Geodesic
Rings with a discrete group of divisor classes
Sbornik. Mathematics, Tome 12 (1970) no. 3, pp. 368-386 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We shall say that a ring $A$ has a DGC (discrete group of classes) if the group of divisor classes is preserved in going to the ring of formal power series, i.e. $C(A)\to C(A[[T]])$ is an isomorphism. We prove the localness and faithfully flat descent of the DGC property. We establish a connection between the DGC property of a ring and its depth. We also give a characterization of two-dimensional rings with DGC and characteristic zero rings with DGC. Finally, it is shown that the discreteness of the group of divisor classes is preserved under regular extensions of rings such as $A[T_1,\dots,T_n]$, $A[[T_1,\dots,T_n]]$, completions, etc. Bibliography: 13 titles. 
@article{SM_1970_12_3_a2,
     author = {V. I. Danilov},
     title = {Rings with a~discrete group of divisor classes},
     journal = {Sbornik. Mathematics},
     pages = {368--386},
     year = {1970},
     volume = {12},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_12_3_a2/}
}
TY  - JOUR
AU  - V. I. Danilov
TI  - Rings with a discrete group of divisor classes
JO  - Sbornik. Mathematics
PY  - 1970
SP  - 368
EP  - 386
VL  - 12
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1970_12_3_a2/
LA  - en
ID  - SM_1970_12_3_a2
ER  - 
%0 Journal Article
%A V. I. Danilov
%T Rings with a discrete group of divisor classes
%J Sbornik. Mathematics
%D 1970
%P 368-386
%V 12
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1970_12_3_a2/
%G en
%F SM_1970_12_3_a2
V. I. Danilov. Rings with a discrete group of divisor classes. Sbornik. Mathematics, Tome 12 (1970) no. 3, pp. 368-386. http://geodesic.mathdoc.fr/item/SM_1970_12_3_a2/

[1] A. Grothendieck, J. Dieudonne, “Éléments de Géométrie algébrique”, Publ. Math. IHES, Paris, 1960, no. 4 ; 1961, no. 11 ; 1966, no. 24; 1966, no. 28 ; 1967, no. 32 | Zbl | MR | Zbl | MR

[2] H. Burbaki, Algebra. Mnogochleny i polya, uporyadochennye gruppy, Nauka, Moskva, 1965 | MR

[3] V. I. Danilov, “O gipoteze Samyuelya”, Matem. sb., 81(123) (1970), 132–144 | MR | Zbl

[4] V. I. Danilov, “Gruppa klassov idealov popolnennogo koltsa”, Matem. sb., 77(119):4 (1968), 533–541 | MR | Zbl

[5] Kh. Khironaka, “Razreshenie osobennostei algebraicheskikh mnogoobrazii nad polyami kharakteristiki nul”, Matematika, 9:6 (1965), 2–71; 10:1 (1966), 3–90 ; 10:2 (1966), 3–59 | MR | MR

[6] D. Mamford, Lektsii o krivykh na algebraicheskoi poverkhnosti, Mir, Moskva, 1968

[7] M. Artin, “Nekotorye chislennye kriterii styagivaemosti krivykh na algebraicheskoi poverkhnosti”, Matematika, 9:3 (1965), 3–14 | MR

[8] N. Bourbaki, Algebre commutative. I, II, VII, Hermann, Paris, 1961, 1965

[9] P. Samuel, “On unique factorization domains”, Illinois J. Math., 5:1 (1961), 1–17 | MR | Zbl

[10] P. Samuel, “Anneaux gradues factoriels et modules reflexifs”, Bull. Soc. Math. France, 92 (1964), 237–249 | MR | Zbl

[11] G. Scheja, “Einige Beispiele iactorieller localer Ringe”, Math. Ann., 172:2 (1967), 124–134 | DOI | MR | Zbl

[12] A. Grothendieck, Seminaire de Geometrie algebrique, IHES, Bures-sur-Ivette, 1962

[13] J.-P. Serre, “Sur la topologie des varietes algebrique en caracteristique $p$”, Symp. Internac. de Topoi. Algebr., Mexico, 1956 | MR

[14] H. Hironaka, “On the characters $\nu$ and $\tau$ of singularities”, J. Math. of Kyoto Univ., 7:1 (1967), 19–44 | MR