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

]]>Fixed link to Eilenberg-Nakayama paper.

]]>It sounds to me that the statement wants to be “every separable field extension of $k$ is a filtered colimit of finite-dimensional separable field extensions (where the latter may in turn may be regarded as separable $k$-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.

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

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

(which had these question marks), because as noted in this article a separable $k$-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.

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

]]>I couldn’t believe this:

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

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 $K$ is anti-equivalent to the category of continuous actions on finite sets of the profinite fundamental group of $K$…

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

]]>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.

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