Proposition 7

a) Let N be a flat A-module. If the family x is completely secant for M, it is so for $M \otimes_A N$.

b) Let $0 \to M' \to M \to M'' \to 0$ be an exact sequence of $A$-modules. If the family $x$ is completely transverse for $M'$ and for $M''$, it is so for $M$.

The complex $K.(x, M \otimes_A N)$ is isomorphic by definition to $K.(x, M) \otimes_A N$; since $N$ is flat, one deduces from this an isomorphism $H.(x, M) \otimes_A N \to H.(x, M \otimes_A N)$ (X, p. 66, cor. 2), whence a).

The complex $K.(x)$ being flat, one has an exact sequence of complexes

assertion b) follows from the associated exact homology sequence.