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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quotient sheaves and valuation rings


Author: Joel Cunningham
Journal: Trans. Amer. Math. Soc. 164 (1972), 227-239
MSC: Primary 16.90; Secondary 13.00
MathSciNet review: 0286845
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Abstract: In this paper a construction of a quotient sheaf of a sheaf of rings is given. This construction is analogous to the Utumi ring of quotients of a ring. For a valuation ring V, a sheaf of rings corresponding to V is introduced and its quotient sheaf is computed. It is shown that this quotient sheaf corresponds to the completion of V in case V is discrete rank one and that V is maximal if and only if its associated sheaf of rings is its own quotient sheaf.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0286845-7
PII: S 0002-9947(1972)0286845-7
Keywords: Sheaf of rings, sheaf of modules, dense extension, quotient sheaf, valuation ring, divisorial ideal, maximal valuation ring
Article copyright: © Copyright 1972 American Mathematical Society