Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Some remarks on Beilinson adeles


Author: Amnon Yekutieli
Journal: Proc. Amer. Math. Soc. 124 (1996), 3613-3618
MSC (1991): Primary 14F40; Secondary 14C30, 13J10
DOI: https://doi.org/10.1090/S0002-9939-96-03644-1
MathSciNet review: 1353408
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [Be] A.A. Beilinson, Residues and adeles, Funkt. Anal. Pril. 14 (1980) no. 1, 44-45; English trans. in Func. Anal. Appl. 14 (1980) no. 1, 34-35. MR 81f:14010
  • [DI] P. Deligne and L. Illusie, Relèvements modulo $p^{2}$ et décomposition du complexe de de Rham, Inv. Math. 89 (1987), 247-270. MR 88j:14029
  • [GH] P. Griffiths and J. Harris, ``Principles of Algebraic Geometry'', Wiley, New York, 1978. MR 80b:14001
  • [Hr] A. Huber, On the Parshin-Beilinson Adeles for Schemes, Abh. Math. Sem. Univ. Hamburg 61 (1991), 249-273. MR 92k:14024
  • [HY1] R. Hübl and A. Yekutieli, Adeles and differential forms, to appear: J. reine angew. Math.
  • [HY2] R. Hübl and A. Yekutieli, Adelic Chern forms and the Bott residue formula, preprint (1994).
  • [Ye1] A. Yekutieli, ``An Explicit Construction of the Grothendieck Residue Complex'' (with an appendix by P. Sastry), Astérisque 208 (1992). MR 94e:14026
  • [Ye2] A. Yekutieli, Smooth formal embeddings, preprint (1995).
  • [Ye3] A. Yekutieli, Residues and differential operators on schemes, preprint (1994).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 14F40, 14C30, 13J10

Retrieve articles in all journals with MSC (1991): 14F40, 14C30, 13J10


Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-96-03644-1
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

American Mathematical Society