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Gave an explicit definition as the quotient ring $A[x] / (ax - 1)$, and mentioned the equivalent definition in terms of multiplicative systems. Gave the example of Laurent polynomials. The explicit definition was already given at localisation of a commutative ring, but the multiplicative system was not given, and I think the construction is fundamental enough to warrant its own page. I will tweak localisation of a commutative ring to link to the new page.
Thank you, that’s a good entry to have! Not least because of the importance of localising in constructive algebra. (For instance, the textbook by Lombardi and Quitté is full of this, and it’s also fundamental to the Zariski topos approach to constructive algebra.)
I’m wondering about the terminology: I grew up with “localising at a prime ideal” (or at a filter), but “localising away from an element”. Elements of the localisation $A_{\mathfrak{p}}$ can be thought to be germs of functions defined on an open neighbourhood of the point $\mathfrak{p}$, while elements of $A[f^{-1}]$ are functions which are defined on $D(f)$, the locus where $f$ is invertible, i.e. away from its zeros.
What do you think?
I agree that technically it should be localizing “away” from an element, as counterintuitive as that is from a purely algebraic perspective.
Thanks for the thoughts! Definitely we should include the classical terminology! Maybe you could add your remarks about germs to the entry, Ingo? I think it would also be good with an entry specific to $A_{p}$ for a prime ideal $p$.
I would like to keep the terminology “localising at” present in the entry, because it does make sense algebraically, and I at least find it easier to remember this way. But as long as it is mentioned, and a redirect is kept if the page changes name, I am happy for the default terminology to be changed.
Why do the “v1, current” links in #1 of this thread have an extra “+” at the end of the URL?
Oh, I see, it’s because the name of the page actually had a space at the end!
Thanks!
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