Lemma 15.55.2. Let $R$ be a ring. Let $\mathcal{A}$ be the abelian category of $R$-modules. There is a canonical isomorphism $\mathop{\mathrm{Ext}}\nolimits _\mathcal {A}(M, N) = \mathop{\mathrm{Ext}}\nolimits ^1_ R(M, N)$ compatible with the long exact sequences of Algebra, Lemmas 10.71.6 and 10.71.7 and the $6$-term exact sequences of Homology, Lemma 12.6.4.

Proof. Omitted. $\square$

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