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Hochschild cohomology can be used to characterise formal smoothness of unital associative algebras; in that such an algebra $A$ is formally smooth if and only if it is of Hochschild cohomological dimension at most $1$.

I was curious, is there a similar characterization of formal smoothness in the category of commutative unital associative algebras, making use of the Hochschild cohomology modules?

I mean in the finite dimensional case: I guess you can say that if A is smooth over k of essentially finite type and is finite dimension $d$, therefore is $d$-calabi-Yau, and so its cohomology (and homology) vanished above index $d$. Therefore it must be of finite hochschild cohomological dimension. However, this all relies on the finite dimension assumption... so whould there be some kind of arbitrary dimensional generalization? Or betteryet an infintiely generated generalization?

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Would you be happy with something like "the Kaehler module is projective as an A-module"? –  Yemon Choi May 18 at 22:38
    
lol nope, besides that would involve hochschild homology.. and no chomology –  CSA May 18 at 22:41
    
So what you want is something like: "if the Harrison/AQ cohomology vanishes in degree 2 and above, plus some other side conditions, then A is formally smooth"? –  Yemon Choi May 18 at 22:57

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up vote 4 down vote accepted

Yes, the Hochschild-Konstant-Rosenberg theorem has a converse. More generally you have vanishing characterizations of smoothness in terms of Hochschild homology (one of them is e.g. Avramov, Luchezar L.; Vigué-Poirrier, Micheline, Hochschild homology criteria for smoothness, Internat. Math. Res. Notices 1992, no. 1, 17–25: If $HH_i(R,R)=0=HH_j(R,R)$ for some positive even integer $i$ and some positive odd integer $j$, then $R$ is smooth, where $R$ is a flat commutative algebra of finite type over a commutative ring; this is the case for instance when $HH_i(R,M)=0$ for all $R$-modules $M$ and all $i\gg 0$). There are also some similar criteria for cohomology (though more vanishing terms are required) (see e.g., Avramov, Luchezar L.; Iyengar, Srikanth, Gaps in Hochschild cohomology imply smoothness for commutative algebras, Math. Res. Lett. 12 (2005), no. 5-6, 789–804).

However, as indicated by Yemon Choi, the main homological tool for these characterizations is André-Quillen (co)homology.

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Of course, these results are characterizations (the converses are clear by HKR theorem). Regarding your last question, if the algebra is not essentially of finite type, you cannot expect formal smoothness (at most regularity), as an algebra of power series shows. –  Vinteuil May 19 at 10:34

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