Integral comparison of Monsky–Washnitzer and overconvergent de Rham–Witt cohomology
By Veronika Ertl and Johannes Sprang
Abstract
The goal of this short note is to extend a result by Christopher Davis and David Zureick-Brown on the comparison between integral Monsky–Washnitzer cohomology and overconvergent de Rham–Witt cohomology for a smooth variety over a perfect field of positive characteristic $p$ to all cohomological degrees independent of the dimension of the base or the prime number $p$.
Résumé. Le but de ce travail est de prolonger un résultat de Christopher Davis et David Zureick-Brown concernant la comparaison entre la cohomologie de Monsky–Washnitzer entière et la cohomologie de de Rham–Witt surconvergente d’une variété lisse sur un corps parfait de charactéristique positive $p$ à tous les degrés cohomologiques indépendent de la dimension de base et du nombre premier $p$.
Introduction
Let $k$ be a perfect field of positive characteristic $p$. As usual denote by $W(k)$ the ring of $p$-typical Witt vectors of $k$, and let $K$ be the fraction field of $W(k)$. By a variety over $k$ we always mean a separated and integral scheme of finite type over the field $k$.
In Reference 2 Christopher Davis, Andreas Langer, and Thomas Zink define for a finitely generated $k$-algebra$\bar{A}$ the overconvergent de Rham–Witt complex $W^\dagger \Omega ^{\scriptscriptstyle \overset{\bullet }{}}_{\bar{A}/k}\subset W\Omega ^{\scriptscriptstyle \overset{\bullet }{}}_{\bar{A}/k}$, which can be globalised to a sheaf on a smooth variety $X$ over $k$. One of their main results is to compare the cohomology of this complex to Monsky–Washnitzer cohomology.
According to Elkik (cf. Reference 7, Sec. 2) there is a smooth finitely generated $W(k)$-algebra$A$ which reduces to $\bar{A}$. Let $\widehat{A}$ be the $p$-adic completion of $A$. The weak completion $A^\dagger$ of $A$ is the smallest $p$-adically saturated subring of $\widehat{A}$ containing $A$ and all series of the form $\sum _{i_1,\ldots ,i_n\geqslant 0}c_{i_1\cdots i_n}x_1^{i_1}\cdots x_n^{i_n}$ with $c_{i_1\cdots i_n}\in W(k)$ and $x_j\in pA^\dagger$. It is a weak formalisation of $\bar{A}$ in the sense of Reference 6, Def. 3.2, and according to Reference 7, (2.4.4) Thm. any two weak formalisations are isomorphic. By construction it is weakly finitely generated. For short it is common to write w.c.f.g. for weakly complete and weakly finitely generated algebras.
Let $\widetilde{\Omega }^{\scriptscriptstyle \overset{\bullet }{}}_{A^\dagger /W(k)}$ be the module of continuous differentials of $A^\dagger$ over $W(k)$. The Monsky–Washnitzer cohomology of $\operatorname {\mathrm{Spec}}\bar{A}$ is then calculated by the rational complex $\widetilde{\Omega }^{\scriptscriptstyle \overset{\bullet }{}}_{A^\dagger /W(k),\mathbb{Q}}:=\widetilde{\Omega }^{\scriptscriptstyle \overset{\bullet }{}}_{A^\dagger /W(k)}\otimes \mathbb{Q}$:
A comparison map between the two complexes in question can be obtained as follows. For a smooth lift of the Frobenius $f:A^\dagger \rightarrow A^\dagger$, which always exists, there is a monomorphism
which has in fact image in the overconvergent subring $W^\dagger (\bar{A})\subset W(\bar{A})$ and induces by the universal property of Kähler differentials and functoriality a comparison map
The main result of Davis, Langer, and Zink regarding this comparison morphism is the following Reference 2, Cor. 3.25.
Davis and Zureick-Brown generalise the first statement, which is an integral result to a comparison independent of the dimension of $\bar{A}$Reference 4, Thm. 1.1.
The goal of this paper is to take the argumentation of Davis and Zureick-Brown a step further and show that in the situation above the comparison map
in all cohomological degrees. A crucial ingredient is the extension of a homotopy result by Monsky and Washnitzer. In Reference 6, Rem. (3), p. 205 they assert that if $A$ and $B$ are w.c.f.g. algebras over $W(k)$, with $A$ flat and $A/pA$ a complete transversal intersection, then two homomorphisms $\psi _1,\psi _2\colon A\rightarrow B$ with the same reduction modulo $p$ induce chain homotopic maps on the associated continuous de Rham complexes
In the first section, we show that this is in fact true without the assumptions that $A/pA$ is a complete transversal intersection and that $B$ is weakly finitely generated. We apply this in two instances.
