# Talk:Hochschild homology

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Field:  Algebra

## Need Motivation and Examples

There needs to be additional motivation and examples for hochschild homology such as computations for artin algebras over a field and the HKR theorem for smooth manifolds/varieties. There should write up a section explaining the relationship of hochschild homology with deformation quantization. Also, it would be nice to discuss the motivic properties with respect to semi-orthogonal decompositions of derived categories, but that would require quite a bit of preperatory articles. — Preceding unsigned comment added by Algebraic geometer (talkcontribs) 05:11, 22 April 2017 (UTC)

It would be more sensible to give the standard usable definition of Hochschild cohomology differential first, rather than starting with Tor and Ext g7c4 16:28, 16 October 2015 (UTC) — Preceding unsigned comment added by G7c4 (talkcontribs)

My hazy (and inexpert) recollection is that the interpretation of Hochschild homology via Loday's construction holds only for commutative unital algebras and symmetric "unit-linked" bimodules. Is there anyone reading, better versed in the relevant details, who could help clarify the corresponding sections of this entry? NowhereDense (talk) 10:14, 27 September 2008 (UTC)

## Symmetric bimodule

What is a "symmetric bimodule"? Is it the same as a Krull symmetric bimodule? (I added a request on the page itself, as well.) Dylan Thurston (talk) 02:30, 23 March 2012 (UTC)

A "symmetic bimodule" $M$ is an $R-R$ bimodule such that the left action equals the right action: that is, $rm=mr$ for all $m\in M$ and $r\in R$. — Preceding unsigned comment added by 67.171.213.135 (talk) 18:48, 17 May 2012 (UTC)