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nLab > Latest Changes: localisation of a commutative ring away from an element

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  1. Gave an explicit definition as the quotient ring A[x]/(ax−1)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.

    v1, current

  2. 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 𝔭A_{\mathfrak{p}} can be thought to be germs of functions defined on an open neighbourhood of the point 𝔭\mathfrak{p}, while elements of A[f −1]A[f^{-1}] are functions which are defined on D(f)D(f), the locus where ff is invertible, i.e. away from its zeros.

    What do you think?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMay 1st 2018

    I agree that technically it should be localizing “away” from an element, as counterintuitive as that is from a purely algebraic perspective.

  4. 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 pA_{p} for a prime ideal pp.

    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.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeMay 1st 2018

    Why do the “v1, current” links in #1 of this thread have an extra “+” at the end of the URL?

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 1st 2018

    Oh, I see, it’s because the name of the page actually had a space at the end!

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeMay 1st 2018

    Changed name to “away from” (and removed trailing space), added remark comparing terminologies, added redirects (e.g. for alternative spelling “localization”).

    diff, v2, current

  8. Thanks!

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 15th 2019

    Added some redirects.

    diff, v4, current

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