On a result of Avramov (Q1110581)

A result of the reviewer [C. R. Acad. Sci., Paris, Sér. I 295, 665-668 (1982; Zbl 0509.13008)] on the triviality of a boundary homomorphism in a certain long exact sequence is used, in conjunction with the technique of André-Quillen homology, to give short proofs of the following two results: If \(\phi: A\to A\) is the Frobenius endomorphism of a local ring of characteristic \(p>0,\) and \(^{\phi}A\) it is the abelian group A, given an A-module structure via \(\phi\), then the following are equivalent: (i) A is regular; (ii) \(^{\phi}A\) is a flat A-module; (iii) \(^{\phi}A\) has finite flat dimension over A. [The equivalence of the first two conditions is a well-known result of \textit{E. Kunz}, cf. Am. J. Math. 91, 772-784 (1969; Zbl 0188.337)]. If \(J\subseteq I\) are ideals of A, such that \(pd_{A/J}(A/I)<\infty\), then \(\mu (I)=\mu (J)+\mu (I/J)\), where \(\mu\) refers to the minimal number of generators. (This result has independently been obtained by the reviewer, using a different approach.)