Koszul Acyclicity Characterises Local Regular Sequences

If $\mathbf{x}$ is empty, its Koszul complex is concentrated in degree zero and the assumed nonzero terminal quotient is exactly the nonzero-module condition in the definition of an empty regular sequence. Thus the equivalence holds in this case. For nonempty $\mathbf{x}$, regularity implies acyclicity by the forward theorem.

Conversely, suppose $\mathbf{x}$ is nonempty and the Koszul complex is acyclic in positive degrees. The local converse lemma makes the last element injective on the quotient by the preceding entries and makes the shorter Koszul complex acyclic. Its terminal quotient is nonzero because it surjects onto $M\mathrm{/}(\mathbf{x})M$. Iterating proves all ordered injectivity conditions, while the final nonzero quotient is assumed; hence $\mathbf{x}$ is regular.