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nLab > Latest Changes: rng

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2014

    added references.

    Any book that develops a bit of algebraic geometry of non-unital commutative rings or one that discusses what would be hte major things that break?

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeAug 19th 2014

    CRngCRng is equivalent to CRing /ℤCRing_{/ \mathbb{Z}} via the augmentation functor that adjoins a unit to a rng, with quasi-inverse given by the augmentation ideal, so CRng opCRng^{op} is equivalent to the category of affine schemes under Specℤ\operatorname{Spec} \mathbb{Z}. I don’t know how helpful that is, though.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2014
    • (edited Aug 20th 2014)

    That is a good point, thanks. I have added that remark to rng.

    Also,I discovered this article and added a pointer to it:

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2014

    CRings over ℤ\mathbb{Z} make one think of E ∞E_\infty-rings over 𝕊\mathbb{S}, which in turn might make one think of ∞\infty-groups over 𝕊\mathbb{S}, of which Sagave showed (here) that they accomodate all ∞\infty-groups of units of E ∞E_\infty-rings and of which Kapranov essentially suggested (here) that they are the source and generalization of all supergeometry.

    On the other hand of course rings over (ℤ,+,⋅)(\mathbb{Z},+,\cdot) are freely (ℤ,+)(\mathbb{Z},+)-graded as abelian groups (which was the point above) and so its not clear if there is indeed a useful relation here.

    Maybe something more interesting happens for E ∞E_\infty-rings over 𝕊\mathbb{S}? Need to think about it.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeAug 20th 2014

    As I was hanging in my student years more around ring theorists than category theorists, I vote for the page title nonunital ring rather than rng. Among my colleagues much more widely spread name, often called just ring.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 20th 2014

    Re #5: I think that’s a good idea. Then rng can redirect to that.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2014

    Very well, I am all in favor of that. I have edited the entry accordingly and in particular I have rewritten the idea-section and its comments on terminology.

    Now maybe Toby will be unhappy, though? If so, we can still roll back or maybe find another compromise.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2014
    • (edited Aug 21st 2014)

    added (here) pointers to

    • Daniel Quillen, K 0K_0 for nonunital rings and Morita invariance, J. Reine Angew. Math., 472:197-217, 1996.

    • Snigdhayan Mahanta, Higher nonunital Quillen K’-theory, KK-dualities and applications to topological T-dualities, J. Geom. Phys., 61 (5), 875-889, 2011. (pdf)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2014

    added to nonunital ring a pointer to prop. 5.2.3.14 in Higher Algebra, which is vastly general but should subsume in particular the equivalence that I was after above:

    E ∞Ring nonunital≃E ∞Ring /𝕊. E_{\infty}Ring^{nonunital} \simeq E_\infty Ring_{/\mathbb{S}} \,.
    • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeAug 22nd 2014
    • (edited Aug 22nd 2014)

    I am not unhappy. But some duplicated discussion on terminology was left at nonunital ring, which I have fixed.

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