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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.

References [Enhancements On Off] (What's this?)

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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