G_a^{perf}-modules and de Rham Cohomology

Abstract

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.

On endomorphisms of the de Rham cohomology functor

“…In the above t is required to be a G a -module map once the target is given the appropriate G a -module structure via restricting scalars along ϕ. In order to understand the map t , we can therefore apply graded Cartier duality [Mon21,§2.4]. We note that W [F ] * = G a and thus we get a map of graded group schemes t * : G a → G a , where the source group scheme G a receives its grading via the G a -module structure induced by restriction of scalars along s. By easy degree considerations, it follows that there exists a unique G amodule map t which fits into the above commutative diagram.…”

“…From a technical point of view, in certain situations, showing that the de Rham cohomology functor has no nontrivial automorphisms has been used as a key tool in [BLM20] and [LL21] to prove that certain constructions are functorially isomorphic. Further, in [Mon21], it was shown that one can reconstruct the theory of crystalline cohomology as the unique deformation of de Rham cohomology theory viewed as a functor defined on smooth F p -schemes.…”

“…(we often omit B to ease the notation). This unwinding construction is a variant of a construction used in [Mon21,§3]. Note the amusing switch of roles played by A and B: the de Rham cohomology theory is a cohomology theory for varieties over A with coefficient ring being B, whereas the stack A 1,dR B is an A-algebra object over B.…”

Some recent advances in topological Hochschild homology

“…Remark 9.4. Theorem 9.3 has very recently been generalized by Mondal [103]: de Rham cohomology of 𝔽 𝑝 -algebras admits a unique deformation over any local Artinian ring with residue field 𝔽 𝑝 (coming from the base-change of crystalline cohomology). Mondal's work relies on some of the stacky ideas studied by Drinfeld [45].…”