# Ring properties which can be checked on sufficiently many Ore localizations

Let $R$ be a non-commutative Ore domain, and let $R[S_i^{-1}]$ be a finite set of Ore localizations. The absence of a geometric theory means there is no obvious candidate for when this set of localizations is a cover' (correct me if I am wrong on this). A weaker requirement is that $$\bigcap_i R[S_i^{-1}] = R,$$ where the intersection is in the skew-field of fractions of $R$.

Question: Assume that $R$ is an intersection of finitely-many Ore localizations $R[S_i^{-1}]$. What ring-theoretic properties of $R$ can be checked on the $R[S_i^{-1}]$?

That is, I am looking for properties $P$ such that, if each $R[S_i^{-1}]$ has property $P$, then $R$ also has property $P$.

Examples of properties I have in mind:

• Finitely-generated
• Noetherian
• Cohen-Macaulay

Properties I can show can be checked on the localizations:

• There are no nontrivial module-finite extension rings contained in the skew-field of fractions. (I don't know the name of this, but in the commutative case, it is one definition of an integrally-closed domain).
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A reasonable replacement for covers is that the product of your localizations be faithfully flat over $R$. This condition is quite useful. – Mariano Suárez-Alvarez Mar 8 '12 at 20:00
To supplement Mariano's comment, one could also demand that $\mathrm{Mod}(R_i) \to \mathrm{Mod}(R)$ defines a cover in the sense of Rosenberg's non-commutative algebraic geometry. Perhaps some properties are already mentioned in his work, you should check it. – Martin Brandenburg Mar 8 '12 at 21:52
I don't do much noncommutative stuff, but it seems to me that an obvious replacement for covers is that the "Cech complex" $0 \to R \to \bigoplus R[S_i^{-1}] \to \bigoplus R[S_i^{-1}, S_j^{-1}] \to \bigoplus R[S_i^{-1}, S_j^{-1}, S_k^{-1}] \to \cdots$` be exact. – David Speyer Mar 8 '12 at 23:05