A Brown representability theorem via coherent functors

By Henning Krause. 2002

99 citations recorded by the source corpus.

Abstract In this paper we discuss the Brown Representability Theorem for triangulated categories having arbitrary coproducts. This theorem is an extremely useful tool and various versions appear in the literature. All of them require a set of objects which generate the category in some appropriate sense. Depending on the proof, there are essentially two types: the first type is based on the analogue of iterated attaching of cells which is used in the topological case; the second type is based on solution sets and applies a variant of Freyd's Adjoint Functor Theorem. Motivated by recent work of Neeman (A. Neeman, Triangulated Categories, Annals of Mathematics Studies, 148, Princeton University Press, Princeton, NJ, 2001) and Franke (On the Brown representability theorem for triangulated categories, Topology, to appear), we prove a new theorem of the first type (Theorem A) and add, as an application, a Brown Representability Theorem for covariant functors (Theorem B). The final Theorem C establishes a filtration of a triangulated category which clarifies the relation between results of the first and the second type.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/22f5b7fd2d0477025e8896ec91e5ba3b8df9c125

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Citations in the corpus

On Neeman's well generated triangulated categories (2001, unknown): “See Lemma 3 in [2].”
On differential graded categories (2006, background): “…is an equivalence between the category of B-C-bimodules and the category of coproduct preserving functors from the category of B-modules to that of C-modules (note that here and in what follows, we need to consider ‘large’ categories and should introduce universes to make our statements rigorous…).”
Silting theory in triangulated categories with coproducts (2015, background): “7] and [35]) and it is algebraic, then there exists a dg category A and a set S of objects in D(A) such that D ∼= LocD(A)(S) .”
Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence (2009, background): “Parts of this paper have been written when I was visiting the Institut des Hautes Études Scientifiques, which I wish to thank for the excellent working conditions.”
Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, unknown): “[48, 49]) effectively applied them to study Brown representability and the telescope conjecture.”
Colocalizing subcategories and cosupport (2010, background): “The notion of a set of perfect generators is analogous; see [17].”
Cohomological quotients and smashing localizations (2003, methodology): “Our theorem provides a bijective correspondence between smashing localizations of S and cohomological quotients of Sc; it improves a similar correspondence [20] – the new ingredient in our proof being a recent variant [21] of Brown’s Representability Theorem [6].”
Torsion pairs in silting theory (2016, unknown): “We also prove that a deﬁnable subcategory V of T is preenveloping, i.e., for any object X in T there is a map φ Recall from [Krause 2002b, Proposition 5.1] that a covariant additive functor F : T → Mod ( (cid:90) ) is said to be coherent if the following equivalent conditions are satisﬁed: (1)…”
Silting Objects (2018, background): “If T is compactly generated, then it satisfies both Brown representability and Brown representability for the dual - see [71] for details.”