(Co)cyclic (co)homology of bialgebroids: An approach via (co)monads

Mathematics – K-Theory and Homology

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

LaTeX file, 39 pages, 4 eps figures. v2: significantly extended, a new section about cyclic homology of groupoids added. v3: f

Scientific paper

10.1007/s00220-008-0540-3

For a (co)monad T_l on a category M, an object X in M, and a functor \Pi: M \to C, there is a (co)simplex Z^*:=\Pi T_l^{* +1} X in C. Our aim is to find criteria for para-(co)cyclicity of Z^*. Construction is built on a distributive law of T_l with a second (co)monad T_r on M, a natural transformation i:\Pi T_l \to \Pi T_r, and a morphism w: T_r X \to T_l X in M. The relations i and w need to satisfy are categorical versions of Kaygun's axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads T_l=T \otimes_R (-) and T_r = (-)\otimes_R T on the category of R-bimodules. The functor \Pi can be chosen such that Z^n= T\hat{\otimes}_R... \hat{\otimes}_R T \hat{\otimes}_R X is the cyclic R-module tensor product. A natural transformation i:T \hat{\otimes}_R (-) \to (-) \hat{\otimes}_R T is given by the flip map and a morphism w: X \otimes_R T \to T\otimes_R X is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. Stable anti Yetter-Drinfel'd modules over certain bialgebroids, so called x_R-Hopf algebras, are introduced. In the particular example when T is a module coring of a x_R-Hopf algebra B and X is a stable anti Yetter-Drinfel'd B-module, the para-cyclic object Z_* is shown to project to a cyclic structure on T^{\otimes_R *+1} \otimes_B X. For a B-Galois extension S \to T, a stable anti Yetter-Drinfel'd B-module T_S is constructed, such that the cyclic objects B^{\otimes_R *+1} \otimes_B T_S and T^ {\hat{\otimes}_S *+1} are isomorphic. As an application, we compute Hochschild and cyclic homology of a groupoid with coefficients, by tracing it back to the group case. In particular, we obtain explicit expressions for ordinary Hochschild and cyclic homology of a groupoid.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

(Co)cyclic (co)homology of bialgebroids: An approach via (co)monads does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with (Co)cyclic (co)homology of bialgebroids: An approach via (co)monads, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and (Co)cyclic (co)homology of bialgebroids: An approach via (co)monads will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-416213

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.