Lemma 46.8.9. Let $A$ be a ring.

1. $\text{Pext}^ i_ A(M, N) = 0$ for $i > 0$ whenever $N$ is pure injective,

2. $\text{Pext}^ i_ A(M, N) = 0$ for $i > 0$ whenever $M$ is pure projective, in particular if $M$ is an $A$-module of finite presentation,

3. $\text{Pext}^ i_ A(M, N)$ is also the $i$th cohomology module of the complex $\mathop{\mathrm{Hom}}\nolimits _ A(P_\bullet , N)$ where $P_\bullet$ is a pure projective resolution of $M$.

Proof. To see (3) consider the double complex

$A^{\bullet , \bullet } = \mathop{\mathrm{Hom}}\nolimits _ A(P_\bullet , I^\bullet )$

Each of its rows is exact except in degree $0$ where its cohomology is $\mathop{\mathrm{Hom}}\nolimits _ A(M, I^ q)$. Each of its columns is exact except in degree $0$ where its cohomology is $\mathop{\mathrm{Hom}}\nolimits _ A(P_ p, N)$. Hence the two spectral sequences associated to this complex in Homology, Section 12.25 degenerate, giving the equality. $\square$

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