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, 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.

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.