Pure resolutions of unbounded complexes of modules

TL;DR

This paper develops pure injective and projective resolutions for unbounded complexes of modules over a commutative ring, enabling a symmetric monoidal structure on the pure derived category.

Contribution

It introduces constructions of pure resolutions for unbounded complexes and establishes a symmetric monoidal structure on the pure derived category.

Findings

Pure injective and projective resolutions can be constructed for unbounded complexes.

A closed symmetric monoidal structure on the unbounded pure derived category is established.

The work extends the theory of pure resolutions to unbounded complexes.

Abstract

Let $R$ be a commutative ring. We show that pure injective resolutions and pure projective resolutions can be constructed for unbounded complexes of $R$-modules. We use these to obtain a closed symmetric monoidal structure on the unbounded pure derived category.