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/

Bibliographie
Cité par

[1] R. Akbarpour; M. Khalkhali Hopf algebra equivariant cyclic homology and cyclic homology of crossed product algebras, J. Reine Angew. Math., Volume 559 (2003), pp. 137-152 | DOI | MR | Zbl

[2] J. C. Baez Higher-dimensional algebra. II. $2$ -Hilbert spaces, Adv. Math., Volume 127 (1997) no. 2, pp. 125-189 | DOI | MR | Zbl

[3] M. Balodi Morita invariance of equivariant and Hopf-cyclic cohomology of module algebras over Hopf algebroids | arXiv

[4] M. Balodi; A. Banerjee Odd Fredholm modules over linear categories and cyclic cohomology (in preparation)

[5] M. Balodi; A. Banerjee; S. Ray Cohomology of modules over $H$ -categories and co- $H$ -categories, Can. J. Math., Volume 72 (2020) no. 5, pp. 1352-1385 | DOI | MR | Zbl

[6] M. Balodi; A. Banerjee; S. Ray On entwined modules over linear categories and Galois extensions, Isr. J. Math., Volume 241 (2021), pp. 623-692 | DOI | MR | Zbl

[7] A. Banerjee On differential torsion theories and rings with several objects, Can. Math. Bull., Volume 62 (2019) no. 4, pp. 703-714 | DOI | MR | Zbl

[8] G. Böhm; S. Lack; R. Street Idempotent splittings, colimit completion, and weak aspects of the theory of monads, J. Pure Appl. Algebra, Volume 216 (2012) no. 2, pp. 385-403 | DOI | MR | Zbl

[9] S. Caenepeel Brauer groups, Hopf algebras and Galois theory, K-Monographs in Mathematics, 4, Kluwer Academic Publishers, 1998, xvi+488 pages | DOI

[10] C. Cibils; A. Solotar Galois coverings, Morita equivalence and smash extensions of categories over a field, Doc. Math., Volume 11 (2006), pp. 143-159 | DOI | MR | Zbl

[11] A. Connes Cohomologie cyclique et foncteurs ${Ext}^{n}$ , C. R. Math. Acad. Sci. Paris, Volume 296 (1983) no. 23, pp. 953-958 | MR | Zbl

[12] A. Connes Noncommutative differential geometry, Publ. Math., Inst. Hautes Étud. Sci., Volume 62 (1985), pp. 257-360 | DOI | Zbl

[13] A. Connes; H. Moscovici Hopf algebras, cyclic cohomology and the transverse index theorem, Commun. Math. Phys., Volume 198 (1998) no. 1, pp. 199-246 | DOI | MR | Zbl

[14] A. Connes; H. Moscovici Cyclic cohomology and Hopf algebras, Lett. Math. Phys., Volume 48 (1999) no. 1, pp. 97-108 Moshé Flato (1937–1998) | DOI | MR

[15] A. Connes; H. Moscovici Cyclic Cohomology and Hopf Algebra Symmetry, Lett. Math. Phys., Volume 52 (2000) no. 1, pp. 1-28 | DOI | MR | Zbl

[16] P. Deligne Catégories tannakiennes, The Grothendieck Festschrift. Vol. II (Progress in Mathematics), Volume 87, Birkhäuser, 1990, pp. 111-195 | Zbl

[17] S. Estrada; S. Virili Cartesian modules over representations of small categories, Adv. Math., Volume 310 (2017), pp. 557-609 | DOI | MR | Zbl

[18] P. M. Hajac; M. Khalkhali; B. Rangipour; Y. Sommerhäuser Hopf-cyclic homology and cohomology with coefficients, C. R. Math. Acad. Sci. Paris, Volume 338 (2004) no. 9, pp. 667-672 | DOI | Numdam | MR | Zbl

[19] P. M. Hajac; M. Khalkhali; B. Rangipour; Y. Sommerhäuser Stable anti-Yetter-Drinfeld modules, C. R. Math. Acad. Sci. Paris, Volume 338 (2004) no. 8, pp. 587-590 | DOI | Numdam | MR | Zbl

[20] M. Hassanzadeh; M. Khalkhali; I. Shapiro Monoidal categories, 2-traces, and cyclic cohomology, Can. Math. Bull., Volume 62 (2019) no. 2, pp. 293-312 | DOI | MR | Zbl

[21] M. Hassanzadeh; D. Kucerovsky; B. Rangipour Generalized coefficients for Hopf cyclic cohomology, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 10 (2014), 093, 16 pages | MR | Zbl

