On a homological characterization of a certain class of local rings

It is shown that the nondegeneracy of the Yoneda product $\operatorname{Ext}_A^p(k,M)\times\operatorname{Ext}_A^{n-p}(M,k)$ ($M$ is either a noetherian module or a complex of finite projective dimension, and $k$ is the residue field) characterizes the regularity of the ring $A$, whereas the isomorphism $\operatorname{Ext}_A^p(k,M)\approx\operatorname{Ext}_A^{n-p} (M,k)$ characterizes the fact that $A$ is Gorenstein.
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