# Neeman's characterization of K(R-Proj) via Bousfield localization

## TL;DR

This paper discusses Neeman's theorem on the compact generation of the homotopy category of projective modules over a ring, providing a new proof using Bousfield localization and recent theoretical ideas.

## Contribution

It offers an alternative proof of Neeman's characterization of $K(R ext{-}Proj)$ using Bousfield localization and recent advances in the field.

## Key findings

- $K(R ext{-}Proj)$ is $oldsymbol{ ext{aleph}_1}$-compactly generated.
- The category $K^+ (R ext{-}proj)$ provides a set of generators.
- Every complex in $K(R ext{-}Proj)$ vanishes in a specific Bousfield localization.

## Abstract

Let $R$ be an associative ring with unit and denote by $K({\rm R \mbox{-}Proj})$ the homotopy category of complexes of projective left $R$-modules. Neeman proved the theorem that $K({\rm R \mbox{-}Proj})$ is $\aleph_1$-compactly generated, with the category $K^+ ({\rm R \mbox{-}proj})$ of left bounded complexes of finitely generated projective $R$-modules providing an essentially small class of such generators. Another proof of Neeman's theorem is explained, using recent ideas of Christensen and Holm, and Emmanouil. The strategy of the proof is to show that every complex in $K({\rm R \mbox{-}Proj})$ vanishes in the Bousfield localization $K({\rm R \mbox{-}Flat})/\langle K^+ ({\rm R \mbox{-}proj}) \rangle.$