Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-08T02:15:06.274Z Has data issue: false hasContentIssue false
  • English
  • Français

On the divisor class groups of a two-dimensional local ring and its form ring

Published online by Cambridge University Press:  22 January 2016

Dario Portelli  and
Walter Spangher
Dario Portelli
Affiliation:
Dipartimento di Scienze matematiche, Università degli Studi di Trieste, Piazzale Europa n. 1, 34100 — TRIESTE, Italy
Walter Spangher
Affiliation:
Dipartimento di Scienze matematiche, Università degli Studi di Trieste, Piazzale Europa n. 1, 34100 — TRIESTE, Italy
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let A be a noetherian ring and let I be an ideal of A contained in the Jacobson radical of A: Rad (A). We assume that the form ring of A with respect to the ideal I: G = Gr (A, I), is a normal integral domain. Hence A is a normal integral domain and one can ask for the links between Cl(A) and Cl(G).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 110 , June 1988 , pp. 137 - 149
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[1] Abhyankar, S. S., Nonprefactorial local ring, Amer. J. Math., 89 (1967), 1137–1146.Google Scholar
[2] Arezzo, M. e Greco, S., Sul gruppo delle classi di ideali, Ann. Se. Norm. Super. Pisa CI. Sci. IV Ser., 21 (1967), 459–483.Google Scholar
[3] Bádescu, L. e Fiorentini, M., Criteri di semifattorialità e di fattorialità per gli anelli locali con applicazioni geometriche, Ann. Mat. Pura Appl. IV Ser., 103 (1975), 211–222.CrossRefGoogle Scholar
[4] Bingener, J. und Storch, U., Zur berechnung der Divisorenklassengruppen kompletter lokaler Ringe, Nova Acta Leopold. Neue Folge, 52/240 (1981), 7–63.Google Scholar
[5] Bourbaki, N., Algèbre Commutative Chi-VII, Paris: Hermann (1961–65).Google Scholar
[6] Boutot, J. F., Schéma de Picard locai, Lect. Notes Math., 632, Berlin, Heidelberg, New York: Springer (1978).Google Scholar
[7] Cavaliere, M. P. and Niesi, G., On Serre’s conditions in the form ring of an ideal, J. Math. Kyoto Univ., 21 (1981), 537–546.Google Scholar
[8] Danilov, V. I., The group of ideal classes of a completed ring, Math. USSR Sb., 6 (1968), 493–500.Google Scholar
[9] Danilov, V. I., On a conjecture of Samuel, Math. USSR Sb., 10 (1970), 127–137.Google Scholar
[10] Danilov, V. I., Rings with a discrete group of divisor classes, Math. USSR Sb., 12 (1970), 368–386.Google Scholar
[11] Faddeev, D. K., Group of divisor classes on the curve defined by the equation X4 + Y4 = 1, Sov. Math. Dokl., 1 (1961), 1149–1151.Google Scholar
[12] Flenner, H., Divisorenklassengruppen quasihomogener Singularitaten, J. Reine Angew. Math., 238 (1981), 128–160.Google Scholar
[13] Fossum, R., The divisor class group of a Krull domain, Berlin-Heidelberg-New York: Springer (1973).CrossRefGoogle Scholar
[14] Greco, S. and Salmon, P., Topics in m-adic Topologies, Berlin-Heidelberg-New York Springer (1971).CrossRefGoogle Scholar
[15] Grothendieck, A., Cohomologie locale des faisceaux cohérents et Théorèmes de Lefschetz locaux et globaux (SGA 2), Amsterdam: North Holland (1968).Google Scholar
[16] Grothendieck, A. et Dieudonné, J., Eléments de géométrie algébrique. Ch. III; Ch. IV (Quartième Partie), Publ. Math. Inst. Hautes Etud. Sci., 11;32 (1961; 1967).Google Scholar
[17] Lipman, J., Rational Singularities with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Hautes Etud. Sci., 36 (1969).Google Scholar
[18] Lipman, J., Rings with discrete divisor class group : theorem of Danilov-Samuel, Am. J. Math., 101 (1979), 203–211.Google Scholar
[19] Matijevich, J., Three local conditions on a graded ring, Trans. Am. Math. Soc, 205 (1975), 275–284.Google Scholar
[20] Portelli, D. e Spangher, W., Condizioni di fattorialità ed anello graduato associato ad un ideale, Ann. Univ. Ferrara Nuova Ser. Sez. VII, 28 (1982), 181–195.Google Scholar
[21] Portelli, D., On the divisor class group of localizations, completions and Veronesean subrings of Z-graded Krull domains, Ann. Univ. Ferrara Nuova Ser. Sez. VII, 30 (1984), 97–118.Google Scholar
[22] Ratliff, L. J. Jr., On Rees localities and Hi-local rings, Pac. J. Math., 60 (1975), 169–194.Google Scholar
[23] Robbiano, L. and Valla, G., Primary powers of a prime ideal, Pac. J. Math., 63 (1976), 491–498.Google Scholar
[24] Rossi, M. E., Altezza e dimensione nell’anello graduato associato ad un ideale, Rend. Semin. Mat. Torino, 36 (1977–78), 305–312.Google Scholar
[25] Salmon, P., Su un problema posto da P. Samuel, Atti Accad. Naz. Lincei Vili Ser. Rend. Cl. Sci. Fis. Mat. Nat., 40 (1966), 801–803.Google Scholar
[26] Samuel, P., On unique factorization domains, 111. J. Math., 5 (1961), 1–17.Google Scholar
[27] Samuel, P., Sur les anneaux factoriels, Bull. Soc. Math. Fr., 89 (1961), 155–173.Google Scholar
[28] Scheja, G., Einige beispiele faktorieller lokaler Ringe, Math. Ann., 172 (1967), 124–134.Google Scholar
[29] Sharp, R. Y., Local cohomology theory in commutative algebra, Q. J. Math. Oxf. II Ser., 21 (1970), 425–434.Google Scholar
[30] Sharp, R. Y., Some results on the vanishing of local cohomology modules, Proc. Lond. Math. Soc. Ill Ser., 30 (1975), 177–195.Google Scholar
[31] Yuan, S., Reflexive modules and Algebra Class group over noetheriaa integrally-closed domains, J. Algebra, 32 (1974), 405–417.Google Scholar