Lemma 2

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings, $N$ a finitely generated $B$-module and $y$ an element of $m_B$. The following two conditions are equivalent:

(i) the $A$-module $N/yN$ is flat and the homothety $y_N$ is injective; (ii) the $A$-module $N$ is flat and the homothety $y_{\kappa_A \otimes N}$ is injective.

When they are satisfied, the homothety $y_{M \otimes_A N}$ is injective for every $A$-module $M$.

Assume that the hypotheses of (i) are satisfied, and let us prove (ii) as well as the last assertion. Let $M$ be an $A$-module. Since the $A$-module $N/yN$ is flat, we deduce from the exact sequence $0 \to N \xrightarrow{y_N} N \to N/yN \to 0$ the exact sequences

where $u = 1_M \otimes y_N$ and $v = \mathrm{Tor}_1^A(1_M, y_N)$; it follows that the homothety of ratio $y$ is injective in $M \otimes_A N$, and bijective in $\mathrm{Tor}_1^A(M, N)$. Assume moreover that the A-module $M$ is finitely generated; then the B-module $\mathrm{Tor}_1^A(M, N)$ is finitely generated (A, X, p. 107, Prop. 6), hence is zero since $y$ belongs to $m_B$ (Nakayama's lemma), which implies that the A-module $N$ is flat (A, X, p. 74, Theorem 2).

(ii) $\Rightarrow$ (i): assume that the hypotheses of (ii) are satisfied. Consider the two exact sequences of B-modules

(1)

(2)

where $p$ and $i$ are the canonical homomorphisms. It follows that the homomorphism $1 \otimes p : \kappa_A \otimes_A N \longrightarrow \kappa_A \otimes_A \mathrm{Im}(y_N)$ is surjective, and (since $N$ is flat) that the kernel of the homomorphism $1 \otimes i : \kappa_A \otimes_A \mathrm{Im}(y_N) \longrightarrow \kappa_A \otimes_A N$ is isomorphic to $\mathrm{Tor}_1^A(\kappa_A, N/yN)$. But the mapping $(1 \otimes i) \circ (1 \otimes p)$, equal to $y_{\kappa_A \otimes_A N}$, is injective by hypothesis; it follows that $1 \otimes p$ is bijective and $1 \otimes i$ injective, and consequently that one has $\mathrm{Tor}_1^A(\kappa_A, N/yN) = 0$. It follows that the A-module $N/yN$ is flat (III, § 5, No. 2, Theorem 1 and No. 4, Proposition 2).

Since $N$ and $N/yN$ are flat over $A$, the same is true of $\mathrm{Im}(y_N)$ (exact sequence (2)). One then deduces from exact sequence (1) that $\kappa_A \otimes_A \mathrm{Ker}(y_N)$ is isomorphic to the kernel of $1 \otimes p$, hence is zero; thus the homothety $y_N$ is injective by Nakayama's lemma.