Some applications of the ultrafilter topology on spaces of valuation domains, Part I
Actes des rencontres du CIRM, Tome 2 (2010) no. 2, pp. 107-109.

In the present note we introduce the basic definitions and the main results concerning the spaces of valuation domains needed in the subsequent Part II.

Publié le :
DOI : 10.5802/acirm.44
Mots clés : Valuation domain, (semi)star operation, prime spectrum, Zariski topology, constructible topology, filter and ultrafilter, Prüfer domain.
@article{ACIRM_2010__2_2_107_0,
     author = {Finocchiaro, Carmelo Antonio and Fontana, Marco},
     title = {Some applications of the ultrafilter topology on spaces of valuation domains, {Part} {I}},
     journal = {Actes des rencontres du CIRM},
     pages = {107--109},
     publisher = {CIRM},
     volume = {2},
     number = {2},
     year = {2010},
     doi = {10.5802/acirm.44},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/acirm.44/}
}
TY  - JOUR
AU  - Finocchiaro, Carmelo Antonio
AU  - Fontana, Marco
TI  - Some applications of the ultrafilter topology on spaces of valuation domains, Part I
JO  - Actes des rencontres du CIRM
PY  - 2010
SP  - 107
EP  - 109
VL  - 2
IS  - 2
PB  - CIRM
UR  - http://archive.numdam.org/articles/10.5802/acirm.44/
DO  - 10.5802/acirm.44
LA  - en
ID  - ACIRM_2010__2_2_107_0
ER  - 
%0 Journal Article
%A Finocchiaro, Carmelo Antonio
%A Fontana, Marco
%T Some applications of the ultrafilter topology on spaces of valuation domains, Part I
%J Actes des rencontres du CIRM
%D 2010
%P 107-109
%V 2
%N 2
%I CIRM
%U http://archive.numdam.org/articles/10.5802/acirm.44/
%R 10.5802/acirm.44
%G en
%F ACIRM_2010__2_2_107_0
Finocchiaro, Carmelo Antonio; Fontana, Marco. Some applications of the ultrafilter topology on spaces of valuation domains, Part I. Actes des rencontres du CIRM, Tome 2 (2010) no. 2, pp. 107-109. doi : 10.5802/acirm.44. http://archive.numdam.org/articles/10.5802/acirm.44/

[1] Paul-Jean Cahen, Alan Loper, and Francesca Tartarone, Integer-valued polynomials and Prüfer v-multiplication domains, J. Algebra 226 (2000), 765–787. | DOI | Zbl

[2] Claude Chevalley et Henri Cartan, Schémas normaux; morphismes; ensembles constructibles, Séminaire Henri Cartan 8 (1955-1956), Exp. No. 7, 1–10.

[3] M. Fontana, K. A. Loper, Cancellation properties in ideal systems: a classification of e.a.b. semistar operations, J. Pure Appl. Algebra 213 (2009), no. 11, 2095–2103. | DOI | MR | Zbl

[4] Marco Fontana and K. Alan Loper, The patch topology and the ultrafilter topology on the prime spectrum of a commutative ring, Comm. Algebra 36 (2008), 2917–2922. | DOI | MR | Zbl

[5] R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York 1972. | Zbl

[6] R. Gilmer and W. Heinzer, Irredundant intersections of valuation rings, Math. Z. 103 (1968), 306–317. | DOI | MR | Zbl

[7] Alexander Grothendieck et Jean Dieudonné, Éléments de Géométrie Algébrique I, IHES 1960; Springer, Berlin, 1970. | DOI | Numdam | Zbl

[8] Melvin Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60. | DOI | MR | Zbl

[9] Roland Huber, Bewertungsspektrum und rigide Geometrie, Regensburger Mathematische Schriften, vol. 23, Universität Regensburg, Fachbereich Mathematik, Regensburg, 1993.

[10] Roland Huber and Manfred Knebusch, On valuation spectra, in “Recent advances in real algebraic geometry and quadratic forms: proceedings of the RAGSQUAD year”, Berkeley, 1990-1991, Contemp. Math. 155, Amer. Math. Soc., Providence, RI, 1994. | DOI | MR

[11] K. Alan Loper, Sequence domains and integer-valued polynomials, J. Pure Appl. Algebra 119 (1997), 185–210. | DOI | MR | Zbl

[12] K. Alan Loper, A classification of all D such that Int(D) is a Prüfer domain, Proc. Amer. Math. Soc. 126 (1998), 657–660. | DOI | Zbl

[13] Niels Schwartz, Compactification of varieties, Ark. Mat. 28 (1990), 333–370. | DOI | MR | Zbl

[14] Niels Schwartz and Marcus Tressl, Elementary properties of minimal and maximal points in Zariski spectra, J. Algebra 323 (2010), 698–728. | DOI | MR | Zbl

[15] John Tate, Rigid analytic spaces, Invent. Math. 12 (1971), 257–269. | DOI | MR | Zbl

[16] O. Zariski, The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc 50 (1944), 683–691. | DOI | MR | Zbl

[17] O. Zariski, P. Samuel, Commutative Algebra, Volume 2, Springer Verlag, Graduate Texts in Mathematics 29, New York, 1975 (First Edition, Van Nostrand, Princeton, 1960). | DOI

Cité par Sources :