Skip to Main Content

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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Coherent functors, with application to torsion in the Picard group
HTML articles powered by AMS MathViewer

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


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.
  • M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58. MR 268188, DOI 10.1007/BF02684596
  • —, Letter to Grothendieck, Nov. 5, 1968.
  • M. Artin, Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 21–71. MR 0260746
  • M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
  • Maurice Auslander, Coherent functors, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 189–231. MR 0212070
  • Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
  • —, Introduction to some methods of algebraic $K$-theory, Conf. Board Math. Sci. Regional Conf. Ser. Math., no. 20, Amer. Math. Soc., Providence, RI, 1974.
  • David Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. With a section by G. M. Bergman. MR 0209285, DOI 10.1515/9781400882069
  • J. E. Bertin, GĂ©nĂ©ralitĂ©s sur les prĂ©schĂ©mas en groupes, SchĂ©mas en Groupes (SĂ©m. GĂ©omĂ©trie AlgĂ©brique, Inst. Hautes Études Sci., 1963/64) Inst. Hautes Études Sci., Paris, 1965, pp. Fasc. 2a, ExposĂ© 6b, 112. MR 0234961
  • N. Bourbaki, Algebra. II. Chapters 4–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1990. Translated from the French by P. M. Cohn and J. Howie. MR 1080964
  • Luther Claborn, Every abelian group is a class group, Pacific J. Math. 18 (1966), 219–222. MR 195889, DOI 10.2140/pjm.1966.18.219
  • Henri Cohen, Un faisceau qui ne peut pas ĂȘtre dĂ©tordu universellement, C. R. Acad. Sci. Paris SĂ©r. A-B 272 (1971), A799–A802 (French). MR 288124
  • —, DĂ©torsion universelle de faisceaux cohĂ©rents, thesis (Docteur $3^\circ$ Cycle), Univ. Paris-XI (Orsay), 1972.
  • Robert M. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York-Heidelberg, 1973. MR 0382254, DOI 10.1007/978-3-642-88405-4
  • L. Fuchs, Abelian groups, Publishing House of the Hungarian Academy of Sciences, Budapest, 1958. MR 0106942
  • Peter Scherk, Bemerkungen zu einer Note von Besicovitch, J. London Math. Soc. 14 (1939), 185–192 (German). MR 29, DOI 10.1112/jlms/s1-14.3.185
  • A. Grothendieck, ÉlĂ©ments de gĂ©omĂ©trie algĂ©brique. III. Étude cohomologique des faisceaux cohĂ©rents. II, Inst. Hautes Études Sci. Publ. Math. 17 (1963), 91 (French). MR 163911
  • —, ÉlĂ©ments de gĂ©omĂ©trie algĂ©brique. IV (part three), Inst. Hautes Études Sci. Publ. Math. 28 (1966).
  • —, ÉlĂ©ments de GĂ©omĂ©trie AlgĂ©brique. I, 2nd ed., Springer-Verlag, New York, 1971. Zbl. 203, 233.
  • R. Guralnick, D. B. Jaffe, W. Raskind and R. Wiegand, On the Picard group: torsion and the kernel induced by a faithfully flat map, J. Algebra (to appear).
  • Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
  • ThĂ©orie des intersections et thĂ©orĂšme de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois-Marie 1966–1967 (SGA 6); DirigĂ© par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
  • Neal Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, Graduate Texts in Mathematics, Vol. 58, Springer-Verlag, New York-Heidelberg, 1977. MR 0466081, DOI 10.1007/978-1-4684-0047-2
  • Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
  • Serge Lang, Algebra, 2nd ed., Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. MR 783636
  • Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
  • Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 538169
  • Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237, DOI 10.1007/978-1-4757-5673-9
Similar Articles
Additional Information
  • David B. Jaffe
  • Affiliation: Department of Mathematics and Statistics, University of Nebraska, Lincoln, Nebraska 68588-0323
  • Email:
  • 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
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 481-527
  • MSC (1991): Primary 14C22, 18A25, 14K30, 18A40
  • DOI:
  • MathSciNet review: 1351490