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.
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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
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
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
Bloch S.: Algebraic K-theory and crystalline cohomology. Inst. Hautes Études Sci. Publ. Math. 47, 187–268 (1978)
Borger, J.: Basic geometry of Witt vectors. I: the affine case. Algebra Number Theory. arXiv:0801. 1691v5
Borger, J.: Λ-rings and the field with one element. arXiv:0906.3146v1
Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin (1990)
Buium A.: Intersections in jet spaces and a conjecture of S. Lang. Ann. Math. (2) 136(3), 557–567 (1992)
Buium A.: Geometry of p-jets. Duke Math. J. 82(2), 349–367 (1996)
Buium, A.: Arithmetic differential equations. In: Mathematical Surveys and Monographs, vol. 118. American Mathematical Society, Providence (2005)
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)
Buium A., Simanca S.R.: Arithmetic Laplacians. Adv. Math. 220(1), 246–277 (2009)
Conrad, B., Lieblich, M., Olsson, M.: Nagata compactification for algebraic spaces. arXiv:0910.5008
Drinfeld V.G.: Coverings of p-adic symmetric domains. Funkcional. Anal. i Priložen. 10(2), 29–40 (1976)
Eisenbud, D.: Commutative algebra. With a view toward algebraic geometry. In: Graduate Texts in Mathematics, vol. 150. Springer-Verlag, New York (1995).
Ekedahl T.: On the multiplicative properties of the de Rham-Witt complex. I. Ark. Mat. 22(2), 185–239 (1984)
Ekedahl T.: On the multiplicative properties of the de Rham-Witt complex. II. Ark. Mat. 23(1), 53–102 (1985)
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)
Giraud, J.: Cohomologie non abélienne. Springer-Verlag, Berlin (1971). Die Grundlehren der mathematischen Wissenschaften, Band 179
Greenberg M.J.: Schemata over local rings. Ann. Math. (2) 73, 624–648 (1961)
Greenberg M.J.: Schemata over local rings. II. Ann. Math. (2) 78, 256–266 (1963)
Grothendieck A.: La théorie des classes de Chern. Bull. Soc. Math. France 86, 137–154 (1958)
Grothendieck A.: Éléments de géométrie algébrique. I. Le langage des schémas. Inst. Hautes Études Sci. Publ. Math. 4, 228 (1960)
Grothendieck A.: Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. 8, 222 (1961)
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)
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)
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)
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)
Hazewinkel, M.: Formal groups and applications. In: Pure and Applied Mathematics, vol. 78. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978)
Illusie L.: Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. École Norm. Sup. (4) 12(4), 501–661 (1979)
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)
Joyal A.: δ-anneaux et vecteurs de Witt. C. R. Math. Rep. Acad. Sci. Canada 7(3), 177–182 (1985)
Knutson, D.: Algebraic spaces. In: Lecture Notes in Mathematics, vol. 203. Springer-Verlag, Berlin (1971)
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
Langer A., Zink T.: De Rham-Witt cohomology for a proper and smooth morphism. J. Inst. Math. Jussieu 3(2), 231–314 (2004)
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)
Raynaud M., Gruson L.: Critères de platitude et de projectivité. Techniques de “platification” d’un module. Invent. Math. 13, 1–89 (1971)
Rydh, D.: Noetherian approximation of algebraic spaces and stacks. arXiv:0904.0227v1
Toën B., Vaquié M.: Algébrisation des variétés analytiques complexes et catégories dérivées. Math. Ann. 342(4), 789–831 (2008)
Toën B., Vezzosi G.: Homotopical algebraic geometry. II: geometric stacks and applications. Mem. Am. Math. Soc. 193(902), x+224 (2008)
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This work was partly supported by Discovery Project DP0773301, a grant from the Australian Research Council.