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Mathematics > Algebraic Geometry

arXiv:1805.05501 (math)
[Submitted on 15 May 2018 (v1), last revised 19 Feb 2020 (this version, v3)]

Title:Revisiting the de Rham-Witt complex

Authors:Bhargav Bhatt, Jacob Lurie, Akhil Mathew
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Abstract:The goal of this paper is to offer a new construction of the de Rham-Witt complex of smooth varieties over perfect fields of characteristic $p>0$.
We introduce a category of cochain complexes equipped with an endomorphism $F$ of underlying graded abelian groups satisfying $dF = pFd$, whose homological algebra we study in detail. To any such object satisfying an abstract analog of the Cartier isomorphism, an elementary homological process associates a generalization of the de Rham-Witt construction. Abstractly, the homological algebra can be viewed as a calculation of the fixed points of the Berthelot-Ogus operator $L \eta_p$ on the $p$-complete derived category. We give various applications of this approach, including a simplification of the crystalline comparison for the $A \Omega$-cohomology theory introduced in [BMS18].
Comments: 158 pages. Final version. Contains a new section on the comparison with crystalline cohomology
Subjects: Algebraic Geometry (math.AG)
Cite as: arXiv:1805.05501 [math.AG]
  (or arXiv:1805.05501v3 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1805.05501

Submission history

From: Akhil Mathew [view email]
[v1] Tue, 15 May 2018 00:24:09 UTC (99 KB)
[v2] Sat, 8 Sep 2018 19:23:36 UTC (100 KB)
[v3] Wed, 19 Feb 2020 02:05:41 UTC (118 KB)
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