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Ok, I’m reading the paper Homology and cohomology of associative algebras. A concise introduction to cyclic homology by Christian Kassel, and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ and $S$ are two Morita equivalent rings then $$ H_*(R,M) \cong H_*(S, Q \otimes_R M \otimes_R P) $$ (here $H_*$ denotes Hochshild homology). He then shows how $$ H_*(M,M) \cong H_*(S,S) \,, $$ and states that

Hochschild cohomology groups are Morita-invariant in a similar way.

I wanted to find a proof or reference for this? Just wanted to be completely sure about this and avoid any confusion, in what similar way are Hochschild cohomology groups Morita-invariant? Please be nice lol.

Jendrik Stelzner
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L-A
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    All is at pag. 19: the functors $\Phi$ and $\Psi$ preserve projective resolutions (of bimodules) and the Hochschild (co)homology is defined as a derived functor. Remember that derived functors do not depend on the choice of resolution, hence the isomorphisms – Avitus Dec 14 '13 at 17:12
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    Check out paragraph 1.5.6 in Loday's "Cyclic Homology". –  Dec 14 '13 at 23:54
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    The proof for homology can be found in Weibel, chapter 9 – Alexander Grothendieck Dec 15 '13 at 22:35
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    Both references are great books – Avitus Dec 16 '13 at 08:58
  • Lol that was staring me right in the face the whole time, I was actually going by Weibel's proof and was curious about the cohomology case and used Kassel as a reference, thank you all for taking the time to answer. – L-A Dec 16 '13 at 19:32
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    Why don't you write down an answer by your own? It would be a nice exercise, and this post would be "completed" :-) – Avitus Dec 17 '13 at 09:28
  • A fun partial answer to this is, if your associative algebra is of finite hochschild cohomological dimension, then you can conclude the result by van den bergh duality – AB_IM Apr 09 '14 at 17:57
  • A concise way to think about it is that a Morita equivalence induces in particular a derived equivalence, and that Hochschild (co)homology is just computing a tensor product or hom-set in this derived category, i.e. a more general statement says that derived Morita equivalent (Rickard's work) algebras have isomorphic Hochschild (co)homology – Pedro Dec 15 '22 at 12:12

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