[22] A. G. Henriques What Chern-Simons theory assigns to a point, Proc. Natl. Acad. Sci. USA, Volume 114 (2017) no. 51, pp. 13418-13423 | DOI | MR | Zbl

[23] A. G. Henriques; D. Penneys Bicommutant categories from fusion categories, Sel. Math., New Ser., Volume 23 (2017) no. 3, pp. 1669-1708 | DOI | MR | Zbl

[24] E. Herscovich; A. Solotar Hochschild-Mitchell cohomology and Galois extensions, J. Pure Appl. Algebra, Volume 209 (2007) no. 1, pp. 37-55 | DOI | MR | Zbl

[25] M. Karoubi; O. Villamayor $K$ -théorie algébrique et $K$ -théorie topologique. I, Math. Scand., Volume 28 (1971), pp. 265-307 | DOI | Zbl

[26] A. Kaygun Bialgebra cyclic homology with coefficients, $K$ -Theory, Volume 34 (2005) no. 2, pp. 151-194 | DOI | MR | Zbl

[27] A. Kaygun Products in Hopf-cyclic cohomology, Homology Homotopy Appl., Volume 10 (2008) no. 2, pp. 115-133 | DOI | MR | Zbl

[28] A. Kaygun The universal Hopf-cyclic theory, J. Noncommut. Geom., Volume 2 (2008) no. 3, pp. 333-351 | DOI | MR | Zbl

[29] A. Kaygun; M. Khalkhali Bivariant Hopf cyclic cohomology, Commun. Algebra, Volume 38 (2010) no. 7, pp. 2513-2537 | DOI | MR | Zbl

[30] B. Keller Deriving DG categories, Ann. Sci. Éc. Norm. Supér., Volume 27 (1994) no. 1, pp. 63-102 | DOI | Numdam | MR | Zbl

[31] B. Keller On differential graded categories, International Congress of Mathematicians. Vol. II, European Mathematical Society, 2006, pp. 151-190 | Zbl

[32] M. Khalkhali; B. Rangipour Cup products in Hopf-cyclic cohomology, C. R. Math. Acad. Sci. Paris, Volume 340 (2005), pp. 9-14 | DOI | Numdam | MR | Zbl

[33] I. Kobyzev; I. Shapiro A Categorical Approach to Cyclic Cohomology of Quasi-Hopf Algebras and Hopf Algebroids, Appl. Categ. Struct., Volume 27 (2019) no. 1, pp. 85-109 | DOI | MR | Zbl

[34] J.-L Loday Cyclic homology, Grundlehren der Mathematischen Wissenschaften, 301, Springer, 1992, xviii+454 pages | DOI | Numdam

[35] W. Lowen Hochschild cohomology with support, Int. Math. Res. Not. (2015) no. 13, pp. 4741-4812 | DOI | MR | Zbl

[36] W. Lowen; M. Van den Bergh Deformation theory of abelian categories, Trans. Am. Math. Soc., Volume 358 (2006) no. 12, pp. 5441-5483 | DOI | MR | Zbl

[37] R. McCarthy The cyclic homology of an exact category, J. Pure Appl. Algebra, Volume 93 (1994) no. 3, pp. 251-296 | DOI | MR | Zbl

[38] B. Mitchell The dominion of Isbell, Trans. Am. Math. Soc., Volume 167 (1972), pp. 319-331 | DOI | MR | Zbl

[39] B. Mitchell Rings with several objects, Adv. Math., Volume 8 (1972), pp. 1-161 | DOI | MR | Zbl

[40] B. Mitchell Some applications of module theory to functor categories, Bull. Am. Math. Soc., Volume 84 (1978) no. 5, pp. 867-885 | DOI | MR | Zbl

[41] G. J. Murphy $C^{*}$ -algebras and operator theory, Academic Press Inc., 1990, x+286 pages

[42] B. Rangipour Cup products in Hopf cyclic cohomology via cyclic modules, Homology Homotopy Appl., Volume 10 (2008) no. 2, pp. 273-286 | DOI | MR | Zbl

[43] H. Schubert Categories, Springer, 1972, xi+385 pages | DOI

[44] B. Toën; M. Vaquié Au-dessous de $Spec ℤ$ , J. $K$ -Theory, Volume 3 (2009) no. 3, pp. 437-500 | DOI | MR | Zbl

[45] F. Xu Hochschild and ordinary cohomology rings of small categories, Adv. Math., Volume 219 (2008) no. 6, pp. 1872-1893 | DOI | MR | Zbl

[46] F. Xu On the cohomology rings of small categories, J. Pure Appl. Algebra, Volume 212 (2008) no. 11, pp. 2555-2569 | DOI | MR | Zbl

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