Quillen's Devissage in Geometry

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.