On q-de Rham cohomology via Λ-rings

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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.
Original languageEnglish
Pages (from-to)425-452
Number of pages28
JournalMathematische annalen
Volume375
Issue number1-2
Early online date5 Feb 2019
DOIs
Publication statusPublished - 8 Oct 2019

Keywords

  • math.AG
  • math.NT

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