Applications of Swan's Bertini theorem to unimodular rows

Abstract

Let R be an affine algebra of dimension d≥4 over a perfect field k of char ≠2 and I be an ideal of R. Then(1)MSd+1(R) is uniquely divisible prime to char k if R is reduced and k is infinite with c.d.(k)≤1.(2)Umd+1(R,I)/Ed+1(R,I) has nice group structure if c.d.2(k)≤2.(3)Umd(R,I)/Ed(R,I) has nice group structure if k is algebraically closed of char k≠2,3 and either (i) k=F‾p or (ii) R is normal.