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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeFeb 5th 2010

    I wrote an entry (short for now) separable algebra. It is a sort of support for the current Galois theory/Tannakian reconstruction/covering space/monodromy interest of Urs.

    • CommentRowNumber2.
    • CommentAuthorJohn Baez
    • CommentTimeMay 18th 2023

    I couldn’t believe this:

    … the Grothendieck Galois theory statement that the category of commutative separable algebras over a field KK is anti-equivalent to the category of continuous actions on finite sets of the profinite fundamental group of KK

    It turns out that an extra word makes it true:

    … the Grothendieck Galois theory statement that the category of commutative separable algebras over a field KK is anti-equivalent to the category of continuous actions on finite sets of the profinite fundamental group of KK

    This is plausible because a commutative separable algebra over KK is like a finite-sheeted covering space of Spec(K)Spec(K).

    diff, v19, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2023
    • (edited May 18th 2023)

    … where the extra word is “commutative” (here) …

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTimeMay 18th 2023

    Added info to the section “In algebraic geometry”. Deleted the sentence:

    Every separable kk-algebra is a filtered colimit of finite-dimensional separable kk-algebras???

    (which had these question marks), because as noted in this article a separable kk-algebra can be given the structure of a Frobenius algebra, and every Frobenius algebra is finite-dimensional, so this claim seems vacuous. Perhaps it arose from confusion between separable algebras and separable fields? Wikipedia says

    If L/K is a field extension, then L is separable as an associative K-algebra if and only if the extension of fields is separable.

    but this seems wrong to me. If K has characteristic zero every extension of it is separable, even those infinite-dimensional over K. Even Wikipedia admits that a separable algebra over K must be a Frobenius algebra over K hence finite-dimensional over K.

    Quite possibly I am confused and perhaps the sentence with question marks should be restored just to prompt someone to clarify things.

    diff, v19, current

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 19th 2023

    It sounds to me that the statement wants to be “every separable field extension of kk is a filtered colimit of finite-dimensional separable field extensions (where the latter may in turn may be regarded as separable kk-algebras in the sense of this article). I think this would fit in with the fundamental group being the profinite completion of the finite Galois groups. But correct me if I’m wrong.

    • CommentRowNumber6.
    • CommentAuthorJohn Baez
    • CommentTimeMay 30th 2023

    Fixed link to Eilenberg-Nakayama paper.

    diff, v21, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2023

    moved the references to the References, instead added links from the text (here)

    diff, v22, current