Charles Weibel: K-theory of line bundles and smooth varieties

The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields.

We give a K-theoretic criterion for a quasi-projective variety to be smooth, generalizing the proof of Vorst's conjecture for affine varieties. If L is a line bundle corresponding to an ample sheaf on X, it suffices that Kq(L) = Kq(X) for all q at most d+1, d the dimension of X. Our proof is in characteristic zero, using sheaf cohomology.