| Let X be a curve over a finite field k of characteristic p. Answering a 25-year old question of Krusemeyer, we prove that SK1(X) is 0 if X is affine. We use this to prove that if X is a reduced projective curve with irreducible components Xi, then SK1(X) ≃⊕ iF*i , where Fi is the algebraic closure of k in the function field of Xi. If X is nonreduced, then SK1( X) may also contain a p-group of bounded exponent as a direct summand. We compute this p-group in several examples. A corollary to the affine result states that if R is a commutative 1-dimensional algebra over a finite field, then every matrix in SL n(R) is a product of elementary matrices, at least for n ≠ 2.;The main method used in the proofs is the exploitation of localization sequences and Mayer-Vietoris exact sequences in excision situations. We relate SK1 of an affine or reduced projective surface to that of its desingularization, pending excision. Finally, we relate the results to the search for an explicit description of the maximal abelian extension of a 2-dimensional global field. |
