Throughout, $X$ is a smooth projective variety of dimension $n$ over $\mathbb{F}_q$, and $F: X \to X$ denotes the $q$-power Frobenius morphism. Fix a projective embedding; let $h \in \operatorname{CH}^1(X)$ be the hyperplane class and $L = h \cup (-) : H^i(X) \to H^{i+2}(X)$ the Lefschetz operator.
The eigenvalues of $F^*$ acting on $H^k_{\operatorname{\acute{e}t}}(X_{\bar{\mathbb{F}}_q},\,\mathbb{Q}_\ell)$ all have absolute value $q^{k/2}$.
We carry over three structural results from Hodge theory (transferred to the $\ell$-adic setting via comparison with Betti cohomology):
For $0 \leq k \leq n$, the cup-product map
is an isomorphism. In particular the Betti numbers satisfy $b_{n-k} = b_{n+k}$.
A class $\alpha \in H^k(X)$ is primitive if $L^{n-k+1}\alpha = 0$. Define the pairing
Then $(-1)^p \cdot Q$ is positive definite on the primitive piece $H^{p,q}_{\mathrm{prim}}$ with $p+q=k$. In particular, for any nonzero primitive $\alpha$ one has $Q(\alpha,\bar\alpha) \neq 0$.
Because Frobenius is a degree-$q$ map on each coordinate, it pulls back the hyperplane class by $q$: $$F^*(h) = q\cdot h \quad\Longrightarrow\quad F^* \circ L \;=\; q\cdot L \circ F^*.$$ The operators $L$ and $F^*$ nearly commute — they differ by a factor of $q$.
It suffices to show that every $F^*$-eigenvalue on $H^n(X)$ has absolute value $q^{n/2}$; the general case follows from Hard Lefschetz.
Let $\alpha \in H^k(X)$ be an $F^*$-eigenvector with $F^*\alpha = \lambda\alpha$ and $k \leq n$. By Hard Lefschetz, $\beta := L^{n-k}\alpha \in H^{2n-k}(X)$ is nonzero. The key relation $F^*\circ L = q\,L\circ F^*$ gives:
So $\beta$ is an eigenvector in $H^{2n-k}$ with eigenvalue $q^{n-k}\lambda$. If the RH holds in degree $2n-k$, then $|q^{n-k}\lambda| = q^{(2n-k)/2}$, which gives $|\lambda| = q^{k/2}$. Poincaré duality handles the case $k > n$ symmetrically.
Take a primitive class $\alpha \in H^{p,q}(X)$ with $p + q = n$ and $F^*\alpha = \lambda\cdot\alpha$. We want $|\lambda|^2 = q^n$.
Frobenius acts on $H^{2n}(X) \cong \mathbb{Q}_\ell(-n)$ by multiplication by $q^n$ (since $F^*$ acts on $\mathbb{Q}_\ell(1)$ by $q^{-1}$, hence on $\mathbb{Q}_\ell(-n)$ by $q^n$). The Poincaré pairing $\langle\cdot,\cdot\rangle : H^n \times H^n \to H^{2n}$ then satisfies
Setting $\alpha' = \bar\alpha$ (complex conjugate via the Betti comparison isomorphism):
The Hodge Index Theorem guarantees $Q(\alpha,\bar\alpha) \neq 0$ since $\alpha$ is a nonzero primitive class. Dividing both sides:
This is Serre’s approach: it uses the Hodge-theoretic comparison isomorphism (valid over $\mathbb{C}$) in an essential way. Deligne’s full proof (1974) works directly over $\mathbb{F}_q$ via the theory of weights in $\ell$-adic sheaves. Serre’s argument nonetheless captures the geometric heart of why the Riemann Hypothesis holds.
To build étale cohomology we need an algebraic replacement for the classical notion of “open cover.” The key players are étale morphisms — the algebraic geometry analogue of local diffeomorphisms.
