# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Coherent functors, with application to torsion in the Picard groupHTML articles powered by AMS MathViewer

by David B. Jaffe
Trans. Amer. Math. Soc. 349 (1997), 481-527 Request permission

## 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.
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