Algebra/Topology seminar – University of Copenhagen

Algebra/Topology seminar

Speaker: Stefan Schwede (Bonn),

Titel: Global algebraic K-theory.

Abstract: K-theory assigns to a symmetric monoidal category a connective spectrum
whose underlying infinite loop space group completes the nerve of
the category, with respect to the E_infty multiplication coming from the
monoidal structure. A prominent example is the category of finitely 
generated projective modules over a ring, under direct sum; this yields
the K-theory spectrum of the ring. Another example are finite sets under 
disjoint unit; by the Barrat-Priddy-Quillen theorem, this yields
the sphere spectrum.

I'll present a construction of a global algebraic K-theory spectrum;
this assigns to a symmetric monoidal category a connective
global stable homotopy type, modeled by a symmetric spectrum.
Such a global homotopy type encodes simultaneous and
compatible equivariant spectra for all finite groups.
For a finite group G, the G-fixed point spectrum models the 
representation K-theory (or Swan K-theory) of G-objects in the given 
category. For finitely generated projective modules over a ring, this
global algebraic K-theory rigidifies all the Swan K-theory spectra
into one object. For finite sets, we obtain the global 
equivariant sphere spectrum.