Title: Cohomological quotients and smashing localizations
Author: Henning Krause
Email: henning@maths.leeds.ac.uk
Abstract: 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.