A noncommutative calculus on the cyclic dual of Ext

Abstract

We show that if the cochain complex computing Ext groups (in the category of modules over Hopf algebroids) admits a cocyclic structure, then the noncommutative Cartan calculus structure on Tor over Ext dualises in a cyclic sense to a calculus on Coext over Cotor. More precisely, the cyclic duals of the chain resp. cochain spaces computing the two classical derived functors lead to complexes that compute the more exotic ones, giving a cyclic opposite module over an operad with multiplication that induce operations such as a Lie derivative, a cap product (or contraction), and a (cyclic) differential, along with higher homotopy operators defining a noncommutative Cartan calculus up to homotopy. In particular, this allows to recover the classical Cartan calculus from differential geometry or the Chevalley-Eilenberg calculus for Lie(-Rinehart) algebras without any finiteness conditions or the use of topological tensor products. CONTENTSIntroduction 1 1. Contramodules over bialgebroids 5 2. The derived functors Cotor and Coext 6 3. The complex computing Ext as a cocyclic module 9 4. The complex computing Coext as a cyclic module 12 5. The noncommutative calculus structure on Coext over Cotor 15 6. Example: Cartan calculi in differential geometry 18 Appendix A. The cyclic category 22 Appendix B. Algebraic operads 23 Appendix C. Noncommutative calculi and opposite operad modules 24 Appendix D. Left and right Hopf algebroids 25 References 27

Centres, trace functors, and cyclic cohomology

“…for arbitrary N, P P U -Mod, which in the same way as in Theorem 3.12 leads to the structure of a cyclic k-module on the complex computing Ext ‚ U pQ, M q if U Ž is A-flat. Since this produces even more unpleasant formulae than those seen so far [Ko2,Prop. 3.5], we refrain from spelling out the details here.…”

“…The category Contramod-U of right U -contramodules is, in general, not monoidal and therefore neither are so U aYD contra´U , the category of anti Yetter-Drinfel'd contramodules nor U saYD contra´U , the category of stable ones. However, in [Ko2,Prop. 3.3] it is shown that Contramod-U is a left module category over U -Comod, the monoidal category of left U -comodules (cf.…”

“…contraaction followed by action, and vice versa again; see, just to name a few, [BePeW,B Ş,Br,BuCaP,Dr,J Ş,HKhRS,Kay,Ko1,PSt,RT,Ye] for these notions in various contexts. For example, as explained in [Ko2], without specifying the technical details here, if U is a left Hopf algebroid (for example, a Hopf algebra or the enveloping algebra A e of an associative algebra or still the enveloping algebra of a Lie algebroid) with respect to which N is a Yetter-Drinfel'd module, M a stable anti Yetter-Drinfel'd module, and P a stable anti Yetter-Drinfel'd contramodule, then (under suitable projectivity resp. flatness assumptions), the (co)chain complexes computing the various derived functors Tor U ‚ pN, M q, Ext ‚ U pN, P q Cotor U ‚ pN, M q, and Coext U ‚ pN, P q can be made into cyclic modules, which, in particular, implies the existence of (co)cyclic differentials of degree ˘1:…”