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Presheaves associated to modules over subrings of Dedekind domains. (English) Zbl 0956.13003

Let \(A\) be a commutative ring with unit. For \(E\subseteq \text{Spec} A\) let \(S_E= \bigcap_{P\in E}(A-P)\). If \(M\) is an \(A\)-module and \(U\) is an open subset of \(\text{Spec} A\), let \(\overline M(U)=S_U^{-1}M\). In general \(\overline M\) is not a sheaf. The paper deals with the question when \(\overline M\) is actually a sheaf. Several characterizations for rings with the property that \(\overline M\) is a sheaf for any \(A\)-module \(M\), are given. It is also shown that the rings of algebraic integers with quadratic quotient field have this property.

MSC:

13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
14A05 Relevant commutative algebra
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