### The negative K-theory of normal surfaces, by Charles A. Weibel

This is a revised version of the original posting dated 22 September, 1999.

We relate the negative K-theory of a normal surface X to a resolution of
singularities. The only two nonzero groups are K_{-2}X, which counts loops in
the exceptional fiber E, and K_{-1}X, which is related to the divisor class
groups of the complete local rings at the singularities of X. Over a perfect
field, X is K_0-regular iff X is is K_{-1}-regular, and this is equivalent to
Pic(E)=Pic(nE) for all nilpotent thickenings nE of E. This property of X is
determined by the cohomology of the canonical line bundle on E.

Our results answer a number of questions raised by Coombes, Srinivas and
Weibel.

The key technical result is that we can resolve the singularities of a
surface as the normalization of a blowup along a local complete
intersection. This apparently fails in higher dimensions, which explains our
restriction to dimension two.

The final version of this article was published in
*Duke Math J.* 108 (2001), 1-35.

Charles A. Weibel <weibel@math.rutgers.edu>