Cohomological quotients and smashing localizations

By Henning Krause. 2003

63 citations recorded by the source corpus.

The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S , a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic K -theory and demonstrates the relevance of the telescope conjecture for derived categories. Another application leads to a derived analogue of an almost module category in the sense of Gabber-Ramero. It is shown that the derived category of an almost ring is of this form.

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

View Paper: https://www.semanticscholar.org/paper/9e010f41c51baf68f9ddda1f46a1c2a805dedc1f

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Citations in the corpus

A tour about existence and uniqueness of dg enhancements and lifts (2016, background): “s follows quite easily from [17, Corollary 5.3]). It is an exercise with the deﬁnition of compact objects to prove that, if this is true, then L is smashing. An approach to Question 5.13 could be via [38] and [39]. 5.3. Strong uniqueness. We ﬁnish this section recalling that a result similar to Coro”
Parametrizing recollement data for triangulated categories (2009, background): “6 of the article [32] of H.”
Homological algebra in bivariant K-theory and other triangulated categories. II (2008, unknown): “In both cases, the morphisms in the relevant localisations can be computed using projective resolutions, so that the localisation is again a category with morphism sets instead of morphism classes.”
Triangulated Categories: Localization theory for triangulated categories (2008, unknown): “ormal properties with a quotient functor. A typical example arises from ﬁnite localization; see Theorem 5.6.1. However, there are examples where Tis compactly generated but 0 = (KerL)c ⊆KerL6= 0; see [25] for details. 5.6. Finite localization. A common type of localization for triangulated categorie”
The telescope conjecture for hereditary rings via Ext-orthogonal pairs (2008, background): “In fact, an analysis of Keller’s argument [18] shows that there are such examples having global dimension 2; see Example 7.”
Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, methodology): “[52, 53 ]) effectively applied them to study Brown representability and the telescope conjecture.”
Noncommutative localisation in algebraic K-theory I (2004, background): “angulated category facts about the functor T. The applications will appear separately. See [26] for K–theoretic consequences, [27] for consequences in L– theory, as well as Krause’s beautiful article [19] which further develops some of our results. In presenting the proofs we tried to keep in mind t”
Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras (2005, background): “For instance they are in bijection with smashing subcategories and are in in the Telescope Conjecture and the classification of thick subcategories of compa jects, see [22,42,44] for details.”
Good tilting modules and recollements of derived module categories (2010, background): “Now, we recall the definition of generalized localizations at complexes of projective modules, which were first discussed in [25] under the terminology “homological localizations”.”