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In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then

  1. it should hopefully be understood by most readers as saying $H^n(A,M)=0$ for all $n\geq 2$ and all $A$-bimodules $M$
  2. this even has an accepted name ("quasi-free", I think, although corrections are welcome)

However, I am looking for terminology to describe a weaker phenomenon.

Question. What should I say if $A$ is a commutative algebra and I am only interested in $H^n(A,M)$ for $M$ a symmetric $A$-bimodule (i.e. $a\cdot m=m\cdot a$)? And what about the corresponding situation for Hochschild homology $H_n(A,M)$?

The context for my question is that I have a working title for some joint work in progress, which to be accurate would read as follows

"A commutative Banach algebra whose Hochschild homology with symmetric coefficients vanishes in degrees two and above, but not necessarily in degree one."

This is fairly ghastly as a title for several reasons, but one step in trying to render it more acceptable would be an accepted or sensible abbreviation for "Hochschild homology with symmetric coefficients". I am tempted by "commutative Hochschild homology" but this instinctively feels like it might already be used to mean something else.

Also, if I were allowing arbitrary bimodules and not just the symmetric ones, then the horrible title would easily be replaced by "an algebra with weak global dimension one". But to my knowledge, although "homology with symmetric coefficients vanishes in all degrees $> n$" clearly has a flavour of being "dimension $\leq n$ in some sense", I don't know of any existing terminology in this vein. Does this restricted version of dimension have an existing name?

By the way, in Banach world the whole correspondence between André–Quillen homology and (a summand of) Hochschild homology breaks down, so I am definitely not talking about AQ here.

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  • $\begingroup$ I will leave the (likely unhelpful) comment that I believe this is equivalent to the concept of being "formally smooth" in the world of ordinary homological algebra that I know. Based on your last sentence you may know this. $\endgroup$
    – Tyler Lawson
    Commented Nov 2, 2015 at 17:53
  • $\begingroup$ I don't have an answer, but +1 for wanting to change that title. For what it is worth, it is entirely reasonable to say $A$ is $n$-acyclic with $M$-coefficients if $H_k(X;M) = 0$ for all $k \geq n$. So maybe "2-acyclicity of a commutative Banach algebra in Hochschild homology with symmetric coefficients"? $\endgroup$
    – Vidit Nanda
    Commented Nov 2, 2015 at 18:02
  • $\begingroup$ @TylerLawson I'm always uncertain which things are definitions and which are results. Smooth means $H^2_s(A,M)=0$ for all commutative $M$, I think? FWIW I'm getting all of $H_2(A,M)=0$, which wouldn't happen for e.g. $k[x,y]$ even though that is smooth $\endgroup$
    – Yemon Choi
    Commented Nov 2, 2015 at 18:28
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    $\begingroup$ Not an answer, but to confirm that "commutative Hochschild homology" is possibly not the best term: The phrase "commutative cohomology" was used by Harrison (1962) for a symmetrized version of Hochschild cohomology over a commutative ring. On the other hand, my understanding is that this and its homological variant have since been called "Harrison (co)homology," as in the following, for instance: sciencedirect.com/science/article/pii/0021869368900628 $\endgroup$
    – Manny Reyes
    Commented Nov 5, 2015 at 15:21
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    $\begingroup$ @MannyReyes Thanks for the reminder. Once upon a time I used to remember Harrison's paper and all that Barr-Gerstenhaber-Schack jazz: arxiv.org/abs/0709.3325 (see the list of references) $\endgroup$
    – Yemon Choi
    Commented Nov 5, 2015 at 15:45

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