Cancellation over affine varieties

A. A. Suslin

Zapiski Nauchnykh Seminarov POMI, Modules and algebraic groups, Tome 114 (1982), pp. 187-195

Résumé

It is proved that if $X$ is a smooth affine curve over a field $F$ of characteristic $\ne\ell$, then the group $SK_1(X)/\ell SK_1(X)$ is isomorphic to a subgroup of the йtale cohomology group $H^3_{et}(X,\mu_e^{\otimes2})$ and if $F$ is algebraically closed, then $SK_1(X)$ is a uniquely divisible group. The following cancellation theorem is obtained from results about $SK_1$ for curves: If $X$ is a normal affine variety of dimension $n$ over a field $F$, and if $\operatorname{char}F>n$ and $c.d._\ell(F)\leqslant1$ for any prime $\ell\leqslant n$ then any stably trivial vector bundle of rank $n$ over $X$ is trivial.