Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-31T23:37:36.689Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LARGE STRUCTURES OF GROTHENDIECK FOUNDED ON FINITE-ORDER ARITHMETIC

Published online by Cambridge University Press:  02 August 2019

COLIN MCLARTY
COLIN MCLARTY*
Affiliation:
Departments of Philosophy and Mathematics, Case Western Reserve University
*
*DEPARTMENTS OF PHILOSOPHY AND MATHEMATICS CASE WESTERN RESERVE UNIVERSITY 10900 EUCLID AVENUE CLEVELAND, OH 44106, USA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck’s SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong.

Keywords

03E3003E5514A9918D99Grothendieck universe.
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 2 , June 2020 , pp. 296 - 325
Copyright
Copyright © Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

The abbreviation SGA 1 refers to Grothendieck (1971) and SGA 4 is Artin et al. (1972). EGA I and IV are respectively Grothendieck & Dieudonné (1971) and Grothendieck & Dieudonné (1964).Google Scholar
Altman, A. & Kleiman, S. (1970). An Introduction to Grothendieck Duality Theory. Springer Lecture Notes in Mathematics, Vol. 146. New York: Springer-Verlag.CrossRefGoogle Scholar
Artin, M. (1962). Grothendieck Topologies: Notes on a Seminar. Cambridge, MA: Harvard University, Department of Mathematics.Google Scholar
Artin, M., Grothendieck, A., & Verdier, J.-L. (1972). Théorie des Topos et Cohomologie Etale des Schémas. Séminaire de géométrie algébrique du Bois-Marie, Vol. 4. New York: Springer-Verlag. Three volumes, cited as SGA 4.Google Scholar
Baer, R. (1940). Abelian groups that are direct summands of every containing abelian group. Bulletin of the American Mathematical Society, 46, 800–806.CrossRefGoogle Scholar
Barr, M. (1974). Toposes without points. Journal of Pure and Applied Algebra, 5, 265–280.CrossRefGoogle Scholar
Blass, A. (1979). Injectivity, projectivity, and the axiom of choice. Transactions of the American Mathematical Society, 255, 31–59.CrossRefGoogle Scholar
Conrad, B. (2000). Grothendieck Duality and Base Charge. Lecture Notes in Mathematics, Vol. 1750. New York: Springer-Verlag.CrossRefGoogle Scholar
Deligne, P. (1977). Cohomologie étale: les points de départ. In Deligne, P., editor. Cohomologie Étale. New York: Springer-Verlag, pp. 4–75.CrossRefGoogle Scholar
Deligne, P. (1998). Quelques idées maîtresses de l’œuvre de A. Grothendieck. Matériaux pour l’Histoire des Mathématiques au XXe Siècle (Nice, 1996). Paris: Société mathématique de France, pp. 11–19.Google Scholar
Demazure, M. & Grothendieck, A. (1970). Schémas en groupes. Séminaire de géométrie algébrique du Bois-Marie, Vol. 3. New York: Springer-Verlag.Google Scholar
Eckmann, B. & Schopf, A. (1953). Über injektive Moduln. Archiv der Mathematik, 4 (2), 75–78.CrossRefGoogle Scholar
Eisenbud, D. (1995). Commutative Algebra. New York: Springer-Verlag.CrossRefGoogle Scholar
Grothendieck, A. (1957a). Sur quelques points d’algèbre homologique. Tôhoku Mathematical Journal, 9, 119–221.Google Scholar
Grothendieck, A. (1957b). Théorèmes de dualité pour les faisceaux algébriques cohérents, exposé 149. Séminaire Bourbaki. Secrétariat mathématique. Paris: Université Paris, pp. 169–193.Google Scholar
Grothendieck, A. (1958). The cohomology theory of abstract algebraic varieties. Proceedings of the International Congress of Mathematicians, 1958. Cambridge: Cambridge University Press, pp. 103–118.Google Scholar
