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The Stacks project

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 ith 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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