  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.
