Algebra/Topology Seminar

Abstract: Given a compactly generated triangulated category, such as the derived category of a ring (spectrum), one usually tries to study it, at least to a first approximation, by understanding the compact objects. In the case that there is also a compatible symmetric monoidal structure, e.g. the ring is commutative, there is a space analogous to the spectrum of a ring which reflects the behaviour of the compact objects. It's natural to ask how this space reflects the behaviour of the whole compactly generated category, but in non-noetherian situations the answer is usually not very well. I'll explain how one can work with all tensor compatible smashing localizations, rather than just the finite ones, to attempt to fix this deficit, and illustrate the situation with some concrete examples. This is based on joint work with Paul Balmer and Henning Krause.