# 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.