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Is there a name for the standard sheaf $\mathscr{O}$ on $\operatorname{Spec}A$ defined to be the set of locally constant functions to the localizations of $A$ ? It is used so often but I don't know the name for it. "Structure sheaf" seems slightly more general than what I'm looking for.

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Dear ashpool, Despite your remark to the contrary, a phrase such as "we let $\mathcal O$ denote the structure sheaf on Spec $A$" seems pretty standard to me. Regards, – Matt E Aug 12 '12 at 14:12
Perhaps OP's objection is that if the phrase "Spec $A$" denotes a topological space, then the phrasing is ambiguous (since lots of rings give rise to the same topological space). In most contexts, the phrase "Spec $A$" denotes the space together with the sheaf, not just the space, so that the phrase "the structure sheaf on Spec $A$" is not ambiguous. – user29743 Aug 12 '12 at 14:15
Dear Matt E, is it true that $\operatorname{Spec}A$ has, as a set, a unique structure sheaf on it? I've long been suspecting that might be the case, and if so, then I guess you are right. – ashpool Aug 12 '12 at 14:22
@ashpool Of course not. Let $k$ be any field, then $\operatorname{Spec} k[x]_{(x)}$ is the Sierpiński space – for any choice of $k$. – Zhen Lin Aug 13 '12 at 2:30

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