Abstract
We show that Aomoto’s q-deformation of de Rham cohomology arises as a natural cohomology theory for Lambda -rings. Moreover, Scholze’s (q-1)-adic completion of q-de Rham cohomology depends only on the Adams operations at each residue characteristic. This gives a fully functorial cohomology theory, including a lift of the Cartier isomorphism, for smooth formal schemes in mixed characteristic equipped with a suitable lift of Frobenius. If we attach p-power roots of q, the resulting theory is independent even of these lifts of Frobenius, refining a comparison by Bhatt, Morrow and Scholze.
Highlights
[13], Scholze discussed the (q − 1)-adic completion of this theory for smooth rings, explaining relations to p-adic Hodge theory and singular cohomology, and conjecturing that it is independent of co-ordinates, so functorial for smooth algebras over a fixed base [13, Conjectures
We show that q-de Rham cohomology with q-connections naturally arises as a functorial invariant of -rings (Theorems 1.17, 1.23 and Proposition 1.25), and that its (q − 1)-adic completion depends only on a P -ring structure (Theorem 2.8), for P the set of residue characteristics; a P -ring has a lift of Frobenius for each p ∈ P
The main idea is to show that the stabilised q-de Rham complex is in a sense given by applying Fontaine’s period ring construction Ainf to the best possible perfectoid approximation to A[ζp∞ ]. This shows (Corollary 3.13) that after attaching all p-power roots of q, q-de Rham cohomology in mixed characteristic is independent of choices, which was already known after base change to a period ring, via the comparisons of [4] between q-de Rham cohomology and their theory A
Summary
We show that q-de Rham cohomology with q-connections naturally arises as a functorial invariant of -rings (Theorems 1.17, 1.23 and Proposition 1.25), and that its (q − 1)-adic completion depends only on a P -ring structure (Theorem 2.8), for P the set of residue characteristics; a P -ring has a lift of Frobenius for each p ∈ P. The main idea is to show that the stabilised q-de Rham complex is in a sense given by applying Fontaine’s period ring construction Ainf to the best possible perfectoid approximation to A[ζp∞ ] As a consequence, this shows (Corollary 3.13) that after attaching all p-power roots of q, q-de Rham cohomology in mixed characteristic is independent of choices, which was already known after base change to a period ring, via the comparisons of [4] between q-de Rham cohomology and their theory A. I would like to thank the anonymous referee for suggesting many improvements
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