Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Hochschild-Mitchell (co)homology of linear categories and one point extensions

Hochschild-Mitchell (co)homology theory for linear categories was introduced by B. Mitchell in [Mit] as a generalization of Hochschild (co)homology theory for algebras. In this talk I shall recall the general definitions of Hochschild-Mitchell theory for linear categories and show how the Hochschild-Mitchell cohomology of a one point extension is related to the Hochschild-Mitchell cohomology of the category by a long exact sequence à la Happel [Hap]. I shall also present a generalization of this result in the context given by two linear categories and a bimodule. Our proof is related to Cibils’ article [Cib], but it is fact simpler, even for the case of algebras. This is joint work with Andrea Solotar [HS]. References [Cib] Cibils, C. Tensor Hochschild homology and cohomology. Interactions between ring theory and representations of algebras (Murcia), pp. 35-51. Lecture Notes in Pure and Appl. Math. 210, Dekker, New-York, 2000. [Hap] Happel, D. Hochschild cohomology of finite-dimensional algebras. Seminaire dAlg`ebre Paul Dubreil et Marie-Paul Malliavin, 39me Ann´ee (Paris, 1987/1988), 108-126, Lecture Notes in Math., 1404, Springer, Berlin, 1989. [HS] Herscovich, E.; Solotar, A. Derived invariance of Hochschild-Mitchell (co)homology and one-point extensions. Journal of Algebra 315, (2007), pp. 852–873. [Mit] Mitchell, B. Rings with several objects. Adv. in Math. 8, (1972), pp. 1–161.