Skip to main content
SpringerLink
Log in

The basic geometry of Witt vectors. II: Spaces

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

This is an account of the algebraic geometry of Witt vectors and arithmetic jet spaces. The usual, “p-typical” Witt vectors of p-adic schemes of finite type are already reasonably well understood. The main point here is to generalize this theory in two ways. We allow not just p-typical Witt vectors but those taken with respect to any set of primes in any ring of integers in any global field, for example. This includes the “big” Witt vectors. We also allow not just p-adic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field. We give similar generalizations of Buium’s formal arithmetic jet functor, which is dual to the Witt functor. We also give concrete geometric descriptions of Witt spaces and arithmetic jet spaces and investigate whether a number of standard geometric properties are preserved by these functors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. Springer-Verlag, Berlin, 1972. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, vol. 269

  2. Théorie des topos et cohomologie étale des schémas. Tome 2. Springer-Verlag, Berlin, 1972. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, vol. 270

  3. Cohomologie l-adique et fonctions L. Lecture Notes in Mathematics, vol. 589. Springer-Verlag, Berlin (1977). Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5), Edité par Luc Illusie

  4. Bloch S.: Algebraic K-theory and crystalline cohomology. Inst. Hautes Études Sci. Publ. Math. 47, 187–268 (1978)

    Google Scholar 

  5. Borger, J.: Basic geometry of Witt vectors. I: the affine case. Algebra Number Theory. arXiv:0801. 1691v5

  6. Borger, J.: Λ-rings and the field with one element. arXiv:0906.3146v1

  7. Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin (1990)

  8. Buium A.: Intersections in jet spaces and a conjecture of S. Lang. Ann. Math. (2) 136(3), 557–567 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Buium A.: Geometry of p-jets. Duke Math. J. 82(2), 349–367 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Buium, A.: Arithmetic differential equations. In: Mathematical Surveys and Monographs, vol. 118. American Mathematical Society, Providence (2005)

  11. Buium A., Poonen B.: Independence of points on elliptic curves arising from special points on modular and Shimura curves. II: local results. Compos. Math. 145(3), 566–602 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Buium A., Simanca S.R.: Arithmetic Laplacians. Adv. Math. 220(1), 246–277 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Conrad, B., Lieblich, M., Olsson, M.: Nagata compactification for algebraic spaces. arXiv:0910.5008

  14. Drinfeld V.G.: Coverings of p-adic symmetric domains. Funkcional. Anal. i Priložen. 10(2), 29–40 (1976)

    MathSciNet  Google Scholar 

  15. Eisenbud, D.: Commutative algebra. With a view toward algebraic geometry. In: Graduate Texts in Mathematics, vol. 150. Springer-Verlag, New York (1995).

  16. Ekedahl T.: On the multiplicative properties of the de Rham-Witt complex. I. Ark. Mat. 22(2), 185–239 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ekedahl T.: On the multiplicative properties of the de Rham-Witt complex. II. Ark. Mat. 23(1), 53–102 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fontaine, J.-M.: Le corps des périodes p-adiques. Astérisque 223, 59–111 (1994). With an appendix by Pierre Colmez, Périodes p-adiques (Bures-sur-Yvette, 1988)

  19. Giraud, J.: Cohomologie non abélienne. Springer-Verlag, Berlin (1971). Die Grundlehren der mathematischen Wissenschaften, Band 179

  20. Greenberg M.J.: Schemata over local rings. Ann. Math. (2) 73, 624–648 (1961)

    Article  MATH  Google Scholar 

  21. Greenberg M.J.: Schemata over local rings. II. Ann. Math. (2) 78, 256–266 (1963)

    Article  MATH  Google Scholar 

  22. Grothendieck A.: La théorie des classes de Chern. Bull. Soc. Math. France 86, 137–154 (1958)

    MathSciNet  MATH  Google Scholar 

  23. Grothendieck A.: Éléments de géométrie algébrique. I. Le langage des schémas. Inst. Hautes Études Sci. Publ. Math. 4, 228 (1960)

    Article  Google Scholar 

  24. Grothendieck A.: Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. 8, 222 (1961)

    Google Scholar 

  25. Grothendieck A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I. Inst. Hautes Études Sci. Publ. Math. 20, 259 (1964)

    MathSciNet  Google Scholar 

  26. Grothendieck A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math. 24, 231 (1965)

    MathSciNet  Google Scholar 

  27. Grothendieck A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math. 28, 255 (1966)

    Article  MathSciNet  Google Scholar 

  28. Grothendieck A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Inst. Hautes Études Sci. Publ. Math. 32, 361 (1967)

    MathSciNet  MATH  Google Scholar 

  29. Hazewinkel, M.: Formal groups and applications. In: Pure and Applied Mathematics, vol. 78. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978)

  30. Illusie L.: Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. École Norm. Sup. (4) 12(4), 501–661 (1979)

    MathSciNet  MATH  Google Scholar 

  31. Illusie, L.: Finiteness, duality, and K ünneth theorems in the cohomology of the de Rham-Witt complex. In: Algebraic Geometry (Tokyo/Kyoto, 1982). Lecture Notes in Mathematics, vol. 1016, pp. 20–72. Springer, Berlin (1983)

  32. Joyal A.: δ-anneaux et vecteurs de Witt. C. R. Math. Rep. Acad. Sci. Canada 7(3), 177–182 (1985)

    MathSciNet  MATH  Google Scholar 

  33. Knutson, D.: Algebraic spaces. In: Lecture Notes in Mathematics, vol. 203. Springer-Verlag, Berlin (1971)

  34. Kunz, E.: Introduction to Commutative Algebra and Algebraic Geometry. Birkhäuser, Boston (1985). Translated from the German by Michael Ackerman, With a preface by David Mumford

  35. Langer A., Zink T.: De Rham-Witt cohomology for a proper and smooth morphism. J. Inst. Math. Jussieu 3(2), 231–314 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lubkin S.: Generalization of p-adic cohomology: bounded Witt vectors. A canonical lifting of a variety in characteristic p ≠  0 back to characteristic zero. Compositio Math. 34(3), 225–277 (1977)

    MathSciNet  MATH  Google Scholar 

  37. Raynaud M., Gruson L.: Critères de platitude et de projectivité. Techniques de “platification” d’un module. Invent. Math. 13, 1–89 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rydh, D.: Noetherian approximation of algebraic spaces and stacks. arXiv:0904.0227v1

  39. Toën B., Vaquié M.: Algébrisation des variétés analytiques complexes et catégories dérivées. Math. Ann. 342(4), 789–831 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Toën B., Vezzosi G.: Homotopical algebraic geometry. II: geometric stacks and applications. Mem. Am. Math. Soc. 193(902), x+224 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Australian National University, Canberra, Australia

    James Borger

Authors
  1. James Borger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James Borger.

Additional information

This work was partly supported by Discovery Project DP0773301, a grant from the Australian Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borger, J. The basic geometry of Witt vectors. II: Spaces. Math. Ann. 351, 877–933 (2011). https://doi.org/10.1007/s00208-010-0608-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-010-0608-1

Mathematics Subject Classification (2010)

Search

Navigation