A noncommutative calculus on the cyclic dual of Ext

We show that if the cochain complex computing Ext groups (in category of modules over Hopf algebroids) admits a cocyclic structure, then noncommutative Cartan calculus structure on Tor dualises in cyclic sense to Coext Cotor. More precisely, duals chain resp. spaces two classical derived functors lead complexes compute more exotic ones, giving opposite module an operad with multiplication induce operations such as Lie derivative, cap product (or contraction), and (cyclic) differential, along higher homotopy operators defining up homotopy. In particular, this allows recover from differential geometry or Chevalley-Eilenberg for Lie(-Rinehart) algebras without any finiteness conditions use topological tensor products.