Lemma 9 ([Finite Artin models]). Let (A, m,k) be an Artin local ring, and let D be a complex of flat A -modules. Suppose H^*(D⊗_A k) is finite dimensional and supported in [a,b] . There is a bounded finite free complex C , with rank_A …

Lemma 9 ([Finite Artin models]). Let (A, m,k) be an Artin local ring, and let D be a complex of flat A -modules. Suppose H^*(D⊗_A k) is finite dimensional and supported in [a,b] . There is a bounded finite free complex C , with rank_A C^j=_k H^j(D⊗_A k), and a strong deformation retraction i:C⟶D, p:D⟶C, pi=1, 1-ip=dh+hd. Thus maps to the original complex, as well as maps between finite models, are available. Any fixed finite collection of chain maps, homotopies, triangles, and perfect pairings can be transported to these models.