Grothendieck, A. (1971). Revêtements Étales et Groupe Fondamental. Séminaire de géométrie algébrique du Bois-Marie, Vol. 1. New York: Springer-Verlag. Cited as SGA 1.CrossRefGoogle Scholar
Grothendieck, A. & Dieudonné, J. (1964). Éléments de Géométrie Algébrique IV: Étude locale des schémas et des morphismes de schémas, Première partie. Publications Mathématiques, Vol. 20. Paris: Institut des Hautes Études Scientifiques.Google Scholar
Grothendieck, A. & Dieudonné, J. (1971). Éléments de Géométrie Algébrique I. New York: Springer-Verlag.Google Scholar
Hartshorne, R. (1966). Residues and Duality, Lecture Notes of a Seminar on the Work of A. Grothendieck given at Harvard 1963–64. Lecture Notes in Mathematics, Vol. 20. New York: Springer-Verlag.Google Scholar
Hartshorne, R. (1977). Algebraic Geometry. New York: Springer-Verlag.CrossRefGoogle Scholar
Johnstone, P. (1977). Topos Theory. London: Academic Press.Google Scholar
Johnstone, P. (2002). Sketches of an Elephant: A Topos Theory Compendium. Oxford: Oxford University Press. To be finished as three volumes.Google Scholar
Kunen, K. (1983). Set Theory: An Introduction to Independence Proofs. Amsterdam: North-Holland.Google Scholar
Lawvere, F. W. (1965). An elementary theory of the category of sets. Lecture notes of the Department of Mathematics, University of Chicago. Reprint with commentary by the author and Colin McLarty in: Reprints in Theory and Applications of Categories, No. 11 (2005) pp. 1–35, on-line at http://138.73.27.39/tac/reprints/articles/11/tr11abs.html.Google Scholar
Lipman, J. & Hashimoto, M. (2009). Foundations of Grothendieck Duality for Diagrams of Schemes. New York: Springer-Verlag.CrossRefGoogle Scholar
Mac Lane, S. (1998). Categories for the Working Mathematician (second edition). New York: Springer-Verlag.Google Scholar
Mac Lane, S. & Moerdijk, I. (1992). Sheaves in Geometry and Logic. New York: Springer-Verlag.Google Scholar
Mathias, A. R. D. (2001). The strength of Mac Lane set theory. Annals of Pure and Applied Logic, 110, 107–234.CrossRefGoogle Scholar
McLarty, C. (2006). Two aspects of constructivism in category theory. Philosophia Scientiae, 6, 95–114. Cahier Spécial.CrossRefGoogle Scholar
Milne, J. (2016). Étale Cohomology. Princeton, NJ: Princeton University Press.Google Scholar
Neeman, A. (2010). Derived categories and Grothendieck duality. In Holm, T., Jørgensen, P., and Rouquier, R., editors. Triangulated Categories. Cambridge: Cambridge University Press, pp. 290–350.CrossRefGoogle Scholar
Shulman, M. (2012). Exact completions and small sheaves. Preprint on the mathematics arXiv, arXiv:1203.4318v2.Google Scholar
Takeuti, G. (1978). A conservative extension of Peano Arithmetic. Two Applications of Logic to Mathematics. Princeton, NJ: Princeton University Press, pp. 77–135.Google Scholar
Takeuti, G. (1987). Proof Theory (second edition). New York: Elsevier Science Ltd.Google Scholar
Tamme, G. (1994). Introduction to Etale Cohomology. New York: Springer-Verlag.CrossRefGoogle Scholar
van Osdol, D. (1975). Homological algebra in topoi. Proceedings of the American Mathematical Society, 50, 52–54.CrossRefGoogle Scholar
Weibel, C. (1994). An Introduction to Homological Algebra. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Wiles, A. (1995). Modular elliptic curves and Fermat’s Last Theorem. Annals of Mathematics, 141, 443–551.CrossRefGoogle Scholar