Categorified Chern character and cyclic cohomology

Abstract

We examine Hopf cyclic cohomology in the same context as the analysis [1,2,3] of the geometry of loop spaces LX in derived algebraic geometry and the resulting close relationship between S 1 -equivariant quasi-coherent sheaves on LX and D X -modules. Furthermore, the Hopf setting serves as a toy case for the categorification of Chern character theory as discussed in [20]. More precisely, this examination naturally leads to a definition of mixed anti-Yetter-Drinfeld contramodules which reduces to that of the usual mixed complexes for the trivial Hopf algebra and generalizes the notion of stable anti-Yetter-Drinfeld contramodules that have thus far served as the coefficients for Hopf-cyclic theories [4]. The cohomology is then obtained as a Hom in this dgcategory between a Chern character object associated to an algebra and an arbitrary coefficient mixed anti-Yetter-Drinfeld contramodule.

Mixed vs Stable Anti-Yetter-Drinfeld Contramodule

“…It is furthermore conjectured that the correct coefficients (generalizing the classical ones) are obtained from the true cyclic homology category; this makes exact the analogy between de Rham and Hopf-cyclic coefficients since the former are shown to be so in [1]. More precisely, in [11], a category of mixed anti-Yetter-Drinfeld contramodules is defined by analogy with the derived algebraic geometry case of [1]. This new generalization is conceptual, and furthermore allows the expression of the Hopf-cyclic cohomology of an algebra A with coefficients in M as an Ext (in this category) between ch(A), the Chern character object associated to A, and M itself.…”

“…This paper is a descendant of [11] where it is shown that the classic stable anti-Yetter-Drinfeld contramodules are simply objects in the naive cyclic homology category of H M, the monoidal category of modules over the Hopf algebra. It is furthermore conjectured that the correct coefficients (generalizing the classical ones) are obtained from the true cyclic homology category; this makes exact the analogy between de Rham and Hopf-cyclic coefficients since the former are shown to be so in [1].…”

“…The comparison in [11] between anti-Yetter-Drinfeld contramodules and the cyclic homology category of H M involves a monad on H M with a central element σ (giving the S 1 -action). It is this description that allows us here to reduce the investigations into the differences between the classical and the new Hopf-cyclic cohomology to the analysis of modules categories over two differential graded algebras (DGAs).…”

“…Let H be a Hopf algebra. From [11] we see that the study of the Hochschild and cyclic homologies of H M, the monoidal category of H-modules, reduces to the study of modules over a certain natural, from the considerations there, monad on H M. Recall that the consideration of Hochschild and cyclic homologies of monoidal categories is motivated by their recently discovered role in the understanding of Hopf-cyclic theory coefficients.…”

“…The anti-Yetter-Drinfeld contramodules then coincide with modules over this monad, while the stable ones consist of those for which the action of σ agrees with that of 1, and the mixed ones introduced in [11] are the homotopic version of this on the nose requirement.…”