In the second section we revisit integral Monsky–Washnitzer cohomology. Bearing in mind Reference 4, Thm. 1.1(1), it remains only to show that the cohomology groups $H^i_{\mathrm{MW}}(\operatorname {\mathrm{Spec}}\bar{A}/W(k)):= H^i(\widetilde{\Omega }^{\scriptscriptstyle \overset{\bullet }{}}_{A^\dagger /W(k)})$ are functorial in $\bar{A}$. Using the homotopy result of the previous section we don’t have to reduce to transversal intersections, as the desired statement follows directly.
We turn our attention to the comparison map in the last section. As mentioned above, the problem is that it depends a priori on the choice of a Frobenius lift $f$. We use again the homotopy result mentioned above to show that two maps
which coincide on the reduction modulo $p$ induce chain homotopic maps on the associated complexes. In this instance it is important that the homotopy be valid in a case where the target is not weakly finitely generated. It then suffices to invoke the universal property of the continuous de Rham complex. Similarly to Reference 4 we now use the fact that any smooth variety can be covered by special affines on which we know that a quasi-isomorphism $\sigma \colon \widetilde{\Omega }^{\scriptscriptstyle \overset{\bullet }{}}_{A_I^\dagger /W(k)}\rightarrow W^\dagger \Omega _{\bar{A}_I/k}^{\scriptscriptstyle \overset{\bullet }{}}$ exists.
In summary we obtain the following result.
1. A homotopy result
The heart of the following proposition is essentially Reference 6, Rem. (3), p. 205 with the additional observation that it is neither necessary to assume that the target of the maps in question is weakly finitely generated nor that the reduction of the source is a complete transversal intersection. While the latter allows us to shorten the proofs in Sections 2 and 3 considerably, the first is crucial because we would like to apply the statement to maps $\psi _1,\psi _2 \colon A^\dagger \rightarrow W^\dagger ({\bar{A}})$ for a smooth $k$-algebra$\bar{A}$ of finite type, and while $W^\dagger (\bar{A})$ is weakly complete Reference 3, Prop. 2.28, it is in general not weakly finitely generated. We recall the proof of Reference 6, Rem. (3), p. 205 with the necessary modifications.
2. Functoriality of integral Monsky–Washnitzer cohomology
Let $\bar{A}$ be a smooth finite $k$-algebra. In Reference 4 Davis and Zureick-Brown prove the existence of an isomorphism
for two different smooth lifts $A$ and $A'$ with weak completions $A^\dagger$ and $(A')^\dagger$ of a non-singular affine $k$-variety$\bar{A}$. This section is nothing but the observation that their argument can also be used to prove functoriality of integral Monsky–Washnitzer cohomology in $\bar{A}$ in order to obtain the following result.
3. An unconditional comparison
In this section, we want to use intrinsic properties of weakly complete weakly finitely generated (w.c.f.g.) algebras to obtain a comparison result between integral Monsky–Washnitzer and overconvergent de Rham–Witt cohomology.
To define a comparison map we consider for a non-singular affine variety $\operatorname {\mathrm{Spec}}\bar{A}$ over $k$, a weak formalisation $A^\dagger$ and a lifting $f\colon A^\dagger \rightarrow A^\dagger$ of the Frobenius morphism $\mathrm{Frob}\colon \bar{A}\rightarrow \bar{A}$.
Recursively, one can define a unique ring homomorphism
such that the ghost components of $s_f(a)$ for $a\in A^\dagger$ are given by $(a,f(a),f^2(a),\ldots )$. As noted in Reference 5, (0.1.3.16) it is functorial in the triple $(\bar{A},A^\dagger ,f)$ in the sense that if $(\bar{A}',{A'}^\dagger , f')$ is another such triple and $\varphi \colon A^\dagger \rightarrow {A'}^\dagger$ is a map commuting with the Frobenius lifts, i.e., the left square of the following diagram commutes, then the right diagram commutes as well:
Let $t_f=W(\pi )\circ s_f \colon A^\dagger \rightarrow W(\bar{A})$ be the composition of $s_f$ with the map induced by the reduction $\pi \colon A^\dagger \rightarrow \bar{A}$. According to Reference 2, Prop. 3.2 it factors through $W^\dagger (\bar{A})$, and one obtains
which by the universal property of the continuous de Rham complex finally results in the desired comparison map between complexes $t_f\colon \widetilde{\Omega }^{\scriptscriptstyle \overset{\bullet }{}}_{A^\dagger /W(k)}\rightarrow W^\dagger \Omega ^{\scriptscriptstyle \overset{\bullet }{}}_{\bar{A}/k}$. One observes right away that the reduction of $t_f$ modulo $p$ is the identity. We aim to show that the induced map on cohomology is an isomorphism which is independent of the choice of Frobenius lift.
Acknowledgments
We are indebted to Kennichi Bannai and Kazuki Yamada for helpful discussions and suggestions related to the content of this paper. Bernard Le Stum’s insight on the topic, which was passed on to us by Christopher J. Davis, led us to consider the paper of Alberto Arabia. We would like to thank both of them for generously sharing their knowledge and giving us important feedback.
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