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Some refinements of the
Deligne–Illusie theorem
Piotr Achinger and Junecue Suh |
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Vol. 17 (2023), No. 2, 465–496
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Abstract |
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We extend the results of Deligne and Illusie on liftings modulo and decompositions of the de Rham complex in several ways. We show that for a smooth scheme over a perfect field of characteristic , the truncations of the de Rham complex in 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 ). Consequently, these truncations are decomposable if admits a lifting to , in which case the first nonzero differential in the conjugate spectral sequence appears no earlier than on page (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 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 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 -split -folds. |
Keywords
de Rham cohomology, Koszul complex, Deligne–Illusie,
lifting modulo $p^2$, conjugate spectral sequence,
$F$-splitting
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Mathematical Subject Classification
Primary: 14F40
Secondary: 14G17
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Milestones
Received: 13 September 2021
Revised: 10 February 2022
Accepted: 25 March 2022
Published: 24 March 2023
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Authors |
| Piotr Achinger | |
| Institute of Mathematics of the
Polish Academy of Sciences Warsaw Poland |
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| Junecue Suh | |
| Mathematics Department University of California Santa Cruz, CA United States |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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