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Let $(A,m,k)$ be commutative noetherian local ring.

Can one detect if $A$ is a Gorenstein ring from the André–Quillen homologies $H_n(A,k,-)$?

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"No" if you want a standard vanishing result, and "no up to now" if you are thinking in another kind of characterization (see however for a related result: Garcia-Soto, Ascent and descent of Gorenstein property, Glasgow Math. J. 46 (2004) 205–210).

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