A Remark on the Generalized Smashing Conjecture
نویسنده
چکیده
Using one of Wodzicki's examples of H-unital algebras [14] we exhibit a ring whose derived category contains a smashing subcategory which is not generated by small objects. This disproves the generalization to arbitrary triangulated categories of a conjecture due 1. Statement of the conjecture We refer to [7] for a nicely written analysis of the following setup: Let S be a triangulated category [13] admitting arbitrary (set-indexed) coproducts. An object X ∈ S is small if the functor Hom (X, ?) commutes with arbitrary coproducts. We denote the full subcategory on the small objects of S by S b. We suppose that S b is equivalent to a small category. A full subcategory of S is localizing if it is a triangulated subcategory in the sense of Verdier which is closed under forming coproducts with respect to S. We
منابع مشابه
Fakultät für Elektrotechnik , Informatik und Mathematik Subcategories of Triangulated Categories and the Smashing Conjecture
In this thesis the global structure of three classes of algebraic triangulated categories is investigated by describing their thick, localizing and smashing subcategories and by analyzing the Smashing Conjecture. We show that the Smashing Conjecture for the stable module category of a self-injective artin algebra A is equivalent to the statement that a class of model categories associated with ...
متن کاملSmashing Subcategories and the Telescope Conjecture – an Algebraic Approach
We prove a modified version of Ravenel’s telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presen...
متن کاملSmashing Subcategories and the Telescope
We give a new characterization of smashing subcategories in a compactly generated triangulated category and prove a modiied version of Ravenel's telescope conjecture in this setting. Our results apply in particular to the stable homotopy category. Our approach, however, is purely algebraic; it is based on an analysis of pure injective objects in a compactly generated triangulated category.
متن کاملThe Seiberg–Witten equations and the Weinstein conjecture
Let M denote a compact, orientable 3–manifold and let a denote a smooth 1–form on M such that a^ da is nowhere zero. Such a 1–form is called a contact form. The associated Reeb vector field is the section, v , of TM that generates the kernel of da and pairs with a to give 1. The generalized Weinstein conjecture in dimension three asserts that v has at least one closed integral curve (see Weinst...
متن کامل