Lemma 46.8.9. Let $A$ be a ring.

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

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

$\text{Pext}^ i_ A(M, N)$ 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 $M$.

## Comments (0)