En–cohomology with coefficients as functor cohomology

Ziegenhagen, Stephanie¹

¹Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden

Algebraic and Geometric Topology, Tome 16 (2016) no. 5, pp. 2981-3004

Résumé

Building on work of Livernet and Richter, we prove that En–homology and En–cohomology of a commutative algebra with coefficients in a symmetric bimodule can be interpreted as functor homology and cohomology. Furthermore, we show that the associated Yoneda algebra is trivial.

Détail

DOI : 10.2140/agt.2016.16.2981

Classification : 13D03, 18G15, 55P48
Keywords: functor homology, $E_n$-homology, iterated bar construction, Hochschild homology, operads

Bibliographie

[1] F W Anderson, K R Fuller, Rings and categories of modules, 13, Springer (1992) | DOI

[2] J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, 347, Springer (1973)

[3] B Fresse, Iterated bar complexes and the poset of pruned trees, (2008)

[4] B Fresse, La catégorie des arbres élagués de Batanin est de Koszul, preprint (2009)

[5] B Fresse, Iterated bar complexes of E–infinity algebras and homology theories, Algebr. Geom. Topol. 11 (2011) 747 | DOI

[6] B Fresse, Functor homology and operadic homology, notes (2014)

[7] B Fresse, S Ziegenhagen, Iterated bar complexes and En–homology with coefficients, J. Pure Appl. Algebra 220 (2016) 683 | DOI

[8] G Ginot, T Tradler, M Zeinalian, Higher Hochschild cohomology, brane topology and centralizers of En–algebra maps, preprint (2012)

[9] A Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. 9 (1957) 119 | DOI

[10] E Hoffbeck, C Vespa, Leibniz homology of Lie algebras as functor homology, J. Pure Appl. Algebra 219 (2015) 3721 | DOI

[11] M Livernet, B Richter, An interpretation of En–homology as functor homology, Math. Z. 269 (2011) 193 | DOI

[12] S Mac Lane, Homology, 114, Academic Press (1963)

[13] J P May, The geometry of iterated loop spaces, 271, Springer (1972)

[14] T Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. 33 (2000) 151 | DOI

[15] T Pirashvili, B Richter, Robinson–Whitehouse complex and stable homotopy, Topology 39 (2000) 525 | DOI

[16] T Pirashvili, B Richter, Hochschild and cyclic homology via functor homology, K–Theory 25 (2002) 39 | DOI

[17] A Robinson, S Whitehouse, Operads and Γ–homology of commutative rings, Math. Proc. Cambridge Philos. Soc. 132 (2002) 197 | DOI

[18] J J Rotman, An introduction to homological algebra, Springer (2009) | DOI

[19] S Ziegenhagen, En–cohomology as functor cohomology and additional structures, PhD thesis, Universität Hamburg (2014)