COHERENT FUNCTORS, WITH APPLICATION TO TORSION IN THE PICARD GROUP

Let A be a commutative noetherian ring. Vile investigate a class of fu nctors from [[commutative A-algebras]] to [[sets]], which we call cohe rent. When such a functor F in fact takes its values in [[abelian grou ps]], we show that there are only finitely many prime numbers p such t hat F-p(A) is infinite, and that none of these primes are invertible i n A. This (and related statements) yield information about torsion in Pic(A). For example, if A is of finite type over Z, we prove that the torsion in Pic(A) is supported at a finite set of primes, and if (p)Pi c(A) is infinite, then the prime p is not invertible in A. These resul ts use the (already known) fact that if such an A is normal, then Pic( A) is finitely generated. We obtain a parallel result fora reduced sch eme X of finite type over Z. We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field.