One of Rick's most influential ideas concern sheaf-theoretic foundations in and around the Lichtenbaum-Quillen conjecture, formulating it as a comparison between "Zariski" and "étale" K-theories (cf, papers on simplicial presheaves and his address in the 1993 CMS meeting). I want to report on an application of the same ideas to give a streamlined proof the following conjecture of Murthy (a theorem, in increasing generality, due to Murthy, Levine, Srinivas, Krishna-Srinivas, Krishna): for an affine variety over an algebraically closed field of dimension d, a vector bundle of rank d splits off a rank 1 summand if and only if top euler class vanishes. The proof relies on a comparison between "Zariski" motivic cohomology (as defined by the speaker and Morrow) and "syntomic" motivic cohomology and pro-excision techniques.

This is joint work with Matthew Morrow.