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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Coherent functors, with application
to torsion in the Picard group


Author: David B. Jaffe
Journal: Trans. Amer. Math. Soc. 349 (1997), 481-527
MSC (1991): Primary 14C22, 18A25, 14K30, 18A40
MathSciNet review: 1351490
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Abstract: Let $A$ be a commutative noetherian ring. We investigate a class of functors from $\langle \langle $commutative $A$-algebras$\rangle \rangle $ to $\langle \langle $sets$\rangle \rangle $, which we call coherent. When such a functor $F$ in fact takes its values in $\langle \langle $abelian groups$\rangle \rangle $, 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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Additional Information

David B. Jaffe
Affiliation: Department of Mathematics and Statistics, University of Nebraska, Lincoln, Nebraska 68588-0323
Email: jaffe@cpthree.unl.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01616-4
PII: S 0002-9947(97)01616-4
Keywords: Coherent functor, representable functor, Picard group
Received by editor(s): July 1, 1994
Received by editor(s) in revised form: September 19, 1995
Additional Notes: Partially supported by the National Science Foundation
Article copyright: © Copyright 1997 American Mathematical Society