Author: Eiko

Tags: arithmetic geometry, algebraic topology, fundamental groups, grothendieck fundamental groups, etale fundamental groups

Time: 2024-11-26 09:49:30 - 2025-01-18 18:22:43 (UTC)

Topics:

• Etale fundamental groups

• Section conjecture

Grothendieck’s fundamental group

This is meant to be a generalization of Galois theory and covering spaces.

Etale Algebras

An etale algebra $A$ over a field $k$ is a finite dimensional $k$-algebra isomorphic to a finite direct product of separable extensions of $k$.

• $\mathrm{Gal}(k^s/k)=G_k$ acts on the left of ${\mathrm{Hom}}_k(A,k^s)$.

Etale Algebras = Galois Sets

Theorem. $A\mapsto {\mathrm{Hom}}_k(A,k^s)$ is an anti-equivalence between the category of finite etale $k$-algebras and the category of finite $G_k$-sets (with continuous $G_k$-action). Here ${\mathrm{Hom}}_k$ is for $k$-algebra maps, not $k$-linear maps.

$$\{\text{Finite etale }k\text{-algebras}\}⟷\{\text{Finite continuous }{G}_k\text{-sets}\}$$ $$A\mapsto {\mathrm{Hom}}_k(A,k^s)\circlearrowright \mathrm{Gal}(k^s/k)$$

(the functor depends on the choice of $k^s$)

Classical Fibre Functor

Let $x\in X$, the functor

$${\mathrm{Fib}}_x:\{\text{Covering }p:Y\to X\}\to \{{\pi }_1(X,x)\text{-sets}\}:p\mapsto p^{-1}(x)$$

is an equivalence of categories, depends on the choice of basepoint $x$. It is representable as

$${\mathrm{Fib}}_x\cong {\mathrm{Hom}}_{\mathrm{Cov}(X)}({\tilde{X}}_x,-)$$

where $\pi :{\tilde{X}}_x\to X$ is the universal cover of $X$ based at $x$, the space of homotopy (homotopy with fixed end points) classes of paths starting from $x$.

Fundamental Groups Are Automorphisms Of Fibres

By Yoneda’s Lemma we have

$${\mathrm{Hom}}_{\mathrm{Hom}(\mathrm{Cov}(X),{\pi }_1-\mathbf{S}\mathbf{e}\mathbf{t}\mathbf{s})}({\mathrm{Fib}}_x,{\mathrm{Fib}}_x)\cong \mathrm{Hom}(h^{{\tilde{X}}_x},h^{{\tilde{X}}_x}){\cong }^{op}\mathrm{Hom}({\tilde{X}}_x,{\tilde{X}}_x)$$

i.e.

$$\mathrm{Aut}({\tilde{X}}_x{)}^{op}\cong \mathrm{Aut}({\mathrm{Fib}}_x)\cong {\pi }_1(X,x)$$

The last isomorphism is called the monodromy action.

Profinite Completion

For a profinite group $G$ acting continuously on a finite set $S$, the action factor through a finite quotient of the group since the finite intersection of stablizers ${\cap }_{s\in S}{G}_s$ are open.

The profinite completion of a group $G$ is the inverse limit of the finite quotients of $G$. For Galois groups or fundamental groups these corresponds to finite covers or finite extensions. You can think of profinite completion as a group that remembers all the finite actions.

The functor (restricted to subcategory of finite coverings)

$${\mathrm{Fib}}_x:\mathrm{FinCov}(X)\to \widehat{{\pi }_1(X,x)}\text{-}\mathbf{S}\mathbf{e}\mathbf{t}\mathbf{s}$$

induces an equivalence between the category of finite covers of $X$ and the category of finite continuous $\widehat{{\pi }_1(X,x)}$-sets.

Algebraic Fundamental Group

Let $S$ be connected scheme, a finite etale map $X\to S$ is also called a finite etale cover of $S$. Its fibre at each $s\in S$ is a finite etale $k(s)$-algebra.

For a geometric point $\overline{s}=\mathrm{Spec}(\mathrm{\Omega })\to S$, the geometric fibre $X{\times }_S\overline{s}$ is a finite etale $\mathrm{\Omega }$-algebra whose underlying set is ${\mathrm{Fib}}_{\overline{s}}(X)$, this is a set valued functor on $\mathrm{FinEt}(S)$.

The algebraic fundamental group is defined as

$${\pi }_1(S,\overline{s}):={\mathrm{Aut}}_{\mathrm{FinEt}(S)\to \mathbf{S}\mathbf{e}\mathbf{t}}({\mathrm{Fib}}_{\overline{s}})$$

• There is a natural action of ${\pi }_1(S,\overline{s})$ on ${\mathrm{Fib}}_{\overline{s}}(X)$ for any finite etale cover $X\to S$.

• Theorem (Grothendieck)

  • The group ${\pi }_1(S,\overline{s})$ is profinite and acts continuously on each fibre ${\mathrm{Fib}}_{\overline{s}}(X)$.

  • This fibre functor induces an equivalence

    $$\mathrm{FinEt}(S)\cong \widehat{{\pi }_1(S,\overline{s})}\text{-}\mathbf{S}\mathbf{e}\mathbf{t}\mathbf{s}$$

    between the category of finite etale covers of $S$ and the category of finite continuous $\widehat{{\pi }_1(S,\overline{s})}$-sets.

• The functor ${\mathrm{Fib}}_{\overline{s}}$ is pro-representable, for some filtered inverse system of finite etale covers $P_i\to S$ we have

  $${\mathrm{Fib}}_{\overline{s}}(X)\cong \mathrm{colim}\mathrm{Hom}(P_i,X)$$

• A path is an isomorphism of fibre functors

  $$\gamma :{\mathrm{Fib}}_{\overline{s}}\cong {\mathrm{Fib}}_{{\overline{s}}^{\prime }}$$

  and path induce a map of fundamental groups

  $$\gamma (\cdot ){\gamma }^{-1}:{\pi }_1(S,\overline{s})\to {\pi }_1(S,{\overline{s}}^{\prime })$$