Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology

Mamta Balodi; Abhishek Banerjee

doi:10.5802/crmath.429

Algèbre

Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology

Mamta Balodi ¹ ; Abhishek Banerjee ¹

¹ Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 617-652.

Résumé

We replace a ring with a small $ℂ$ -linear category $𝒞$ , seen as a ring with several objects in the sense of Mitchell. We introduce Fredholm modules over this category and construct a Chern character taking values in the cyclic cohomology of $𝒞$ . We show that this categorified Chern character is homotopy invariant and is well-behaved with respect to the periodicity operator in cyclic cohomology. For this, we also obtain a description of cocycles and coboundaries in the cyclic cohomology of $𝒞$ (and more generally, in the Hopf cyclic cohomology of a Hopf-module category) by means of DG-semicategories equipped with a trace on endomorphism spaces.

Reçu le : 2022-01-03
Révisé le : 2022-09-21
Accepté le : 2022-09-24
Publié le : 2023-03-02

DOI : 10.5802/crmath.429

Classification : 18E05, 47A53, 53C99, 58B34

Affiliations des auteurs :

Mamta Balodi ¹ ; Abhishek Banerjee ¹

¹ Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Mamta Balodi and Abhishek Banerjee},
     title = {Fredholm modules over categories, {Connes} periodicity and classes in cyclic cohomology},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {617--652},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.429},
     language = {en},
}

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Mamta Balodi; Abhishek Banerjee. Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 617-652. doi : 10.5802/crmath.429. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.429/

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