5
$\begingroup$

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).
$\endgroup$
3
  • 2
    $\begingroup$ A reasonable replacement for covers is that the product of your localizations be faithfully flat over $R$. This condition is quite useful. $\endgroup$ Mar 8, 2012 at 20:00
  • $\begingroup$ 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. $\endgroup$ Mar 8, 2012 at 21:52
  • $\begingroup$ 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. $\endgroup$ Mar 8, 2012 at 23:05

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.