Are All Localizing Subcategories of Stable Homotopy Categories Coreflective?

We prove that, in a triangulated category with combinatorial models, every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large-cardinal axiom (Vopěnka’s principle) is assumed true. It follows that, under the same assumptions, orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories. The existence of such a bijection was left as an open problem by Hovey, Palmieri and Strickland in their axiomatic study of stable homotopy categories and also by Neeman in the context of well-generated triangulated categories. Introduction The main purpose of this article is to address a question asked in [37, p. 35] of whether every localizing subcategory (i.e., a full triangulated subcategory closed under coproducts) of a stable homotopy category T is the kernel of a localization on T (or, equivalently, the image of a colocalization). We prove that the answer is affirmative if T arises from a combinatorial model category, assuming the truth of a large-cardinal axiom from set theory called Vopěnka’s principle [2], [38]. A model category (in the sense of Quillen) is called combinatorial if it is cofibrantly generated [33], [36] and its underlying category is locally presentable [2], [27]. Many triangulated categories of interest admit combinatorial models, including derived categories of rings and the homotopy category of spectra. More precisely, we show that, if K is a stable combinatorial model category, then every semilocalizing subcategory C of the homotopy category Ho(K) is coreflective under Vopěnka’s principle, and the coreflection is exact if C is localizing. We call C semilocalizing if it is closed under coproducts, cofibres and extensions, but not necessarily under fibres. Examples include kernels of nullifications in the sense of [11] or [19] on the homotopy category of spectra. We also prove that, under the same hypotheses, every semilocalizing subcategory C is singly generated ; that is, there is an object A such that C is the smallest semilocalizing subcategory containing A. The same result is true for localizing subcategories. The Date: April 16, 2012. 2010 Mathematics Subject Classification. 18E30, 18G55, 55P42, 55P60, 03E55. ∗ The two first-named authors were supported by the Spanish Ministry of Education and Science under MEC-FEDER grants MTM2007-63277 and MTM2010-15831, and by the Generalitat de Catalunya as members of the team 2009 SGR 119. The third-named author was supported by the Ministry of Education of the Czech Republic under the project MSM 0021622409 and by the Czech Science Foundation under grant 201/11/0528. He gratefully acknowledges the hospitality of the University of Barcelona and the Centre de Recerca Matemàtica.