CIRM - Videos & books Library - Commutative algebra for Artin approximation

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Auteurs : Hauser, Herwig (Auteur de la conférence)
CIRM (Editeur )

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approximation classical Artin approximation Néron desingularization commutative rings power series $m$-adic topology solution varieties convergent power series Banach space inductive limit topology Weierstrass division theorem

Résumé : In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach spaces), and give a conceptual proof of the Weierstrass division theorem.
Lecture two covers smooth, unramified and étale morphisms between noetherian rings. The relation of these notions with the concepts of submersion, immersion and diffeomorphism from differential geometry is given.
In the third lecture, we investigate ring extensions between the three power series rings and describe the respective flatness properties. This allows us to prove approximation in the linear case.
The last lecture is devoted to the geometry of solution sets in power series spaces. We construct in the case of one $x$-variable an isomorphism of an $m$-adic neighborhood of a solution with the cartesian product of a (singular) scheme of finite type with an (infinite dimensional) smooth space, thus extending the factorization theorem of Grinberg-Kazhdan-Drinfeld.
CIRM - Chaire Jean-Morlet 2015 - Aix-Marseille Université

Codes MSC :
13J05 - Power series rings [See also 13F25]

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2015-01-26_Hauser_part1.mp4

Informations sur la Rencontre

Nom de la Rencontre : Jean-Morlet Chair - Doctoral school : Introduction to Artin approximation and the geometry of power series
Organisateurs de la Rencontre : Hauser, Herwig ; Rond, Guillaume
Dates : 26/01/15 - 30/01/15
Année de la rencontre : 2015
URL de la Rencontre : https://www.chairejeanmorlet.com/1254.html

Données de citation

DOI : 10.24350/CIRM.V.18681103
Citer cette vidéo: Hauser, Herwig (2015). Commutative algebra for Artin approximation - Part 1. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18681103
URI : http://dx.doi.org/10.24350/CIRM.V.18681103

Voir Aussi

[Multi angle] Commutative algebra for Artin approximation - Part 3 / Auteur de la conférence Hauser, Herwig.
[Multi angle] Commutative algebra for Artin approximation - Part 2 / Auteur de la conférence Hauser, Herwig.

Bibliographie

[1] Atiyah, M.F., & Macdonald, I.G. (1969). Introduction to commutative algebra. Reading (Mass.) ; London ; Don Mills (Ont.): Addison-Wesley - https://www.zbmath.org/?q=an:0175.03601

[2] Bourbaki, N. Eléments de mathématique. Algèbre commutative. Chapitre 1 - 10. Berlin: Springer - https://www.zbmath.org/?q=an:1290.00001

[3] Eisenbud, D. (1995). Commutative algebra. With a view toward algebraic geometry. Berlin: Springer-Verlag. (Graduate Texts in Mathematics, 150) - http://dx.doi.org/10.1007/978-1-4612-5350-1

[4] Hochster, M. (2010). Lecture notes for Math 615, winter 2010 - http://www.math.lsa.umich.edu/~hochster/615W10/615.pdf

[5] Krantz, S.G., and Parks, H.R. (2013). The implicit function theorem. History, theory, and applications. Reprint of the 2003 edition. New York: Birkhäuser/Springer - http://dx.doi.org/10.1007/978-1-4614-5981-1

[6] Lang, S. (2002). Algebra. 3rd revised ed. New York: Springer-Verlag. (Graduate Texts in Mathematics, 211) - http://dx.doi.org/10.1007/978-1-4613-0041-0

[7] Matsumura, H. (1989). Commutative ring theory. 2nd ed. Cambridge: Cambridge University Press. (Cambridge Studies in Advanced Mathematics, 8) - https://www.zbmath.org/?q=an:0666.13002

[8] Milne, J.S. (1980). Étale cohomology. Princeton, New Jersey: Princeton University Press. (Princeton Mathematical Series, 33) - https://www.zbmath.org/?q=an:0433.14012

[9] Ruiz, J.M. (1993). The basic theory of power series. Wiesbaden: Vieweg+Teubner Verlag. (Advanced Lectures in Mathematics) - http://www.springer.com/mathematics/geometry/book/978-3-528-06525-6

[10] Scheidemann, V. (2005). Introduction to complex analysis in several variables. Basel: Birkhäuser Verlag - http://dx.doi.org/10.1007/3-7643-7491-8

[11] Shafarevich, I.R. (1994). Basic algebraic geometry 1. Varieties in projective space. 3rd ed. Berlin: Springer - http://dx.doi.org/10.1007/978-3-642-37956-7

[12] Swan, R.G. (1998). Neron-Popescu Desingularization. In M.-C. Kang (Ed.), Lectures in algebra and geometry. Proceedings of the international conference on algebra and geometry, National Taiwan University, Taipei, Taiwan, December 26-30, 1995 (pp. 135-192). Cambridge, MA: International Press - https://www.zbmath.org/?q=an:0954.13003

[13] The Stacks Project Authors. Stacks Project - http://math.columbia.edu/algebraic_geometry/stacks-git

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