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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.
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 $Spec(K)$.
… where the extra word is “commutative” (here) …
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.
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.
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