Exercise 1, § 5

One says that an A-complex C is pure if for every right A-module P one has $H(P \otimes_A C) = 0$.

a) A complex homotopic to zero is pure; a complex of flat modules, with zero homology and bounded on the right, is pure. Give an example of a complex of free modules of rank 1, with zero homology but not pure.

b) Now prove that the following conditions are equivalent:

α) C is pure.

β) For every A-module M of finite presentation, one has $H(\mathrm{Homgr}_A(M, C)) = 0$.

γ) There exists a filtering inductive system $(C^{(\alpha)})_{\alpha \in J}$ of complexes homotopic to zero such that $C = \varprojlim C^{(\alpha)}$. (To see that β) implies γ), reduce to the case where $C_i = 0$ for $i < i_0$ and write $C_{i_0}$ as an inductive limit of modules of finite presentation.)