Coherent functors, with application to torsion in the Picard group

Abstract:

Let $A$ be a commutative noetherian ring. We investigate a class of functors from $\lBrack$commutative $A$-algebras$\rBrack$ to $\lBrack$sets$\rBrack$, which we call coherent. When such a functor $F$ in fact takes its values in $\lBrack$abelian groups$\rBrack$, we show that there are only finitely many prime numbers $p$ such that ${}_pF(A)$ is infinite, and that none of these primes are invertible in $A$. This (and related statements) yield information about torsion in $\operatorname {Pic}(A)$. For example, if $A$ is of finite type over $\mathbb {Z}$, we prove that the torsion in $\operatorname {Pic}(A)$ is supported at a finite set of primes, and if ${}_p\operatorname {Pic}(A)$ is infinite, then the prime $p$ is not invertible in $A$. These results use the (already known) fact that if such an $A$ is normal, then $\operatorname {Pic}(A)$ is finitely generated. We obtain a parallel result for a reduced scheme $X$ of finite type over $\mathbb {Z}$. We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field.