Some remarks on Beilinson adeles

Amnon Yekutieli

doi:10.1090/s0002-9939-96-03644-1

Some remarks on Beilinson adeles

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Abstract

Let X be a scheme of finite type over a field k. Denote by A_x^̇ the sheaf of Beilinson adeles with values in the algebraic De Rham complex Ω_X/k^̇. Then Ω_X/k^̇ → A_x^̇ is a flasque resolution. So if X is smooth, A_x^̇ calculates De Rham cohomology. In this note we rewrite the proof of Deligne-Illusie for the degeneration of the Hodge spectral sequence in terms of adeles. We also give a counterexample to show that the filtration A_X^̇,≥q does not induce Hodge decomposition.

Original language	English
Pages (from-to)	3613-3618
Number of pages	6
Journal	Proceedings of the American Mathematical Society
Volume	124
Issue number	12
DOIs	https://doi.org/10.1090/s0002-9939-96-03644-1
State	Published - 1 Jan 1996
Externally published	Yes

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/s0002-9939-96-03644-1

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AB - Let X be a scheme of finite type over a field k. Denote by Aẋ the sheaf of Beilinson adeles with values in the algebraic De Rham complex ΩX/k̇. Then ΩX/k̇ → Aẋ is a flasque resolution. So if X is smooth, Aẋ calculates De Rham cohomology. In this note we rewrite the proof of Deligne-Illusie for the degeneration of the Hodge spectral sequence in terms of adeles. We also give a counterexample to show that the filtration AẊ,≥q does not induce Hodge decomposition.

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Some remarks on Beilinson adeles

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