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On Witt vector cohomology for singular varieties

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Pierre Berthelot, Spencer Bloch, Hélène Esnault

Algebraic Geometry · Submitted 17 Oct 2005

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope $< 1$ part of the former.

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arXiv IDmath/0510349 (version 1)
Pith IDUIUT3FZO, Pith's permanent record of this paper and its review
CitationH\'el\`ene Esnault, Pierre Berthelot, Spencer Bloch. "On Witt vector cohomology for singular varieties." arXiv:math/0510349 (2005). https://pith.science/paper/UIUT3FZO
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,380 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.34 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.23 · compares with prior results 0.87 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Over a perfect field $k$ of characteristic $p > 0$, we construct a ``Witt vector cohomology with compact supports'' for separated $k$-schemes of finite type, extending (after tensorisation with $\mathbb{Q}$) the classical theory for proper $k$-schemes. We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope $< 1$ part of the former. Over a finite field, this allows one to compute congruences for the number of rational points in special examples. In particular, the congruence modulo the cardinality of the finite field of the number of rational points of a theta divisor on an abelian variety does not depend on the choice of the theta divisor. This answers positively a question by J.-P. Serre.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Over a perfect field $k$ of characteristic $p > 0$, we construct a ``Witt vector cohomology with compact supports'' for separated $k$-schemes of finite type, extending (after tensorisation with $\mathbb{Q}$) the classical theory for proper $k$-schemes. contribution 1.00 0.96 0.25 asserted 1.00
s1 We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope $< 1$ part of the former. result 0.62 0.96 0.55 asserted 1.00
s2 Over a finite field, this allows one to compute congruences for the number of rational points in special examples. interpretation 0.79 0.65 0.06 hedged possibility 0.56
s3 In particular, the congruence modulo the cardinality of the finite field of the number of rational points of a theta divisor on an abelian variety does not depend on the choice of the theta divisor. result 0.96 0.83 0.20 asserted 1.00
s4 This answers positively a question by J.-P. Serre. interpretation 0.53 0.70 0.19 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.86 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Over a perfect field $k$ of characteristic $p > 0$, we construct a ``Witt vector cohomology with compact supports'' for separated $k$-schemes of finite type, extending (after tensorisation with $\mathbb{Q}$) the classical theory for proper $k$-schemes. We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope $< 1$ part of the former. Over a finite field, this allows one to compute congruences for the number of rational points in special examples. In particular, the congruence modulo the cardinality of the finite field of the number of rational points of a theta divisor on an abelian variety does not depend on the choice of the theta divisor. This answers positively a question by J.-P. Serre.

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