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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 $HC^{\bullet}(A) \cong HC^{\bullet}(B)$?

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    $\begingroup$ Did you consult the literature? The Hochschild and cyclic case is in several books, e. g. "Cyclic homology" by Loday. The K-theoretic case is straightforward: K-theory of a ring can be defined solely in terms of the category of modules over the ring, and Morita equivalence is more or less the same as equivalence of the categories of modules. $\endgroup$ – მამუკა ჯიბლაძე Sep 26 '17 at 17:02
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    $\begingroup$ One quick way to do the Hochschild case is to use the fact that $HH_*(A/k) \cong \mathrm{Tor}_*^{A \otimes A^{op}}(A,A)$, in other words Tor in the category of A-bimodules. Now, the Morita equivalence yields an equivalence between categories of bimodules by $M \mapsto Q \otimes_AM\otimes_AP$ and similarly the other way. This sends $A$ to $B$, so the two Tor groups are the same. Same argument for Ext and $HH^*$. $\endgroup$ – Dylan Wilson Sep 26 '17 at 22:33
  • $\begingroup$ The map $K(A) =KK(\mathbb{C},A)\rightarrow K(B) = KK(\mathbb{C},B)$ could be realized by multiplication with the Kasparov element $[P,0] \in KK(A,B)$, where $P$ is the Morita $A,B$-bimodule. $\endgroup$ – hänsel Dec 1 '17 at 15:26
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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 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.

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