An Algebraic Proof of Quillen's Resolution Theorem for K1

In his 1973 paper [4] Quillen proved a resolution theorem for the K-Theory of an exact category; his proof was homotopic in nature. By using the main result of Nenashev's paper [3], we are able to give an algebraic proof of Quillen's Resolution Theorem for K1 of an exact category. We view this as an advance towards the goal of giving an essentially algebraic subject an algebraic foundation.

Essential Dimensions of Algebraic Groups and a Resolution Theorem for G-Varieties

Let G be an algebraic group and let X be a generically free G-variety. We show that X can be transformed, by a sequence of blowups with smooth G-equivariant centers, into a G-variety Xʹ with the following property: the stabilizer of every point of Xʹ is isomorphic to a semidirect product U × A of a unipotent group U and a diagonalizable group A.As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.