Lemma 10.99.6. Let $R$ be a local ring with maximal ideal $\mathfrak m$ and residue field $\kappa = R/\mathfrak m$. Let $M$ be an $R$-module. If $\text{Tor}_1^ R(\kappa , M) = 0$, then for every finite length $R$-module $N$ we have $\text{Tor}_1^ R(N, M) = 0$.
Proof. By descending induction on the length of $N$. If the length of $N$ is $1$, then $N \cong \kappa $ and we are done. If the length of $N$ is more than $1$, then we can fit $N$ into a short exact sequence $0 \to N' \to N \to N'' \to 0$ where $N'$, $N''$ are finite length $R$-modules of smaller length. The vanishing of $\text{Tor}_1^ R(N, M)$ follows from the vanishing of $\text{Tor}_1^ R(N', M)$ and $\text{Tor}_1^ R(N'', M)$ (induction hypothesis) and the long exact sequence of Tor groups, see Lemma 10.75.2. $\square$
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