Quillen's Devissage in Geometry

Inna Zakharevich,
Tue, Jun 11, 2019
PIMS, University of British Columbia
Workshop on Arithmetic Topology

In this talk we discuss a new perspective on Quillen's devissage theorem. Originally, Quillen proved devissage for algebraic $ K $-theory of abelian categories. The theorem showed that given a full abelian subcategory $ \mathcal{A} $ of an abelian category $ \mathcal{B} $, $ K(\mathcal{A})\simeq K(\mathcal{B}) $ if every object of $ \mathcal{B} $ has a finite filtration with quotients lying in $ \mathcal{A} $. This allows us, for example, to relate the $ K $-theory of torsion $ \mathbf{Z} $-modules to the $ K $-theories of $ \mathbf{F}_p $-modules for all $ p $. Generalizations of this theorem to more general contexts for $ K $-theory, such as Walhdausen categories, have been notoriously difficult; although some such theorems exist they are generally much more complicated to state and prove than Quillen's original. In this talk we show how to translate Quillen's algebraic approach to a geometric context. This translation allows us to construct a devissage theorem in geometry, and prove it using Quillen's original insights.