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

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, Guillermo Horacio
Publicado: 2003
Materias:
Commutator filtration
Cyclic homology
Grothendieck topology
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0010437X_v136_n2_p171_Cortinas
http://hdl.handle.net/20.500.12110/paper_0010437X_v136_n2_p171_Cortinas
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling paper:paper_0010437X_v136_n2_p171_Cortinas2023-06-08T14:34:20Z De Rham and Infinitesimal Cohomology in Kapranov's Model for Noncommutative Algebraic Geometry Cortiñas, Guillermo Horacio Commutator filtration Cyclic homology Grothendieck topology 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. 2003 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0010437X_v136_n2_p171_Cortinas http://hdl.handle.net/20.500.12110/paper_0010437X_v136_n2_p171_Cortinas
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Commutator filtration
Cyclic homology
Grothendieck topology
spellingShingle Commutator filtration
Cyclic homology
Grothendieck topology
Cortiñas, Guillermo Horacio
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.
author Cortiñas, Guillermo Horacio
author_facet Cortiñas, Guillermo Horacio
author_sort Cortiñas, Guillermo Horacio
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 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0010437X_v136_n2_p171_Cortinas
http://hdl.handle.net/20.500.12110/paper_0010437X_v136_n2_p171_Cortinas
work_keys_str_mv AT cortinasguillermohoracio derhamandinfinitesimalcohomologyinkapranovsmodelfornoncommutativealgebraicgeometry
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