Universal Extension Classes for GL2
In this note we give a new existence proof for the universal extension classes for GL2 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 SL2 relative to its first Frobenius kernel stabilizes at the E2-page. Consequently, we obtain a new proof that if G is an infinitesimal subgroup scheme of GL2, then the cohomology ring H∙(G,k) of G is a finitely-generated noetherian k-algebra.