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arXiv · 1606.07976

Totally acyclic approximations

Petter A. Bergh·David A. Jorgensen·W. Frank Moore
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Abstract

Let $R$ be a commutative local ring. We study the subcategory of the homotopy category of $R$-complexes consisting of the totally acyclic $R$-complexes. In particular, in the context where $Q\to R$ is a surjective local ring homomorphism such that $R$ has finite projective dimension over $Q$, we define an adjoint pair of functors between the homotopy category of totally acyclic $R$-complexes and that of $Q$-complexes, which are analogous to the classical adjoint pair between the module categories of $R$ and $Q$. We give detailed proofs of the adjunction in terms of the unit and counit. As a consequence, one obtains a precise notion of approximations of totally acyclic $R$-complexes by totally acyclic $Q$-complexes.

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Petter A. Bergh, David A. Jorgensen, W. Frank Moore. 2016-06-25. Totally acyclic approximations. https://arxiv.org/abs/1606.07976

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