Quotient Sheaves and Valuation Rings

descriptionPublicationkeyboard_double_arrow_right Article 01 Jan 1972Publisher:JSTORJournal:Transactions of the American Mathematical Society, volume 164, page 227 (issn: 0002-9947,

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Authors: Cunningham, Joel;

doi: 10.2307/1995970 , 10.1090/s0002-9947-1972-0286845-7

Quotient Sheaves and Valuation Rings

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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.

Keywords

Associative rings of functions, subdirect products, sheaves of rings

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