منابع مشابه
Brown Representability and Flat Covers
We exhibit a surprising connection between the following two concepts: Brown representability which arises in stable homotopy theory, and flat covers which arise in module theory. It is shown that Brown representability holds for a compactly generated triangulated category if and only if for every additive functor from the category of compact objects into the category of abelian groups a flat c...
متن کاملDerived categories, resolutions, and Brown representability
These notes are based on a series of five lectures given during the summer school “Interactions between Homotopy Theory and Algebra” held at the University of Chicago in 2004.
متن کاملAuslander-reiten Theory via Brown Representability
We develop an Auslander-Reiten theory for triangulated categories which is based on Brown’s representability theorem. In a fundamental article [3], Auslander and Reiten introduced almost split sequences for the category of finitely generated modules over an artin algebra. These are short exact sequences which look almost like split exact sequences, but many authors prefer to call them Auslander...
متن کاملBrown Representability Does Not Come for Free
We exhibit a triangulated category T having both products and coproducts and a triangulated subcategory S ⊂ T which is both localizing and colocalizing, and for which neither a Bousfield localization nor a colocalization exists. It follows that neither the category S nor its dual satisfy Brown representability. Our example involves an abelian category whose derived category does not have small ...
متن کاملFailure of Brown Representability in Derived Categories
Let T be a triangulated category with coproducts, T ⊂ T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors {T}op −→ Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this ...
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ژورنال
عنوان ژورنال: Journal of Pure and Applied Algebra
سال: 2001
ISSN: 0022-4049
DOI: 10.1016/s0022-4049(00)00002-5