Example

Let $k$ be a ring; take $A = k[X_1, ..., X_n]$ and $x = (X_1, ..., X_n)$. The sequence $x$ is A-regular and the proposition gives again the acyclicity in degrees $> 0$ of the complex $S_k(k^n) \otimes_k \Lambda_k(k^n)$ (cf. X, p. 152, prop. 3). Likewise in the ring of formal series $\hat{A} = k[[X_1, ..., X_n]]$, the sequence $(X_1, ..., X_n)$ is $\hat{A}$-regular, hence completely secant for $\hat{A}$.