De Rham and Infinitesimal Cohomology in Kapranov's Model for Noncommutative Algebraic Geometry

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The title refers to the nilcommutative of NC-schemes introduced by M. Kapranov in 'Noncommutative Geometry Based on Commutator Expansions', J. Reine Angew. Math 505 (1998) 73-118. The latter are noncommutative nilpotent thickenings of commutative schemes. We also consider the parallel theo...

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Autor principal: Cortiñas, G.
Formato: Artículo publishedVersion
Publicado: 2003
Materias:
Commutator filtration
Cyclic homology
Grothendieck topology
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_0010437X_v136_n2_p171_Cortinas
https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_0010437X_v136_n2_p171_Cortinas_oai
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spelling I28-R145-paper_0010437X_v136_n2_p171_Cortinas_oai2024-08-16 Cortiñas, G. 2003 The title refers to the nilcommutative of NC-schemes introduced by M. Kapranov in 'Noncommutative Geometry Based on Commutator Expansions', J. Reine Angew. Math 505 (1998) 73-118. The latter are noncommutative nilpotent thickenings of commutative schemes. We also consider the parallel theory of nil-Poisson of NP-schemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for NC- and NP-schemes. The variants include nilcommutative and nil-Poisson versions of the de Rham complex as well as of the cohomology of the infinitesimal site introduced by Grothendieck in Crystals and the de Rham Cohomology of Schemes, Dix exposés sur la cohomologie des schémas, Masson, Paris (1968), pp. 306-358. It turns out that each of these noncommutative variants admits a kind of Hodge decomposition which allows one to express the cohomology groups of a noncommutative scheme Y as a sum of copies of the usual (de Rham, infinitesimal) cohomology groups of the underlying commutative scheme X (Theorems 6.1, 6.4, 6.7). As a byproduct we obtain new proofs for classical results of Grothendieck (Corollary 6.2) and of Feigin and Tsygan (Corollary 6.8) on the relation between de Rham and infinitesimal cohomology and between the latter and periodic cyclic homology. Fil:Cortiñas, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. application/pdf http://hdl.handle.net/20.500.12110/paper_0010437X_v136_n2_p171_Cortinas info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar Compos. Math. 2003;136(2):171-208 Commutator filtration Cyclic homology Grothendieck topology De Rham and Infinitesimal Cohomology in Kapranov's Model for Noncommutative Algebraic Geometry info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_0010437X_v136_n2_p171_Cortinas_oai
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-145
collection Repositorio Digital de la Universidad de Buenos Aires (UBA)
topic Commutator filtration
Cyclic homology
Grothendieck topology
spellingShingle Commutator filtration
Cyclic homology
Grothendieck topology
Cortiñas, G.
De Rham and Infinitesimal Cohomology in Kapranov's Model for Noncommutative Algebraic Geometry
topic_facet Commutator filtration
Cyclic homology
Grothendieck topology
description The title refers to the nilcommutative of NC-schemes introduced by M. Kapranov in 'Noncommutative Geometry Based on Commutator Expansions', J. Reine Angew. Math 505 (1998) 73-118. The latter are noncommutative nilpotent thickenings of commutative schemes. We also consider the parallel theory of nil-Poisson of NP-schemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for NC- and NP-schemes. The variants include nilcommutative and nil-Poisson versions of the de Rham complex as well as of the cohomology of the infinitesimal site introduced by Grothendieck in Crystals and the de Rham Cohomology of Schemes, Dix exposés sur la cohomologie des schémas, Masson, Paris (1968), pp. 306-358. It turns out that each of these noncommutative variants admits a kind of Hodge decomposition which allows one to express the cohomology groups of a noncommutative scheme Y as a sum of copies of the usual (de Rham, infinitesimal) cohomology groups of the underlying commutative scheme X (Theorems 6.1, 6.4, 6.7). As a byproduct we obtain new proofs for classical results of Grothendieck (Corollary 6.2) and of Feigin and Tsygan (Corollary 6.8) on the relation between de Rham and infinitesimal cohomology and between the latter and periodic cyclic homology.
format Artículo
Artículo
publishedVersion
author Cortiñas, G.
author_facet Cortiñas, G.
author_sort Cortiñas, G.
title De Rham and Infinitesimal Cohomology in Kapranov's Model for Noncommutative Algebraic Geometry
title_short De Rham and Infinitesimal Cohomology in Kapranov's Model for Noncommutative Algebraic Geometry
title_full De Rham and Infinitesimal Cohomology in Kapranov's Model for Noncommutative Algebraic Geometry
title_fullStr De Rham and Infinitesimal Cohomology in Kapranov's Model for Noncommutative Algebraic Geometry
title_full_unstemmed De Rham and Infinitesimal Cohomology in Kapranov's Model for Noncommutative Algebraic Geometry
title_sort de rham and infinitesimal cohomology in kapranov's model for noncommutative algebraic geometry
publishDate 2003
url http://hdl.handle.net/20.500.12110/paper_0010437X_v136_n2_p171_Cortinas
https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_0010437X_v136_n2_p171_Cortinas_oai
work_keys_str_mv AT cortinasg derhamandinfinitesimalcohomologyinkapranovsmodelfornoncommutativealgebraicgeometry
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