Chapter 13 Monsky-Washnitzer cohomology
Original lecture date: November 19, 2019.
In this lecture, we explain how to adapt algebraic de Rham cohomology to obtain the Monsky–Washnitzer cohomology of a smooth affine variety over a finite field. The construction globalizes naturally to smooth nonaffine varieties; the generalization to nonsmooth varieties is more difficult, and is part of Berthelot’s theory of rigid cohomology which we do not discuss in detail here.
Section 13.1
Definition 13.1.1.
Throughout this lecture, let \(k\) be a finite field. Let \(W(k)\) be the the ring of Witt vectors of \(k\text{;}\) all you need to know about this ring is that it is a finite étale algebra over \(\ZZ_p\) with \(W(k)/pW(k) \cong k\text{.}\) Let \(K\) be the fraction field of \(W(k)\text{,}\) which is obtained by inverting \(p\text{.}\)
Let \(X = \Spec(\overline{A})\) be a smooth affine scheme over \(k\text{.}\) Since a Weil cohomology theory has to have coefficient field of characteristic \(0\) (otherwise it cannot completely control the zeta function), if we want to use differential forms it will have to be over some characteristic-\(0\) lift of \(k\text{.}\)
Theorem 13.1.2. Elkik, Arabia.
There exists a smooth affine scheme \(\frakX = \Spec(A)\) over \(W(k)\) with \(\frakX \times_{W(k)} k \cong X\text{.}\)
Proof.
A crude summary of Elkik’s proof is the following. Since \(X\) is smooth, there is no local obstruction to finding at least a formal lift (i.e., a lift to the formal scheme \(\Spf W(k)\)). Since \(X\) is affine, there is also no global obstruction. To complete the proof, one must show that the existence of a formal lift implies the existence of an algebraic lift. A streamlined presentation and generalization of Elkik’s result has been given by Arabia [5].
Our first candidate for the cohomology of \(X\) is the de Rham cohomology of the generic fiber of a lift:
\begin{equation*}
H^i(\frakX_{W(k)[1/p]}, \Omega^\bullet).
\end{equation*}
Unfortunately, this is not independent of the choice of the lift \(\frakX\text{,}\) so we need to try something else.
Definition 13.1.3.
Let \(\hat{A}\) be the \(p\)-adic completion of \(A\text{.}\) We define the module of continuous Kähler differentials of \(\hat{A}\) to be
\begin{equation*}
\Omega_{\hat{A}[\frac{1}{p}]/K}:=\varprojlim \Omega_{(A/p^n)/(W(k)/p^n)}\otimes_{W(k)}K.
\end{equation*}
Since \(A\) is smooth, this is is a finite projective \(\hat{A}[\frac{1}{p}]\)-module.
Our second candidate for the cohomology of \(X\) is the cohomology of the resulting de Rham complex:
\begin{equation*}
H^{\bullet}(\Omega_{\hat{A}[\frac{1}{p}]/K}^{\bullet}).
\end{equation*}
Unfortunately, this turns out not to be finite-dimensional over \(K\text{!}\)
Example 13.1.4.
Take \(\overline{A} = k[X]\text{,}\) \(A = W(k)[X]\text{.}\) Then \(\hat{A}\) is the ring \(W(k) \langle X \rangle\) of null power series (also called strictly convergent power series) over \(W(k)\text{,}\) and similarly
\begin{equation*}
\hat{A}\left[\frac{1}{p}\right]=K\langle X\rangle=\left\{\sum_{n=0}^{\infty} a_{n} X^{n} \in K \llbracket X \rrbracket|a_n\rightarrow 0 \text{ for the }p\text{-adic topology}\right\}.
\end{equation*}
It is easy to see that \(\sum p^nX^{p^n-1}\) belongs to \(\hat{A}[\frac{1}{p}]\) but its antiderivative \(\sum X^{p^n}\) does not. By similar considerations, one may show that \(H^1(\Omega_{\hat{A}[\frac{1}{p}]/K}^{\bullet})\) is infinite-dimensional over \(K\text{.}\)
Definition 13.1.5.
