Lemma 15.89.6. Let $R$ be a ring. Let $I = (f_1, \ldots , f_ n)$ be a finitely generated ideal of $R$. For any $R$-module $N$ set
\[ H_1(N, f_\bullet ) = \frac{\{ (x_1, \ldots , x_ n) \in N^{\oplus n} \mid f_ i x_ j = f_ j x_ i \} }{\{ f_1x, \ldots , f_ nx) \mid x \in N\} } \]
For any $R$-module $N$ there exists a canonical short exact sequence
\[ 0 \to \mathop{\mathrm{Ext}}\nolimits _ R(R/I, N) \to H_1(N, f_\bullet ) \to \mathop{\mathrm{Hom}}\nolimits _ R(K, N) \]
where $K$ is as in Lemma 15.89.5.
Proof.
The notation above indicates the $\mathop{\mathrm{Ext}}\nolimits $-groups in $\text{Mod}_ R$ as defined in Homology, Section 12.6. These are denoted $\mathop{\mathrm{Ext}}\nolimits _ R(M, N)$. Using the long exact sequence of Homology, Lemma 12.6.4 associated to the short exact sequence $0 \to I \to R \to R/I \to 0$ and the fact that $\mathop{\mathrm{Ext}}\nolimits _ R(R, N) = 0$ we see that
\[ \mathop{\mathrm{Ext}}\nolimits _ R(R/I, N) = \mathop{\mathrm{Coker}}(N \longrightarrow \mathop{\mathrm{Hom}}\nolimits (I, N)) \]
Using the short exact sequence of Lemma 15.89.5 we see that we get a complex
\[ N \to \mathop{\mathrm{Hom}}\nolimits (M, N) \to \mathop{\mathrm{Hom}}\nolimits _ R(K, N) \]
whose homology in the middle is canonically isomorphic to $\mathop{\mathrm{Ext}}\nolimits _ R(R/I, N)$. The proof of the lemma is now complete as the cokernel of the first map is canonically isomorphic to $H_1(N, f_\bullet )$.
$\square$
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