64.27 Fundamental groups

This material is discussed in more detail in the chapter on fundamental groups. See Fundamental Groups, Section 58.1. Let $X$ be a connected scheme and let $\overline{x}\to X$ be a geometric point. Consider the functor

$\begin{matrix} F_{\overline{x}}: & \text{ finite étale } \atop \text{ schemes over } X & \longrightarrow & \text{ finite sets} \\ & Y/X & \longmapsto & F_{\overline{x}}(Y) = \left\{ \text{ geom points }\overline y \atop \text{ of } Y \text{ lying over }\overline{x}\right\} = Y_{\overline{x}} \end{matrix}$

Set

$\pi _1(X, \overline{x}) = Aut(F_{\overline{x}}) = \text{ set of automorphisms of the functor }F_{\overline{x}}$

Note that for every finite étale $Y \to X$ there is an action

$\pi _1(X, \overline{x}) \times F_{\overline{x}}(Y) \to F_{\overline{x}}(Y)$

Definition 64.27.1. A subgroup of the form $\text{Stab}(\overline y\in F_{\overline{x}}(Y))\subset \pi _1(X, \overline{x})$ is called open.

Theorem 64.27.2 (Grothendieck). Let $X$ be a connected scheme.

1. There is a topology on $\pi _1(X, \overline{x})$ such that the open subgroups form a fundamental system of open nbhds of $e\in \pi _1(X, \overline x)$.

2. With topology of (1) the group $\pi _1(X, \overline{x})$ is a profinite group.

3. The functor

$\begin{matrix} \text{ schemes finite } \atop \text{ étale over }X & \to & \text{ finite discrete continuous } \atop \pi _1(X, \overline{x})\text{-sets} \\ Y / X & \mapsto & F_{\overline{x}}(Y) \text{ with its natural action} \end{matrix}$

is an equivalence of categories.

Proof. See [SGA1]. $\square$

Proposition 64.27.3. Let $X$ be an integral normal Noetherian scheme. Let $\overline y\to X$ be an algebraic geometric point lying over the generic point $\eta \in X$. Then

$\pi _ x(X, \overline\eta ) = Gal(M/\kappa (\eta ))$

($\kappa (\eta )$, function field of $X$) where

$\kappa (\overline\eta )\supset M\supset \kappa (\eta ) = k(X)$

is the max sub-extension such that for every finite sub extension $M\supset L\supset \kappa (\eta )$ the normalization of $X$ in $L$ is finite étale over $X$.

Proof. Omitted. $\square$

Change of base point. For any $\overline{x}_1, \overline{x}_2$ geom. points of $X$ there exists an isom. of fibre functions

$\mathcal{F}_{\overline{x}_1} \cong \mathcal{F}_{\overline{x}_2}$

(This is a path from $\overline{x}_1$ to $\overline{x}_2$.) Conjugation by this path gives isom

$\pi _1(X, \overline{x}_1) \cong \pi _1(X, \overline{x}_2)$

well defined up to inner actions.

Functoriality. For any morphism $X_1\to X_2$ of connected schemes any $\overline{x}\in X_1$ there is a canonical map

$\pi _1(X_1, \overline{x}) \to \pi _1(X_2, \overline{x})$

(Why? because the fibre functor ...)

Base field. Let $X$ be a variety over a field $k$. Then we get

$\pi _1(X, \overline{x}) \to \pi _1(Spec(k), \overline{x}) =^{\text{prop}} Gal(k^{\text{sep}}/k)$

This map is surjective if and only if $X$ is geometrically connected over $k$. So in the geometrically connected case we get s.e.s. of profinite groups

$1 \to \pi _1(X_{\overline{k}}, \overline{x}) \to \pi _1(X, \overline{x}) \to Gal(k^{\text{sep}}/k) \to 1$

($\pi _1(X_{\overline{k}}, \overline{x})$: geometric fundamental group of $X$, $\pi _1(X, \overline{x})$: arithmetic fundamental group of $X$)

Comparison. If $X$ is a variety over $\mathbf{C}$ then

$\pi _1(X, \overline{x}) = \text{ profinite completion of } \pi _1(X(\mathbf{C})(\text{ usual topology}), x)$

(have $x\in X(\mathbf{C})$)

Frobenii. $X$ variety over $k$, $\# k < \infty$. For any $x \in X$ closed point, let

$F_ x\in \pi _1(x, \overline{x}) = \text{Gal}(\kappa (x)^{\text{sep}}/\kappa (x))$

be the geometric frobenius. Let $\overline\eta$ be an alg. geom. gen. pt. Then

$\pi _1(X, \overline\eta ) \leftarrow ^{\cong } \pi _1(X, \overline{x}) {\text{functoriality} \atop \leftarrow } \pi _1(x, \overline{x})$

Easy fact:

