On q-de Rham cohomology via Λ-rings

2016/08/25 by Pridham, J. P.
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1608.07142

Abstract

We show that Aomoto's q-deformation of de Rham cohomology arises as a natural cohomology theory for Λ-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.

Related

• p-adic derived de Rham cohomology
• Canonical q-deformations in arithmetic geometry
• q-deformation with (φ, Γ) structure of the de Rham cohomology of the Legendre family of elliptic curves
• Dwork cohomology, de Rham cohomology, and hypergeometric functions
• A p-adic Cartier isomorphism between the A_inf-cohomology and de Rham-Witt complexes for semistable formal schemes