Mathematical Research Letters

Volume 16 (2009)

Number 1

Brown representability does not come for free

Pages: 1 – 5



Carles Casacuberta (Universitat de Barcelona)

Amnon Neeman (The Australian National University)


We exhibit a triangulated category $\mathcal{T}$ having both products and coproducts, and a triangulated subcategory $\mathcal{S} \subset \mathcal{T}$ which is both localizing and colocalizing, for which neither a Bousfield localization nor a colocalization exists. It follows that neither the category $\mathcal{S}$ nor its dual satisfy Brown representability. Our example involves an abelian category whose derived category does not have small Hom-sets.

Published 1 January 2009