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.



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> How to describe explicitly an isomorphisms $K(A) \cong K(B)$, $HH^{\bullet}(A) \cong HH^{\bullet}(B)$ and $HC^{\bullet}(A) \cong HC^{\bullet}(B)$?
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Morita equivalence and isomorphisms in cohomology theories
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:


1. Starting from the category of modules $\text{Mod}(A)$ we can isolate the subcategory of [tiny](https://qchu.wordpress.com/2015/05/07/tiny-objects/) 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.
2. 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.
3. 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.


