Uses of Frobenius elements in mathematics

Stream: maths

[Kevin Buzzard]

For Fermat I need Frob_p in Gal(Q-bar/Q) and more generally Frob_P in Gal(F-bar/F) for F a number field. But people use these things in function fields as well. In #14155 I try and write down quite a generic "Frobenius", which takes as input an extension of commutative rings B/A and maximal ideals Q/P of these rings, and under the assumptions that B/Q (and A/P) is finite, and that the fixed points of Aut(B/A) on B are A, produces an element Frob_Q of Aut(B/A) which fixes Q and induces x mapsto x^{#(A/P)} on B/Q. This is certainly enough for the number field case, I can let L/K be a finite Galois field extension of number fields and prove all these hypotheses, and galRestrict is an isomorphism Gal(L/K)=Aut(B/A). More generally it will work for full rings of S-integers for S an (arbitrary) set of places (nonempty in the function field case) for both number fields and function fields over finite fields, in the finite Galois setting. It will also work for finite Galois extensions of p-adic fields and more generally finite Galois extensions of nonarch local fields.

Are there any settings where Frobenius is used by mathematicians, which is not covered by this B/A abstraction?

[Joël Riou]

In SGA 1 V 1.1, there is a more general setup which is very useful in algebraic geometry. If $G$ is a finite group acting on an $A$-algebra $B$ in such a way that $A$ is the invariant subring, the prime spectrum of $A$ identifies (topologically) to the quotient of $Spec(B)$ by the action of $G$ (e.g. $G$ acts transitively on the set of prime ideals of $B$ lying over a given prime ideal of $A$). If $x$ in $Spec(B)$ is over $y$, the extension $k(x)/k(y)$ is quasi-galois, and more importantly for your application, the stabilizer of $x$ (the decomposition group) maps surjectively onto $Gal(k(x)/k(y))$. (In your application, $k(x)$ and $k(y)$ are finite and you are lifting the Frobenius.)

I believe this is very important for the étale fundamental group. SGA refers to Bourbaki, Algèbre commutative, Chap. 5, paragraphs 1 and 2 (theorem 2).

[Kevin Buzzard]

Aha, you're saying that surjectivity is true more generally. In which case #14155 isn't done in the right generality.

[Joël Riou]

Yes. The proofs are slightly more difficult in the general situation.

[Antoine Chambert-Loir]

(deleted)

[Kevin Buzzard]

There's a discussion of this generalisation in the FLT steam btw