A morphism $f: X \to Y$ (locally of finite presentation) is unramified if the sheaf of relative Kähler differentials vanishes:
Equivalently: For every point $x \in X$ with $y = f(x)$, the residue field extension $\kappa(y) \hookrightarrow \kappa(x)$ is finite and separable, and $\mathfrak{m}_y\cdot\mathcal{O}_{X,x} = \mathfrak{m}_x$.
A morphism $f: X \to Y$ is étale if it is simultaneously:
The map $\operatorname{Spec}\,R[t]/(g)[1/g'] \to \operatorname{Spec}\,R$ “extracts a root of $g$” and inverts the derivative. The condition that $g'$ is a unit modulo $g$ means exactly that $g$ has no repeated roots in any fiber. Étale morphisms are precisely those locally of this form.
Any open immersion $j: U \hookrightarrow X$ is étale. It is flat and unramified (residue fields are unchanged). Open immersions are the prototypical étale morphisms.
The map $[n]: \mathbb{G}_m \to \mathbb{G}_m$, $t \mapsto t^n$, is étale when $n$ is invertible in the base. Here $g(s) = s^n - t$ with derivative $ns^{n-1}$, a unit when $n \in R^\times$.
If $L/k$ is a finite separable field extension, then $\operatorname{Spec}\,L \to \operatorname{Spec}\,k$ is étale (finite étale). Separability is exactly the unramifiedness condition.
The standard étale cover extracting an $n$-th root: $$\operatorname{Spec}\,\mathbb{Z}[1/n][t^{\pm1}]/(t^n - a)\;\to\;\operatorname{Spec}\,\mathbb{Z}[1/n].$$ Inverting $n$ ensures $nt^{n-1}$ is a unit.
The open immersion $\mathbb{G}_m \hookrightarrow \mathbb{A}^1$ is étale but not proper, hence not finite étale. Finite étaleness requires properness in addition.
The absolute Frobenius $F_X : X \to X$ in characteristic $p$ is never étale. Residue field extensions $\kappa(x)^p \hookrightarrow \kappa(x)$ are purely inseparable — the opposite of separable.
Étale morphisms are stable under:
Let $X$ and $Y$ be varieties over an algebraically closed field $k$, and let $f: X \to Y$ be étale. For a closed point $x \in X$, with $y = f(x)$, the induced map on complete local rings is an isomorphism:
Under these hypotheses, this is the algebraic analogue of “local diffeomorphisms induce isomorphisms on completions.”
Let $R$ be a ring and consider a standard étale $R$-algebra
where $g \in R[t]$ is monic and the image of $g'$ in $B$ is a unit. Prove that
is étale. What to check: locally of finite presentation, flat, and unramified.
Assume $n$ is invertible in the base. Prove that
is étale. Hint: $\frac{d(t^n)}{dt} = n t^{n-1}$ (over a field $k$, this condition is $\operatorname{char}(k) \nmid n$).
For composable morphisms
prove the 2-out-of-3 assertion stated above: if $\varphi$ and $\varphi \circ \psi$ are étale, then $\psi$ is étale.
By Stacks Project, Tag 0616, it is enough to prove that $\psi$ is locally of finite presentation and formally étale.
Locally of finite presentation. By Stacks Project, Tag 01TQ, this can be checked on affine open neighborhoods. Fix $x\in X$ and choose affine opens $U=\operatorname{Spec}C$, $V=\operatorname{Spec}B$, and $W=\operatorname{Spec}A$ around $x$, $\psi(x)$, and $\varphi(\psi(x))$, respectively, such that $\psi(U)\subset V$ and $\varphi(V)\subset W$. Since $\varphi\circ\psi$ is étale, $A\to C$ is of finite presentation. Since $\varphi$ is étale, $A\to B$ is of finite presentation, hence of finite type. Applied to
assertion (4) of Stacks Project, Tag 00F4 shows that $B\to C$ is of finite presentation. Tag 01TQ now implies that $\psi$ is locally of finite presentation.
