal{W}_n) as the standard sheaf cohomology (on the Zariski site of XX). Let Λ=W(k)\Lambda=W(k), then since 𝒲 n\mathcal{W}_n are Λ\Lambda-modules annihilated by p nΛp^n\Lambda, we get that H q(X,𝒲 n)H^q(X, \mathcal{W}_n) are also Λ\Lambda-modules annihilated by p nΛp^n\Lambda.

In fact, all of our old operators FF, VV, and RR still act on H q(X,𝒲 n)H^q(X, \mathcal{W}_n). They are easily seen to satisfy the formulas F(λw)=F(λ)F(w)F(\lambda w)=F(\lambda)F(w), V(λw)=F 1(λ)V(w)V(\lambda w)=F^{-1}(\lambda)V(w), and R(λw)=λR(w)R(\lambda w)=\lambda R(w) for λΛ\lambda\in \Lambda. If XX is projective then H q(X,𝒲 n)H^q(X, \mathcal{W}_n) is a finite Λ\Lambda-module.

What is usually referred to with Witt cohomology is H q(X,𝒲)H^q(X, \mathcal{W}) which is defined to be limH q(X,𝒲 n)lim H^q(X, \mathcal{W}_n). Note that even if XX is projective, this limit does not have to be a finite type Λ\Lambda-module.

References

  • J.P. Serre, Sur la topologie des variétés algébriques en caractéristique p, Symposium de Topologie Algébrique, Mexico (August, 1956)

Revised on August 8, 2012 00:49:30 by Toby Bartels (98.16.182.220)