An A-infinity algebra A is smooth a’la Kontsevich if it is perfect as an A–A-bimodule. I am wondering about the standard tricks to show smoothness of given algebras. A relatively basic example should be the following. I have a guess that the following Z/2Z-graded A-infinity algebras over C should be smooth even though their “underlying associative algebra” isn’t. The algebra An is a vector space with basis {1,e}, where e is in even degree. The “classical” multiplications are 1 acts as a unit and e∗e=0. There is only one higher multiplication, e⊗2n↦1, n>1. If I did my math right, this defines an A-infinity algebra. The basis for the guess of smoothness is that the Hochschild homology is finite dimensional, which would be a corollary of perfection, but that doesn’t quite prove it without some other statement about the perfection of finite-dimensional modules over A⊗Aop(which would also imply the statement directly of course).

I have struggled quite a bit unsuccessfully to brute force this. I have tried to find a suitable dg-algebra equivalent to A and then to compute explicitly A⊗Aop and then write down a resolution of A. This was too gritty for me, though maybe a more insightful person could make it work. Is the fact true? Can anyone give an explanation/proof?

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**Attribution***Source : Link , Question Author : Daniel Pomerleano , Answer Author : Community*