Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology
We replace a ring with a small $\mathbb{C}$-linear category $\mathcal{C}$, 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 $\mathcal{C}$. We show that this categor...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-03-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.429/ |
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| author | Balodi, Mamta Banerjee, Abhishek |
| author_facet | Balodi, Mamta Banerjee, Abhishek |
| author_sort | Balodi, Mamta |
| collection | DOAJ |
| description | We replace a ring with a small $\mathbb{C}$-linear category $\mathcal{C}$, 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 $\mathcal{C}$. 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 $\mathcal{C}$ (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. |
| format | Article |
| id | doaj-art-ab98b846e65e4d82b387dcee6e1f579e |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-03-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-ab98b846e65e4d82b387dcee6e1f579e2025-02-07T11:07:00ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-03-01361G361765210.5802/crmath.42910.5802/crmath.429Fredholm modules over categories, Connes periodicity and classes in cyclic cohomologyBalodi, Mamta0Banerjee, Abhishek1Department of Mathematics, Indian Institute of Science, Bangalore - 560012, IndiaDepartment of Mathematics, Indian Institute of Science, Bangalore - 560012, IndiaWe replace a ring with a small $\mathbb{C}$-linear category $\mathcal{C}$, 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 $\mathcal{C}$. 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 $\mathcal{C}$ (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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.429/ |
| spellingShingle | Balodi, Mamta Banerjee, Abhishek Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology Comptes Rendus. Mathématique |
| title | Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology |
| title_full | Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology |
| title_fullStr | Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology |
| title_full_unstemmed | Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology |
| title_short | Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology |
| title_sort | fredholm modules over categories connes periodicity and classes in cyclic cohomology |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.429/ |
| work_keys_str_mv | AT balodimamta fredholmmodulesovercategoriesconnesperiodicityandclassesincycliccohomology AT banerjeeabhishek fredholmmodulesovercategoriesconnesperiodicityandclassesincycliccohomology |