Universal Extension Classes for GL 2

Abstract

In this note we give a new existence proof for the universal extension classes for GL 2 previously constructed by Friedlander and Suslin via the theory of strict polynomial functors. The key tool in our approach is a calculation of Parker showing that, for suitable choices of coefficient modules, the Lyndon-Hochschild-Serre spectral sequence for SL 2 relative to its first Frobenius kernel stabilizes at the E 2 -page. Consequently, we obtain a new proof that if G is an infinitesimal subgroup scheme of GL 2 , then the cohomology ring H • (G, k) of G is a finitely-generated noetherian k-algebra.