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nLab > Latest Changes: divided power algebra

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeJul 10th 2021
    • (edited Apr 11th 2023)

    [deleted]

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2021
    • (edited Jul 10th 2021)

    Thanks. I have added some more hyperlinks (associative algebra, ring, ideal, …) . Your “operations” and “maps” should probably be linear maps. (?)

    We ought to have an entry divided power, but unfortunately we don’t. Maybe you’d enjoy creating it?

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeJul 10th 2021
    • (edited Apr 11th 2023)

    [deleted]

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2021

    Okay, probably “divided power” should just redirect here.

    diff, v4, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 11th 2021

    Added a reference to some informal discussion

    diff, v5, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2021
    • (edited Jul 11th 2021)

    for definiteness, I have changed

    … a collection of maps

    to

    ..an indexed set of functions (of underlying sets)

    Also I added more hyperlinks to the references (author names, DOI-s, publisher pages,…)

    Finally, I fixed the typesetting of the indices: due to an Instiki speciality, the source code nm gets rendered as “nmnm”. One needs a whitespace n m to tell Instiki that these are two distinct variables such as to obtain the desired italicized “nmn m”.

    diff, v6, current

    • CommentRowNumber7.
    • CommentAuthorHurkyl
    • CommentTimeJul 11th 2021

    Proof that n!γ n(x)=x nn! \gamma_n(x) = x^n

    diff, v8, current

    • CommentRowNumber8.
    • CommentAuthorHurkyl
    • CommentTimeJul 11th 2021

    In the definition of a divided power RR-algebra, the base ring RR is never used. Is there a reason to define this for AA an RR-algebra rather than just for AA a (possibly noncommutative) ring?

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 12th 2021

    The definition given doesn’t agree with the stacks project, and the hypotheses on the data of the maps in the PD-structure doesn’t typecheck. I’m not sure what is meant, else I’d fix it.

    • CommentRowNumber10.
    • CommentAuthorHurkyl
    • CommentTimeJul 12th 2021
    • (edited Jul 12th 2021)

    I hadn’t noticed that either; IIRC you’re just supposed to have x,y∈Ix,y \in I? Oh I see that’s been changed already.

    Another thing to note (after making that correction) is that II is a commutative rng, since you can swap xx and yy in γ 2(x+y)=γ 2(x)+γ 1(x)γ 1(y)+γ 2(y) \gamma_2(x+y) = \gamma_2(x) + \gamma_1(x) \gamma_1(y) + \gamma_2(y).

    This meshes with the fact I thought when I saw divided power structures, they were on commutative rings with an ideal. Should we narrow the definition to AA being commutative, or is there an application with noncommutative AA?

    • CommentRowNumber11.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeJul 12th 2021
    • (edited Apr 11th 2023)

    [deleted]

    • CommentRowNumber12.
    • CommentAuthorHurkyl
    • CommentTimeJul 12th 2021

    I’ve gone ahead and made the RR-algebra case a secondary definition, and changed the definition to take AA to be a commutative ring; feel free to adjust it if the noncommutative case really is of interest.

    I think that the properties and constructions only really work well when II is in the center of AA, so I expect substantial savings by making this change.

    diff, v13, current

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 12th 2021
    • (edited Jul 12th 2021)

    The definition is still a bit odd. in https://stacks.math.columbia.edu/tag/07GL, we see that the maps γ n\gamma_n are meant to be endomorphisms (as sets) of the ideal II. But in the nLab page they are maps I→AI\to A. I’m not sure about the difference between a divided power algebra (as at the nLab) and a divided power structure (as at the Stacks Project), which only applies in the case of a divided power ring. Can someone sort this out?

    • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 12th 2021

    Made doi text point to doi link in the reference to Berthelot’s LNM volume.

    diff, v14, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2021

    It looks like item 3 in our entry forces γ\gamma to take values in II after all…

    Maybe this is meant to be following the Wikipedia entry (here) that also insists on the codomain of γ n\gamma_n being AA.

    But, yeah, it looks odd. I suggest to follow the StacksProject, make the codomain II and also fix the clauses 1 and 2 to have x∈Ix \in I instead of x∈Ax \in A.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2021

    Ah, the point is whether the indexing starts at 0 or 1, since γ 0=1\gamma_0 = 1 indeed can’t take values in II unless I=AI = A.

    • CommentRowNumber17.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 12th 2021

    To me that seems like one should have maps γ n:I→I\gamma_n\colon I \to I for n≥1n\geq1 and then just define γ 0\gamma_0 by convention to be 1.

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 12th 2021

    Cleaned up definition according to recent comments.

    diff, v16, current

    • CommentRowNumber19.
    • CommentAuthorHurkyl
    • CommentTimeJul 12th 2021

    Moved the statement of the γ 0\gamma_0 convention further up since we adopt it in all of the identities, not just one of them. I’ve also added a remark on the different handling of γ\gamma in different sources.

    diff, v17, current

    • CommentRowNumber20.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 16th 2022

    Added a few things on divided powers in symmetric monoidal categories. And some very recent references.

    diff, v18, current

    • CommentRowNumber21.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 16th 2022
    • (edited Aug 16th 2022)

    Don’t succeed to put all my edit because of crazy errors… That’s ok now, it was the error of putting a tikzcd diagram into $$

    • CommentRowNumber22.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 16th 2022

    Added relation with symmetric powers, including completely new observations on the potential interpretation of symmetric and divided powers in a graded (differential) linear logic.

    diff, v18, current

    • CommentRowNumber23.
    • CommentAuthorJ-B Vienney
    • CommentTimeNov 11th 2022

    Added the reference Luis Narváez Macarro, Hasse-Schmidt derivations, divided powers and differential smoothness, 2009

    diff, v20, current

    • CommentRowNumber24.
    • CommentAuthorJ-B Vienney
    • CommentTimeNov 19th 2022
    • (edited Nov 19th 2022)

    Added that permutations σ:A ⊗n→A ⊗n\sigma:A^{\otimes n} \rightarrow A^{\otimes n} are defined in the entry symmetric monoidal category.

    diff, v21, current

1 to 24 of 24

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