Cohomological quotients and smashing localizations
Krause, Henning
Krause
Henning
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 Verdierâ€™s construction. Slightly simplifying this concept,
the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localiza-
tions. 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
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1191-1246
1191-1246
Project MUSE
2005