On Perfectly Generating Projective Classes in Triangulated Categories

Abstract

We say that a projective class in a triangulated category with coproducts is perfect if the corresponding ideal is closed under coproducts of maps. We study perfect projective classes and the associated phantom and cellular towers. Given a perfect generating projective class, we show that every object is isomorphic to the homotopy colimit of a cellular tower associated to that object. Using this result and the Neeman's Freyd-style representability theorem, we give a new proof of Brown Representability Theorem.

Key Words:

• Brown representability
• Perfect projective class
• Triangulated category with coproducts

2000 Mathematics Subject Classification:

• 18E15
• 18E35
• 18E40
• 16D90

ACKNOWLEDGMENTS

The author would like to thank to Henning Krause for his interest in this work. He is also very indebted to an anonymous referee, which pointed out a number of mistakes in a preliminary version of this article, leading to a radical change of the article. The author was supported by the grant PN2CD-ID-489.

Notes

Communicated by L. Small.