As prelude for the general case, let us see how to modify this example to eliminate the issue we have just seen. If we think of \(K\langle X\rangle\) as the rigid analytic functions on the closed unit disc, the problem is that antidifferentiation preserves the radius of convergence but not the convergence at the boundary. (This is of course backwards from what happens in classical analysis, where it is differentiation of power series that has a similar problem.) To remedy this issue, we consider instead functions which are holomorphic on some larger disc: taking these together yields the ring
\begin{align*}
K\langle X\rangle^{\dagger}&=\left\{\sum_{n=0}^{\infty} a_{n} X^{n}|a_n\in K, \lim_{n\rightarrow \infty}\sup|a_n|^{\frac{1}{n}}\lt 1\right\}\\
&=\varinjlim_{\rho>1}\left\{\sum a_nX^n|a_n\in K, \lim_{n\rightarrow \infty}|a_n|\rho^n=0\right\}.
\end{align*}
One may verify easily that the sequence
\begin{equation*}
0 \to K \langle X \rangle^\dagger \stackrel{d}{\to} K \langle X \rangle^\dagger\,dx \to 0
\end{equation*}
has cohomology \(K\) in degree 0 (the constants) and 0 in degree 1 (differentiation is surjective).
Returning to the general case, we introduce the following definition.
Definition 13.1.6.
Let \(R\) be a ring and let \(I=(x_1,\dots,x_n)\subset R\) be a finitely generated ideal. Let \(\hat{R} := \varprojlim_{m \to \infty} R/I^m\) denote the \(I\)-adic completion of \(R\text{.}\) Note that for any \(y_1,\dots,y_n \in R\) and any power series \(c(X) = \sum_J c_J X_1^{j_1} \cdots X_n^{j_n} \in R \llbracket X_1,\dots,X_n\rrbracket\) for which there exists a function \(f(x)\) with \(\lim_{x \to \infty} f(x) = \infty\) and
\begin{equation*}
c_J \in I^{f(j_1 + \cdots + j_n)} \qquad (j_1,\dots,j_n \geq 0),
\end{equation*}
the evaluation \(c(y_1,\dots,y_n)\) makes sense as an element of \(R\text{.}\)
We define the weak completion of \(R\) with respect to \(I\) to be the smallest subring \(R^\dagger\) of \(\hat{R}\) with the following property: if \(y_1,\dots,y_n \in R^\dagger\) and \(c(X) = \sum_J c_J X_1^{j_1} \cdots X_n^{j_n} \in R \llbracket X_1,\dots,X_n\rrbracket\) is a power series for which there exists a constant \(C>0\) with
\begin{equation*}
c_J \in I^{\lfloor C(j_1 + \cdots + j_n) \rfloor} \qquad (j_1,\dots,j_n \geq 0),
\end{equation*}
then \(c(y_1,\dots,y_n) \in R^\dagger\text{.}\)
Example 13.1.7.
For \(R = W(k)[X_1,\dots,X_n]\text{,}\)
\begin{equation*}
R^\dagger = W(k) \langle X_1,\dots,X_n \rangle^\dagger :=
\varinjlim_{\rho>1}\left\{\sum a_I X^I|a_I\in W(k), \lim_{n\rightarrow \infty}|a_I|\rho^{i_1+\cdots+i_n}=0\right\}
\end{equation*}
is the ring of rigid analytic functions on all possible polydiscs of radius \(>1\text{.}\)
Theorem 13.1.8. Fulton; see [49].
Remark 13.1.9.
Corresponding to the passage from adically complete rings to formal schemes, one may use weak completions to define weak formal schemes. This was done by Meredith [96].
Definition 13.1.10.
With notation as before, let \(A^\dagger\) be the weak completion of \(A\) with respect to the ideal \((p)\text{.}\) We may compute \(A^\dagger\) by choosing a surjection \(W(k)[X_1,\dots,X_n] \to A\) and then taking
\begin{equation*}
A^\dagger = A \otimes_{W(k)[X_1,\dots,X_n]} W(k)\langle X_1,\dots,X_n\rangle^\dagger.
