Abstract. For a Cohen-Macaulay ring R, we exhibit the equivalence of the bounded derived categories of certain resolving subcategories, which, amongst other results, yields an equivalence of the bounded derived category of finite length and finite projective dimension modules with the bounded derived category of projective modules with finite length homologies. This yields isomorphisms of K-theory and Witt groups (amongst other invariants) and improves on terms of associated spectral sequences and Gersten complexes.

a b s t r a c tLet X = Spec(A) be a smooth, affine variety of dimension n ≥ 2 over the field R of real numbers. Let P be a projective A-module of rank n such that its nth Chern class C n (P) ∈ CH 0 (X) is zero. In this set-up, Bhatwadekar-Das-Mandal showed (amongst many other results) that P A ⊕ Q in the case that either n is odd or the topological space X (R) of real points of X does not have a compact, connected component. In this paper, we prove that similar results hold for smooth, affine varieties over an Archimedean real closed field R.

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We consider bounded complexes P • of finitely generated projective A-modules whose homologies have finite projective dimension and are locally Cohen-Macaulay. We give a necessary and sufficient condition so that its dual P * • also has the same property.

Let X = Spec( A) be a smooth, affine variety of dimension n 2 over the field R of real numbers. Let P be a projective A-module of rank n such that its nth Chern class C n (P ) ∈ CH 0 (X) is zero.In this set-up, Bhatwadekar-Das-Mandal showed (amongst many other results) that P A ⊕ Q in the case that either n is odd or the topological space X(R) of real points of X does not have a compact, connected component. In this paper, we prove that similar results hold for smooth, affine varieties over an arbitrary real closed field R. The proof is algebraic and does not make use of Tarski's principle, nor of the earlier result for R.