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A General Theory of One-dimensional Local Rings

Published online by Cambridge University Press:  18 May 2009

D. G. Northcott
D. G. Northcott
Affiliation:
The University, Sheffield.
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The development of the theory of local rings has been greatly stimulated by the importance of the applications to algebraic geometry, but it is none the less true that this stimulus has produced a theory which, on aesthetic grounds, is somewhat unsatisfactory. In the first place, if a local ring Q arises in the ordinary way from a geometric problem, then Qwill have the same characteristic as its residue field. It is partly for this reason that our knowledge of equicharacteristic local rings is much more extensive than it is of those local rings which present the case of unequal characteristics. Again, in the geometric case, the integral closure of Q in its quotient field will be a finite Q-module. Here, once more, we have a special situation which it would be desirable to abandon from the point of view of a general abstract theory.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 2 , Issue 4 , February 1956 , pp. 159 - 169
Copyright
Copyright © Glasgow Mathematical Journal Trust 1956

References

REFERENCES

(1)Akizuki, Y., Einige Bemerhungen über primäre Integritätsbereiche mit Teilerkettensatz, Proc. Phys. Math. Soc. Japan. 17 (1935), 327–336.Google Scholar
(2)Chovalley, C., La notion d'anneau da décomposition, Nagoya Math. J. 7 (1954), 21–33.Google Scholar
(3)Cohen, I. S., On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54–106.Google Scholar
(4)Grell, H., Verzweigungstheorie in allgemeinen Ordnungeti algebraischer Zahlkörper, Math.Z., 40 (1935), 629–657.Google Scholar
(5)Krull, W., Ein Satz über primärc Integritätsbereiche, Math. Ann. 103 (1930), 450–465.Google Scholar
(6)Nagata, M., Some remarks on Local Rings II, Memoirs of College of Science, Kyoto Univ. 28, Mathematics No. 2 (1953), 109–120.Google Scholar
(7)Nagata, M., On the theory of Henselian rings II, Nagoya Math. J. 7 (1954), 1–19.Google Scholar
(8)Northcott, D. G., Ideal Theory, Cambridge Tracts No. 42.Google Scholar
(9)Samuel, P., La notion de multiplied en algebre et en géométrie, Thesis, Paris (1951).Google Scholar
(10)Walker, R. J., Algebraic Curves (Princeton Math. Series No. 13).Google Scholar