Geodesic
On rings with a discrete divisor class group
Sbornik. Mathematics, Tome 17 (1972) no. 2, pp. 228-236 
V. I. Danilov. On rings with a discrete divisor class group. Sbornik. Mathematics, Tome 17 (1972) no. 2, pp. 228-236. http://geodesic.mathdoc.fr/item/SM_1972_17_2_a4/
@article{SM_1972_17_2_a4,
     author = {V. I. Danilov},
     title = {On~rings with a~discrete divisor class group},
     journal = {Sbornik. Mathematics},
     pages = {228--236},
     year = {1972},
     volume = {17},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1972_17_2_a4/}
}
TY  - JOUR
AU  - V. I. Danilov
TI  - On rings with a discrete divisor class group
JO  - Sbornik. Mathematics
PY  - 1972
SP  - 228
EP  - 236
VL  - 17
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1972_17_2_a4/
LA  - en
ID  - SM_1972_17_2_a4
ER  - 
%0 Journal Article
%A V. I. Danilov
%T On rings with a discrete divisor class group
%J Sbornik. Mathematics
%D 1972
%P 228-236
%V 17
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1972_17_2_a4/
%G en
%F SM_1972_17_2_a4

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the conjecture: $C(A)=C(A[[T]])$ for a local ring $A$ if and only if the divisor class group of the strict henselization $C(^\mathrm{sh}A)$ has a finite number of generators. This conjecture is proved in two cases: 1) $A$ has characteristic $0$, 2) $A$ is an equicharacteristic ring of an isolated singularity. Bibliography: 15 titles.

[1] A. Grothendieck, J. Dieudonne, Éléments de géométrie algébrique, Publ. Math. IHES, 24, 1965 ; 32, 1967 | MR | MR

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

[3] V. I. Danilov, “Koltsa s diskretnoi gruppoi klassov divizorov”, Matem. sb., 83(125) (1970), 372–389 | MR | Zbl

[4] M. Artin, “Algebraicheskaya approksimatsiya struktur nad polnymi lokalnymi koltsami”, Matematika, 14:3 (1970), 3–89 | MR

[5] U. Storch, “Über die Divisorenklassengruppen normaler komplex-analytischer Algebren”, Math. Ann., 183:2 (1969), 93–104 | DOI | MR | Zbl

[6] N. Bourbaki, Algebre commutative, chap. VII, 1965

[7] X. 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

[8] A. Grothendieck, Fondement de la géométrie algebrique (Extraits du Seminaire Bourbaki, 1957–1962), Secretariat Math., Paris | MR | Zbl

[9] S. Leng, A. Neron, “Ratsionalnye tochki abelevykh mnogoobrazii nad funktsionalnymi polyami”, Matematika, 7:4 (1963), 13–33

[10] Zh. Serr, Algebraicheskie gruppy i polya klassov, Mir, Moskva, 1968

[11] M. Demazure, A. Grothendieck, Schémas et groups, Lecture Notes in Mathematics, Berlin, 1970

[12] E. Oort, “Sur le schéme de Picard”, Bull. Soc. Math. France, 90 (1962), 1–14 | MR | Zbl

[13] D. Mamford, “Topologiya normalnykh osobennostei algebraicheskoi poverkhnosti i kriterii prostoty”, Matematika, 10:6 (1966), 3–24 | MR

[14] P. Samuel, “Classes de diviseurs et dérivées logarithmiques”, Topology, 3 (1964), 81–96 | DOI | MR | Zbl

[15] P. Salmon, “Su un problema posto da P. Samuel”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 40 (1966), 801–803 | MR | Zbl

[16] M. Artin, “On the solutions of analytic equations”, Invent. Math., 5 (1968), 277–291 | DOI | MR | Zbl