Formally étale. Use the lifting criterion of Tag 049S. Let $i:T_0\hookrightarrow T$ be a closed immersion of affine schemes defined by a square-zero ideal, and consider a commutative diagram
After composing $b$ with $\varphi$, formal étaleness of $\varphi\circ\psi$ gives a unique map $a:T\to X$ extending $a_0$ such that
The maps $\psi\circ a$ and $b$ from $T$ to $Y$ agree on $T_0$ and have the same composite with $\varphi$. They are therefore two lifts in the square-zero lifting problem for $\varphi$. Since $\varphi$ is formally étale, uniqueness gives $\psi\circ a=b$. Thus $a$ is a lift for the original diagram. It is unique because any such lift is also a lift for $\varphi\circ\psi$, whose formal étaleness gives uniqueness. Hence $\psi$ is formally étale.
Therefore $\psi$ is locally of finite presentation and formally étale, so Tag 0616 implies that $\psi$ is étale.
AI reference: ChatGPT conversation used to refine the presentation of this proof.
Let $k$ be an algebraically closed field, let $X$ and $Y$ be varieties over $k$, and let $f: X \to Y$ be étale. If $x \in X$ is a closed point and $y = f(x)$, prove that the induced map
is an isomorphism. Hint: use the criterion/formal lifting property for formal étaleness, applying it to successive infinitesimal neighborhoods.
A sheaf on a topological space $X$ assigns sections to open sets such that local sections agree on overlaps and glue uniquely. Grothendieck observed that to set this up, you need only three ingredients:
A Grothendieck topology on a category $\mathcal{C}$ assigns, to each object $U$, a set of covering families $\{U_i \to U\}_{i \in I}$, satisfying:
A site is a pair $(\mathcal{C},\tau)$ of a category with a Grothendieck topology.
For a topological space $X$: take $\mathcal{C} = \mathbf{Open}(X)$ (open subsets, ordered by inclusion) and declare $\{U_i \hookrightarrow U\}$ a cover iff $\bigcup_i U_i = U$. Sheaves on this site are exactly the classical sheaves on $X$.
For a smooth manifold $M$: take $\mathcal{C}$ to consist of open subsets $U \subset M$ admitting a diffeomorphism $U \cong \mathbb{R}^n$, with covering families being smooth atlases (surjective families of local diffeomorphisms). This site sees the smooth structure, not just the topology.
For a scheme $X$, the small étale site has:
Every Zariski open cover is an étale cover (open immersions are étale), but the étale site has many more covers, including finite separable field extensions of residue fields.
| Site | Category $\mathcal{C}$ | Covering families | What it sees |
|---|---|---|---|
| Zariski | Open subsets of $X$ | Surjective open covers | Geometry of opens; misses arithmetic |
| Manifold / smooth | Coordinate patches $U \cong \mathbb{R}^n$ | Smooth atlases | Smooth and analytic structure |
| Étale $X_{\operatorname{\acute{e}t}}$ | Étale $X$-schemes $U \to X$ | Surjective étale families | Separable extensions; arithmetic of residue fields |
A presheaf on a site $(\mathcal{C},\tau)$ is a contravariant functor $\mathcal{F}: \mathcal{C}^{\mathrm{op}} \to \mathbf{Ab}$ (or sets, rings, …). It is a sheaf if for every covering family $\{U_i \to U\}$, the sequence
is an equalizer: sections on $U$ are uniquely determined by compatible sections on the covering pieces.
The Zariski topology is too coarse: constant sheaves like $\mathbb{Z}/n\mathbb{Z}$ have zero cohomology in all positive degrees on any connected normal scheme, completely missing the topological content. The étale topology is the minimal refinement that (1) makes finite separable extensions into covers so the topology sees arithmetic, and (2) still supports a workable cohomological formalism. Over $\mathbb{C}$, étale cohomology of constant sheaves recovers singular cohomology.
With sites and sheaves in hand, we will define étale cohomology $H^i_{\operatorname{\acute{e}t}}(X, \mathcal{F})$ as the right derived functors of global sections on the étale site. The key comparison theorem:
for $X$ smooth and proper over $\mathbb{C}$. This is what justifies étale cohomology as the correct algebro-geometric replacement for singular cohomology over arbitrary fields.