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 Mat. Sb. (N.S.), 1970, Volume 83(125), Number 3(11), Pages 372–389 (Mi msb3517)

Rings with a discrete group of divisor classes

V. I. Danilov

Abstract: 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 $A[T_1,…,T_n]$, $A[[T_1,…,T_n]]$, completions, etc.
Bibliography: 13 titles.

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English version:
Mathematics of the USSR-Sbornik, 1970, 12:3, 368–386

Bibliographic databases:

UDC: 513.015.7
MSC: 13F15, 20K30, 14C20

Citation: V. I. Danilov, “Rings with a discrete group of divisor classes”, Mat. Sb. (N.S.), 83(125):3(11) (1970), 372–389; Math. USSR-Sb., 12:3 (1970), 368–386

Citation in format AMSBIB
\Bibitem{Dan70} \by V.~I.~Danilov \paper Rings with a~discrete group of divisor classes \jour Mat. Sb. (N.S.) \yr 1970 \vol 83(125) \issue 3(11) \pages 372--389 \mathnet{http://mi.mathnet.ru/msb3517} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=282980} \zmath{https://zbmath.org/?q=an:0222.13015} \transl \jour Math. USSR-Sb. \yr 1970 \vol 12 \issue 3 \pages 368--386 \crossref{https://doi.org/10.1070/SM1970v012n03ABEH000926} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. I. Danilov, “On rings with a discrete divisor class group”, Math. USSR-Sb., 17:2 (1972), 228–236
2. Hartshorne Robin, Ogus Arthur, “On the Factoriality of Local Rings of Small Embedding Codimension”, Communications in Algebra, 1:5 (1974), 415
3. Frank R DeMeyer, Timothy J Ford, “On the Brauer group of surfaces”, Journal of Algebra, 86:1 (1984), 259
4. Lieven Le Bruyn, Alain Verschoren, “Maximal orders having a discrete normalizing class group”, Journal of Algebra, 100:2 (1986), 430
5. Phillip Griffith, “Some results in local rings on ramification in low codimension”, Journal of Algebra, 137:2 (1991), 473
6. Claudia Miller, “Recovering divisor classes via their (T)-adic filtrations”, Journal of Pure and Applied Algebra, 127:3 (1998), 257
7. Sandra Spiroff, “The limiting behavior on the restriction of divisor classes to hypersurfaces”, Journal of Pure and Applied Algebra, 186:1 (2004), 77
8. Phillip Griffith, Sandra Spiroff, “Restriction of divisor classes to hypersurfaces in characteristic p”, Journal of Algebra, 275:2 (2004), 801
9. Phillip Griffith, “Approximate liftings in local algebra and a theorem of Grothendieck”, Journal of Pure and Applied Algebra, 196:2-3 (2005), 185
10. Kazuhiko Kurano, Ei-ichi Sato, Anurag K. Singh, Kei-ichi Watanabe, “Multigraded rings, diagonal subalgebras, and rational singularities”, Journal of Algebra, 322:9 (2009), 3248
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