On $q$-de Rham cohomology via $\Lambda$-rings
Algebraic Geometry
2019-01-10 v4 Number Theory
Abstract
We show that Aomoto's -deformation of de Rham cohomology arises as a natural cohomology theory for -rings. Moreover, Scholze's -adic completion of -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 -power roots of , the resulting theory is independent even of these lifts of Frobenius, refining a comparison by Bhatt, Morrow and Scholze.
Cite
@article{arxiv.1608.07142,
title = {On $q$-de Rham cohomology via $\Lambda$-rings},
author = {J. P. Pridham},
journal= {arXiv preprint arXiv:1608.07142},
year = {2019}
}
Comments
24 pp, v3 new functoriality results in section 3; v4 significantly expanded following referee's comments, plus some new results - final version, to appear in Math. Annalen