Further Results on the Structure of (Co)Ends in Finite Tensor Categories

摘要

Let C be a finite tensor category, and letMbe an exact left C-module category. The action of C onMinduces a functor. : C. Rex(M), where Rex(M) is the category of k-linear right exact endofunctors onM. Our key observation is that. has a right adjoint.ra given by the end.ra ( F) = similar to M.M Hom( M, F( M)) (F. Rex(M)). As an application, we establish the following results: (1) We give a description of the composition of the induction functor C* M. Z(C* M) and Schauenburg's equivalenceZ(C* M) similar to Z(C). (2) We introduce the space CF(M) of `class functions' ofMand initiate the character theory for pivotal module categories. (3) We introduce a filtration for CF(M) and discuss its relation with some ring-theoretic notions, such as the Reynolds ideal and its generalizations. (4) We show that Ext similar to C(1,.ra(idM)) is isomorphic to the Hochschild cohomology ofM. As an application, we showthat themodular group acts projectively on the Hochschild cohomology of a modular tensor category.