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arXiv · 1608.07142

On $q$-de Rham cohomology via $\Lambda$-rings

J. P. Pridham
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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.

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J. P. Pridham. 2016-08-25. On $q$-de Rham cohomology via $\Lambda$-rings. https://arxiv.org/abs/1608.07142

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