Morita equivalence and isomorphisms in cohomology theories Let $A,B$ be two unital algebras. We say that $A,B$ are Morita equivalent if there are $A-B$ and $B-A$ bimodules $P,Q$ such that 
$$P \otimes_{B} Q \cong A, Q \otimes_A P \cong B$$
(as $A-A$ and $B-B$ bimodules).
Suppose that $A,B$ are Morita equivalent. Then one can show that $K$-theory and cyclic and Hochschild cohomologies of them are isomorphic.

How to describe explicitly an isomorphisms $K(A) \cong K(B)$, $HH^{\bullet}(A) \cong HH^{\bullet}(B)$ and $HC^{\bullet}(A) \cong HC^{\bullet}(B)$?

 A: The conceptual point is that all of these invariants are Morita invariant because they can be defined directly in terms of the category of modules. Explicitly:


*

*Starting from the category of modules $\text{Mod}(A)$ we can isolate the subcategory of tiny or compact projective objects: those modules $M$ such that $\text{Hom}(M, -)$ preserves all colimits. These turn out to be precisely the finitely generated projective modules, from which one can define K-theory as usual. So given a Morita equivalence $Q \otimes_A (-) : \text{Mod}(A) \cong \text{Mod}(B)$ the induced map on f.g. projectives comes from restricting to tiny objects, and then the induced map on K-theory comes from passing to the Grothendieck group.

*Hochschild cohomology of $A$ is the derived endomorphisms of the identity functor $\text{Mod}(A) \to \text{Mod}(A)$. So given a Morita equivalence $\text{Mod}(A) \cong \text{Mod}(B)$ we get an induced equivalence on (cocontinuous) endofunctor categories $\text{Bimod}(A, A) \cong \text{Bimod}(B, B)$ sending the identity to the identity, which furthermore induces an equivalence on self-Ext.

*I am less familiar with the details of the cyclic case but morally the only additional thing to do is to incorporate the natural (derived) $S^1$-action on Hochschild stuff. 
