Mixed vs Stable Anti-Yetter–Drinfeld Contramodules

We examine the cyclic homology of the monoidal category of modules over a finite dimensional Hopf algebra, motivated by the need to demonstrate that there is a difference between the recently introduced mixed anti-Yetter–Drinfeld contramodules and the usual stable anti-Yetter–Drinfeld contramodules. Namely, we show that Sweedler's Hopf algebra provides an example where mixed complexes in the category of stable anti-Yetter–Drinfeld contramodules (previously studied) are not equivalent, as differential graded categories to the category of mixed anti-Yetter–Drinfeld contramodules (recently introduced).