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Some refinements of the Deligne–Illusie theorem

Piotr Achinger and Junecue Suh

Vol. 17 (2023), No. 2, 465–496
Abstract

We extend the results of Deligne and Illusie on liftings modulo p2 and decompositions of the de Rham complex in several ways. We show that for a smooth scheme X over a perfect field k of characteristic p > 0, the truncations of the de Rham complex in max (p1,2) consecutive degrees can be reconstructed as objects of the derived category in terms of its truncation in degrees at most one (or, equivalently, in terms the obstruction class to lifting modulo p2). Consequently, these truncations are decomposable if X admits a lifting to W2(k), in which case the first nonzero differential in the conjugate spectral sequence appears no earlier than on page max (p,3) (these corollaries have been recently strengthened by Drinfeld, by Bhatt and Lurie, and by Li and Mondal). Without assuming the existence of a lifting, we describe the gerbes of splittings of two-term truncations and the differentials on the second page of the conjugate spectral sequence, answering a question of Katz.

The main technical result used in the case p > 2 belongs purely to homological algebra. It concerns certain commutative differential graded algebras whose cohomology algebra is the exterior algebra, dubbed by us abstract Koszul complexes, of which the de Rham complex in characteristic p is an example.

In the Appendix, we use the aforementioned stronger decomposition result to prove that Kodaira–Akizuki–Nakano vanishing and Hodge–de Rham degeneration both hold for F-split (p+1)-folds.

Keywords
de Rham cohomology, Koszul complex, Deligne–Illusie, lifting modulo $p^2$, conjugate spectral sequence, $F$-splitting
Mathematical Subject Classification
Primary: 14F40
Secondary: 14G17
Milestones
Received: 13 September 2021
Revised: 10 February 2022
Accepted: 25 March 2022
Published: 24 March 2023
Authors
Piotr Achinger
Institute of Mathematics of the Polish Academy of Sciences
Warsaw
Poland
Junecue Suh
Mathematics Department
University of California
Santa Cruz, CA
United States

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