\end{equation*}
Using such a presentation, we may also define the module of continuous Kähler differentials \(\Omega_{A^{\dagger}[\frac{1}{p}]/K}\) (for \(A = W(k)[X_1,\dots,X_n]\) it will again be freely generated by \(dX_1,\dots,dX_n\)) and then define the Monsky–Washnitzer cohomology to be
\begin{equation*}
H^i_{\MW}(X):=\mathrm{H}^{i}(\Omega_{A^{\dagger}[\frac{1}{p}]/K}^{\bullet}).
\end{equation*}
To see that this gives something well-defined, we need to verify that it is independent of the choice of lifting and the presentation.
Theorem 13.1.11. Monsky–Washnitzer.
The Monsky–Washnitzer cohomology groups of \(X\) are independent of the choice of lift \(A\) and of the presentation of \(A\) as a finitely generated \(W(k)\)-algebra. Moreover, they are (contravariantly) functorial in \(X\text{.}\)
Proof.
Let us sketch some ideas behind the proof. One starts with a form of the Poincaré lemma: the natural map
\begin{equation*}
H^i_{\MW}(X) \to H^i_{\MW}(X\times_k \AAA_k^1)
\end{equation*}
is an isomorphism (using a particular presentation of \(X\) and the corresponding presentation of \(X\times_k \AAA_k^1\) by adding one more variable). This can then be applied in the following ways.
-
Adding generators to a presentation does not change cohomology. Given two presentations, we may then combine their generators to see that the cohomology groups given by the two presentations may be naturally identified.
-
Given two different lifts of the same morphisms, we may “interpolate” between the two to see that they define the same morphism in cohomology.
This gives everything we need, except that it is not yet apparent that one can always lift a morphism at all (even if one does not specify in advance a lift of either the source or target). This again follows from the theorem of Arabia [5].
Remark 13.1.12.
The proof of the Poincaré lemma gives more than just an isomorphism in cohomology, but also a chain homotopy witness for the fact that the composition \(H^i_{\MW}(X) \to H^i_{\MW}(X\times_k \AAA_k^1) \to H^i_{\MW}(X)\) is the identity (mapping \(X\) to \(X\times_k \AAA_k^1\) via the zero section). This makes it possible to globalize the definition of Monsky–Washnitzer cohomology to accommodate general smooth schemes over \(k\text{.}\)
The following is not a priori clear, and indeed was unknown to Monsky–Washnitzer.
Theorem 13.1.13. Berthelot.
Proof.
What makes Monsky–Washnitzer computable in practice is the following comparison theorem with the de Rham cohomlogy of the generic fiber.
Theorem 13.1.14.
If \(X\) is nice enough, then
\begin{equation*}
H^{i}_{\dR}(\frakX_{K})\stackrel{\simeq}{\rightarrow} H^{i}_{\MW}(X).
\end{equation*}
Here “nice enough” means that \(X\) admits a smooth proper compactification \(\overline{X}\) such that the complement \(\overline{X}-X\) corresponds to a normal crossings divisor and \(\overline{X}\) admits a smooth proper lifting over \(W(k)\text{,}\) in which the complement of a certain relative normal crossings divisor lifts \(X\text{.}\)
When \(X\) is not nice, we could use de Jong’s resolution of singularities and excision to reduce to the nice case.
Lemma 13.1.15. Excision.
Suppose \(X\) is smooth and \(Z\subset X\) is pure and smooth of codimension \(d\text{,}\) set \(U=X-Z\text{,}\) then we have a short exact sequence
\begin{equation*}
\cdots \rightarrow H_{\dR}^{i-2d}(Z)\rightarrow H_{\dR}^{i}(X)\rightarrow H_{\dR}^{i}(U)\rightarrow H_{\dR}^{i-2d+1}(Z)\rightarrow\cdots
\end{equation*}
The final piece needed to make Monsky–Washnitzer cohomology into a Weil cohomology theory is the Lefschetz trace formula for Frobenius. We will state and prove this in the next lecture.
