Totally acyclic approximations

TL;DR

This paper investigates the structure of totally acyclic complexes over local rings, establishing an adjoint pair of functors that facilitate approximations between complexes over related rings, advancing understanding in homological algebra.

Contribution

It introduces an adjoint pair of functors between categories of totally acyclic complexes over related rings, providing a framework for approximations in homotopy categories.

Findings

Defined an adjoint pair of functors between homotopy categories of totally acyclic complexes.

Established a notion of approximations of complexes over R by those over Q.

Provided detailed proofs of the adjunction in terms of unit and counit.

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.