Remark 1

Let A be a local ring. The A-module $\operatorname{Tor}_1^A(\kappa_A, \kappa_A)$ is isomorphic to $\mathfrak{m}_A / \mathfrak{m}_A^2$ (A, X, p. 72, Example). Hence when A is noetherian, in order that $\operatorname{Tor}_1^A(\kappa_A, \kappa_A)$ be zero, it is necessary and sufficient that $\mathfrak{m}_A$ be zero, that is to say, that A be a field. Cor. 1 therefore implies that a noetherian local ring of homological dimension 0 is a field.