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nLab > Latest Changes: Isbell duality - table

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 23rd 2018

    the table didn’t have the basic examples, such as Gelfand duality and Milnor’s exercise. Added now.

    diff, v4, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 26th 2018

    added one more line to the table (super L ∞L_\infty-algebras ↔\leftrightarrow differential graded-commutative superalgebras)

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 26th 2018
    • (edited Jul 26th 2018)

    added pointer to the equivalence between fully faithfulness of affine schemes in opposites of rings.

    Undecided now whether to display mostly full inclusions or the equivalences onto their essential image. Table may deserve further reworking..

    diff, v6, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2021
    • (edited Jan 22nd 2021)

    In revision 12 from October 2020, an anonymous edit tried to hyperlink the words “Gelfand-Kolmogorov”, but failed. I have fixed this now.

    Of course the issue is that the hyperlinks are inside a \text-environment inside a maths environment, and so necessarlily (?) Instiki syntax doesn’t work here – but HTML still does…

    … or at least it does with Firefox. I suppose most other browsers show just broken links on this page here?

    diff, v13, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 22nd 2021

    I’m just seeing “Unknown node type: a” in Chrome.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2021
    • (edited Oct 19th 2021)

    I have changed the table header from “Isbell duality” to “duality between algebra and geometry”. The former was meant to be read as a synonym for the latter, but I can see how that’s too ideosyncratic not to be confusing (as per the discussion here).

    For the moment the page title (which is not displayed in the pages where this table is !include-ed) remains the same. Because I am unsure if the software would correctly handle !redirects inside !includes, and since I don’t have the time now to search for and change all the !include-commands to this table.

    diff, v14, current

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeSep 12th 2024
    • (edited Sep 12th 2024)

    The table lists affine schemes as a (edit:) subcategory of finitely generated commutative algebras.

    The category of affine schemes is antiequivalent to the category of all commutative rings. One can work over some ground ring kk if wanted, but finite generation is not needed for anything unless it is required at both sides. Some people consider just “affine algebras” over some kk, but usually when talking the case of varieties.

    However, spectrum can also be understood as a right adjoint to the global sections functor on the category of locally affine ringed spaces.

    I do not know what the author meant with putting these finiteness requirements in the table.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2024

    Hm, I forget why I put in that qualifiers. Let’s remove it.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeSep 12th 2024

    Done.

1 to 9 of 9

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