We compute the moduli of endomorphisms of the de Rham and crystalline cohomology functors, viewed as a cohomology theory on smooth schemes over truncated Witt vectors. As applications of our result, we deduce Drinfeld's refinement of the classical Deligne-Illusie decomposition result for de Rham cohomology of varieties in characteristic p > 0 that are liftable to W 2 , and prove further functorial improvements.

We prove that algebraic de Rham cohomology as a functor defined on smooth Fp-algebras is formally étale in a precise sense. To prove this, we define and study the notion of a pointed G perf a -module and its refinement which we call a quasi-ideal in G perf a following Drinfeld. Our results show that given de Rham cohomology, one obtains the theory of crystalline cohomology as its unique functorial deformation.1. We reduce the problem to the case where the base Artinian local ring A is F p [ǫ]/ǫ 2 . 2. Given any deformation dR ′ of dR, we extract a quasi-ideal from dR ′ denoted as r(dR ′ ) which is a deformation of u * W [F ]. 3. We show that dR ′ is essentially determined by the quasi-ideal r(dR ′ ). 4. We show that any deformation of u * W [F ] to F p [ǫ]/ǫ 2 as a pointed G perf a -module is uniquely isomorphic to the trivial deformation obtained by base change. This is proven in Proposition 2.5.11. Therefore r(dR ′ ) is necessarily the trivial deformation of u * W [F ] and by 3, dR ′ is necessarily the trivial deformation dR⊗F p [ǫ]/ǫ 2 as well.Other approaches to Theorem 1.1.1. Our approach to Theorem 1.1.1 uses QRSP algebras in an essential way in order to not deal with deformation theory of coconnective E ∞ -rings. Our construction of the unwinding functor Un is also devised in a way to work with the category Fun(QRSP, Alg Fp ). However, in principle, this is not absolutely necessary. Below we attempt to loosely explain other possible approaches that could be seen as more natural.

We prove that if G is a finite flat group scheme or a p-divisible group, then the second crystalline cohomology of its classifying stack H 2 crys (BG) recovers the Dieudonné module of G. We also prove mixed characteristic analogues of this result using prismatic cohomology.

In this short paper, we use Tannakian reconstruction techniques to prove a result that explains how to reconstruct the stacky approach to de Rham cohomology from the classical theory algebraic de Rham cohomology via an application of the adjoint functor theorem.

We prove that if G is a finite flat group scheme of p-power rank over a perfect field of characteristic p, then the second crystalline cohomology of its classifying stack $H^2_{\text {crys}}(BG)$ recovers the Dieudonné module of G. We also provide a calculation of the crystalline cohomology of the classifying stack of an abelian variety. We use this to prove that the crystalline cohomology of the classifying stack of a p-divisible group is a symmetric algebra (in degree $2$ ) on its Dieudonné module. We also prove mixed-characteristic analogues of some of these results using prismatic cohomology.