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Some remarks on Beilinson adeles

Author: Amnon Yekutieli
Journal: Proc. Amer. Math. Soc. 124 (1996), 3613-3618
MSC (1991): Primary 14F40; Secondary 14C30, 13J10
MathSciNet review: 1353408
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Abstract: Let $X$ be a scheme of finite type over a field $k$. Denote by $\cal {A}^{{\textstyle \cdot }}_X$ the sheaf of Beilinson adeles with values in the algebraic De Rham complex $\Omega ^{{\textstyle \cdot }}_{X/k}$. Then $\Omega ^{{\textstyle \cdot }} _{X/k}\rightarrow \cal {A}^{{\textstyle \cdot }}_X$ is a flasque resolution. So if $X$ is smooth, $\cal {A}^{{\textstyle \cdot }}_X$ 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 $\cal {A}^{{\textstyle \cdot },\geq q}_X$ does not induce Hodge decomposition.

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Amnon Yekutieli
Affiliation: Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Isreal

Received by editor(s): May 24, 1995
Additional Notes: This research was partially supported by an Allon Fellowship. The author is an incumbent of the Anna and Maurice Boukstein Career Development Chair
Communicated by: Eric M. Friedlander
Article copyright: © Copyright 1996 American Mathematical Society