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nLab > Latest Changes: linear equation

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 17th 2012

    I finally started linear equation. But am too tired now to really do it justice…

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 18th 2012

    I personally don’t think it’s a good idea to launch in with the type-theoretic notation right in the Idea section. It will look intimidating or rebarbative to many. At the same time, I hesitate to even say so, because I fear a long argument ahead.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 18th 2012

    No argument here. I left the entry unfinished. Am tired now. Feel free to rework.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 18th 2012
    • (edited Sep 18th 2012)

    Okay, thanks. I might like to see what plans you have for this article first, before attempting to rework it in any way. One thing I don’t understand yet is why attention is being constrained to RR-module maps of the form K⊗ RN:R m⊗ RN→R n⊗ RNK \otimes_R N: R^m \otimes_R N \to R^n \otimes_R N.

    To ask it more encouragingly, perhaps: are you planning to say something about syzygy in this article? Zoran once created a gray link to syzygy that has yet to be written.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 18th 2012
    • (edited Sep 18th 2012)

    are you planning to say something about syzygy

    Yes, that’s where I was headed re projective resolutions. But then I definitely ran out of steam, being distracted by a few other things.

    I might like to see what plans you have for this article first,

    I’d think the plan for nnLab entries with titles like “linear equation” should be universal: give an elementary introduction first and then flesh out the category-theoretic and higher-category theoretic aspects.

    But don’t worry if you don’t feel like getting into this and just wanted to point out that the current version is a stub. I’ll get to this more seriously by next week. But now I first have to take care of one last round of polishing mapping cone and then completing the entries on the diagram-chasing lemmas.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeSep 18th 2012
    • (edited Sep 18th 2012)

    I had noted the grey link to identity among the relations and thus to homotopy syzygy and homological szyzgy. Creating those is on my list of things to do. I am not sure of the direction that Urs is thinking of for syzygy and equation, but the non-Abelian group theoretic ideas of Ronnie, Loday, Kapranov etc. may be useful and I can add material on these if and when it is needed. (There is stuff in the Menagerie, though probably not in the version I have available from my pages in the Lab.) Of course, these are non linear syzygies in general.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 18th 2012
    • (edited Sep 18th 2012)

    I have started syzygy. But let’s discuss that in it’s thread.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 18th 2012

    I’d think the plan for nLab entries with titles like “linear equation” should be universal: give an elementary introduction first

    Ostensibly we agree then. If I were commissioned to write an elementary introduction, I would be inclined to write something extremely simple, like “in the context of modules MM, NN over a commutative ring RR, a linear equation is an equation of the form f=gf = g where f,g∈Mod R(M,N)f, g \in Mod_R(M, N).” (That’s just an example.) But since that kind of utterance is very close to being self-evident, the main question would be: what’s the point or goal of such an article? What is it that one really wants to say here? That’s why I’m trying to draw you out.

    My earlier complaint was that a universal, elementary introduction (and especially the “Idea” section) should state things very simply, and avoid things like type-theoretic jargon with turnstiles and type declarations and so on (which is probably not the real point of the article anyway, and is probably distracting or confusing). What’s the point of the article?

    And let me ask again, why did you want to restrict attention to maps of the form f=K⊗ RNf = K \otimes_R N where KK is an RR-linear map R m→R nR^m \to R^n?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 18th 2012
    • (edited Sep 18th 2012)

    Hey Todd,

    from #2 I gather you don’t want a long argument, and I don’t want it either. So far it’s a stub entry, it’s just something I happened to produce late at night one day. If nobody else feels like working on it, I’ll expand it next week to full beauty and then announce it again.

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 18th 2012

    I’m just asking questions, Urs. But I can wait. No argument here either.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 24th 2012

    I did end up editing linear equation a bit more last week. Looks like I am not going to touch it further for the time being.

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeMay 20th 2024

    Linear equations over skewfields were studied in

    • Oystein Ore, Linear equations in non-commutative fields, Ann. Math. 32:3 (1931) 463–477 doi

    and more lately in the theory of quasideterminants, see there.

    diff, v5, current

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