Algebra/Topology Seminar by Irakli Patchkoria

Abstract: In 80s Bousfield gave an algebraic classification of isomorphism classes of K-local spectra at an odd prime, where K is the (complex) K-theory. This used Adams operations and the Adams spectral sequence. Using a more general approach to Adams spectral sequences and the ideas from the theory of perverse sheaves, Franke in 90s constructed certain functors and claimed that they give an algebraic description of the whole homotopy category of K-local spectra at an odd prime. The proof of this claim contains a gap though. We will explain Franke's ideas and how to fix this gap in certain cases. It will in particular follow that Franke's claim is correct for primes greater than 3. We will also show how Franke's methods can be used to provide algebraic models for the derived homotopy categories of certain K-theory spectra, truncated Brow-Peterson spectra and Johnson-Wilson spectra.