Abstract
Throughout this chapter, we denote by R a complete discrete valuation ring with maximal ideal \(\mathfrak{m}\) and residue field k. For every integer n⩾0, we set \(R_{n} = R/\mathfrak{m}^{n+1}\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In (ÉGA II) the terminology fibré vectoriel (vector bundle) is used instead, but we prefer to reserve this name for the case where \(\mathcal{E}\) is locally free, as is common in the literature.
Bibliography
M. Artin (1969), Algebraic approximation of structures over complete local rings. Inst. Hautes Études Sci. Publ. Math. 36, 23–58
V.G. Berkovich (1993), Étale cohomology for non-Archimedean analytic spaces. Publ. Math. Inst. Hautes Études Sci. 78, 5–161
S. Bosch, W. Lütkebohmert, M. Raynaud (1995), Formal and rigid geometry. III. The relative maximum principle. Math. Ann. 302(1), 1–29
J. Denef, F. Loeser (1999), Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 135(1), 201–232
A. Ducros (2009), Les espaces de Berkovich sont excellents. Ann. Inst. Fourier (Grenoble) 59(4), 1443–1552
A. Ducros (2018), Families of Berkovich spaces. arXiv:1107.4259v5
D. Eisenbud (1995), Commutative Algebra with a View Towards Algebraic Geometry. Graduate Texts in Mathematics, vol. 150 (Springer, Berlin)
R. Elkik (1973/1974), Solutions d’équations à coefficients dans un anneau hensélien. Ann. Sci. École Norm. Sup. 6, 553–603
M.J. Greenberg (1966), Rational points in Henselian discrete valuation rings. Inst. Hautes Études Sci. Publ. Math. 31, 59–64
A. Grothendieck, J. Dieudonné (1961a), Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Publ. Math. Inst. Hautes Études Sci. 8, 5–222. Quoted as (ÉGA II)
S. Ishii, A.J. Reguera (2013), Singularities with the highest Mather minimal log discrepancy. Math. Z. 275(3–4), 1255–1274
M. Kontsevich (1995), Motivic integration. Lecture at Orsay. http://www.lama.univ-savoie.fr/~raibaut/Kontsevich-MotIntNotes.pdf
E. Looijenga (2002), Motivic measures. Astérisque, 276, 267–297. Séminaire Bourbaki, vols. 1999/2000
J.S. Milne (1980), Étale Cohomology. Mathematical Notes, vol. 33 (Princeton University Press, Princeton)
D. Popescu (1986), General Néron desingularization and approximation. Nagoya Math. J. 104, 85–115
D. Popescu (2000), Artin Approximation. Handbook of Algebra, vol. 2 (North-Holland, Amsterdam), pp. 321–356
J. Sebag (2004a), Intégration motivique sur les schémas formels. Bull. Soc. Math. Fr. 132(1), 1–54
M. Spivakovsky (1999), A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms. J. Am. Math. Soc. 12(2), 381–444
R.G. Swan (1998), Néron-Popescu desingularization, in Algebra and Geometry (Taipei, 1995). Lectures on Algebraic Geometry, vol. 2 (International Press, Cambridge), pp. 135–192
L.A. Tarrío, A.J. López, M.P. Rodríguez (2007), Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes. Commun. Algebra 35(4), 1341–1367
B. Teissier (1995), Résultats récents sur l’approximation des morphismes en algèbre commutative (d’après André, Artin, Popescu et Spivakovsky). Astérisque, 227, pp. Exp. No. 784, 4, 259–282. Séminaire Bourbaki, vols. 1993/1994
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Science+Business Media, LLC, part of Springer Nature
About this chapter
Cite this chapter
Chambert-Loir, A., Nicaise, J., Sebag, J. (2018). Structure Theorems for Greenberg Schemes. In: Motivic Integration. Progress in Mathematics, vol 325. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-7887-8_5
Download citation
DOI: https://doi.org/10.1007/978-1-4939-7887-8_5
Published:
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-1-4939-7885-4
Online ISBN: 978-1-4939-7887-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)