$\begin{matrix} \pi _1(X, \overline\eta ) & \to ^{\deg } \pi _1(\mathop{\mathrm{Spec}}(k), \overline\eta ) * & = Gal(k^{sep}/k) \\ & & || \\ & & \widehat{\mathbf{Z}}\cdot F_{\mathop{\mathrm{Spec}}(k)} \\ F_ x & \mapsto & \deg (x)\cdot F_{\mathop{\mathrm{Spec}}(k)} \end{matrix}$

Recall: $\deg (x) = [\kappa (x):k]$

Fundamental groups and lisse sheaves. Let $X$ be a connected scheme, $\overline{x}$ geom. pt. There are equivalences of categories

$\begin{matrix} (\Lambda \text{ finite ring}) & \text{fin. loc. const. sheaves of } \atop \Lambda \text{-modules of }X_{\acute{e}tale} & \leftrightarrow & \text{ finite (discrete) }\Lambda \text{-modules} \atop \text{ with continuous }\pi _1(X, \overline{x})\text{-action} \\ (\ell \text{ a prime}) & \text{ lisse }\ell \text{-adic} \atop \text{ sheaves} & \leftrightarrow & \text{finitely generated }\mathbf{Z}_\ell \text{-modules }M\text{ with continuous} \atop \pi _1(X, \overline{x})\text{-action where we use } \ell \text{-adic topology on }M \end{matrix}$

In particular lisse $\mathbf{Q}_ l$-sheaves correspond to continuous homomorphisms

$\pi _1(X, \overline{x}) \to \text{GL}_ r(\mathbf{Q}_ l), \quad r\geq 0$

Notation: A module with action $(M, \rho )$ corresponds to the sheaf $\mathcal{F}_\rho$.

Trace formulas. $X$ variety over $k$, $\# k < \infty$.

1. $\Lambda$ finite ring $(\# \Lambda , \# k)=1$

$\rho : \pi _1(X, \overline{x})\to \text{GL}_ r(\Lambda )$

continuous. For every $n\geq 1$ we have

$\sum _{d|n}d\left( \sum _{x\in |X|, \atop \deg (x)=d} \text{Tr}(\rho (F_ x^{n/d}))\right) = \text{Tr}\left( (\pi _ x^ n)^* |_{R\Gamma _ c(X_{\overline{k}}, \mathcal{F}_\rho )}\right)$
2. $l\neq char(k)$ prime, $\rho : \pi _1(X, \overline{x})\to \text{GL}_ r(\mathbf{Q}_ l)$. For any $n\geq 1$

$\sum _{d|n} d\left( \sum _{x\in |X| \atop \deg (x)=d} \text{Tr} \left( \rho (F_ x^{n/d}) \right) \right) = \sum _{i = 0}^{2\dim X} (-1)^ i \text{Tr}\left( \pi _ X^* |_{H_ c^ i(X_{\overline{k}}, \mathcal{F}_\rho )}\right)$

Weil conjectures. (Deligne-Weil I, 1974) $X$ smooth proj. over $k$, $\# k = q$, then the eigenvalues of $\pi _ X^*$ on $H^ i(X_{\overline{k}}, \mathbf{Q}_ l)$ are algebraic integers $\alpha$ with $|\alpha |=q^{1/2}$.

Deligne's conjectures. (almost completely proved by Lafforgue + $\ldots$) Let $X$ be a normal variety over $k$ finite

$\rho : \pi _1(X, \overline{x}) \longrightarrow \text{GL}_ r(\mathbf{Q}_ l)$

continuous. Assume: $\rho$ irreducible $\det (\rho )$ of finite order. Then

1. there exists a number field $E$ such that for all $x\in |X|$(closed points) the char. poly of $\rho (F_ x)$ has coefficients in $E$.

2. for any $x\in |X|$ the eigenvalues $\alpha _{x, i}$, $i = 1, \ldots , r$ of $\rho (F_ x)$ have complex absolute value $1$. (these are algebraic numbers not necessary integers)

3. for every finite place $\lambda$( not dividing $p$), of $E$ (maybe after enlarging $E$ a bit) there exists

$\rho \lambda : \pi _1(X, \overline{x}) \to \text{GL}_ r(E_\lambda )$

compatible with $\rho$. (some char. polys of $F_ x$'s)

Theorem 64.27.4 (Deligne, Weil II). For a sheaf $\mathcal{F}_\rho$ with $\rho$ satisfying the conclusions of the conjecture above then the eigenvalues of $\pi _ X^*$ on $H_ c^ i(X_{\overline{k}}, \mathcal{F}_{\rho })$ are algebraic numbers $\alpha$ with absolute values

$|\alpha |=q^{w/2}, \text{ for }w\in \mathbf{Z},\ w\leq i$

Moreover, if $X$ smooth and proj. then $w = i$.

Proof. See [WeilII]. $\square$

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