A morphism $\operatorname{Spec}(L) \to \operatorname{Spec}(k)$ is etale if and only if $L/k$ is a finite separable field extension (or a finite product of such). The map $\operatorname{Spec}(\mathbb{Q}(\sqrt{2})) \to \operatorname{Spec}(\mathbb{Q})$ is etale. The map $\operatorname{Spec}(\mathbb{F}_p(t^{1/p})) \to \operatorname{Spec}(\mathbb{F}_p(t))$ is not etale (it is purely inseparable).

More generally, $\operatorname{Spec}(B) \to \operatorname{Spec}(A)$ is finite etale if and only if $B$ is a finite projective $A$-module and the trace form $B \otimes_A B \to A$ is non-degenerate.

The morphism $\operatorname{Spec}(\mathbb{Z}[t]/(t^2 - 5)) \to \operatorname{Spec}(\mathbb{Z})$ is etale away from the prime $2$ (since the derivative $2t$ is invertible away from $2$) but is not etale at $2$ (the polynomial $t^2 - 5 \equiv t^2 - 1 = (t-1)(t+1) \pmod{2}$ has a repeated root modulo $2$... actually, $(t-1)$ and $(t+1)$ are the same modulo $2$, so $t^2 - 5 \equiv (t-1)^2 \pmod{2}$).

So $\operatorname{Spec}(\mathbb{Z}[t]/(t^2 - 5)[1/2]) \to \operatorname{Spec}(\mathbb{Z}[1/2])$ is finite etale of degree 2. The fiber over a prime $p \neq 2$ splits into two points if $5$ is a square modulo $p$ (i.e., the Legendre symbol $(5/p) = 1$).

Every open immersion $U \hookrightarrow X$ is etale (it is flat, unramified, and locally of finite presentation). In fact, open immersions are precisely the etale morphisms that are also monomorphisms.

Conversely, an etale morphism of finite type between integral schemes with the same function field is an open immersion (by Zariski's main theorem).

Let $C$ be a smooth projective curve over an algebraically closed field $k$, and let $f: D \to C$ be a finite morphism of smooth curves. Then $f$ is etale if and only if $f$ is unramified, i.e., the ramification divisor $R = \sum_{P \in D} (e_P - 1) \cdot P$ is zero (every point has ramification index $1$).

By the Hurwitz formula $2g(D) - 2 = \deg(f)(2g(C) - 2) + \deg(R)$, an etale cover of degree $d$ satisfies $g(D) - 1 = d(g(C) - 1)$. For $C = \mathbb{P}^1$ (genus $0$), no nontrivial etale covers exist when $\operatorname{char}(k) = 0$ (since $g(D) - 1 = d(0-1) = -d < 0$). Over $\operatorname{Spec}(\mathbb{Z})$, Minkowski's theorem says there are no nontrivial unramified extensions, reflecting the same phenomenon.

A local ring $(R, \mathfrak{m})$ is henselian if Hensel's lemma holds: for every monic polynomial $f \in R[t]$, if $\bar{f} = \bar{g}\bar{h}$ in $(R/\mathfrak{m})[t]$ with $\gcd(\bar{g}, \bar{h}) = 1$, then $f = gh$ with $g, h$ lifting $\bar{g}, \bar{h}$.

Equivalently, $(R, \mathfrak{m})$ is henselian if and only if every finite $R$-algebra is a product of local rings, if and only if every etale neighborhood of the closed point $\operatorname{Spec}(R/\mathfrak{m}) \hookrightarrow \operatorname{Spec}(R)$ has a section.

The strict henselization $R^{\text{sh}}$ is the colimit of all local-etale $R$-algebras, and the stalk of an etale sheaf $\mathcal{F}$ at a geometric point over $\mathfrak{m}$ is $\mathcal{F}_{\bar{x}} = \varinjlim \mathcal{F}(\operatorname{Spec}(R'))$ where $R'$ runs over etale neighborhoods, converging to $\mathcal{F}(\operatorname{Spec}(R^{\text{sh}}))$.

Let $E$ be an elliptic curve over a field $k$. For each integer $n$ with $\gcd(n, \operatorname{char}(k)) = 1$, the multiplication-by-$n$ map $[n]: E \to E$ is a finite etale morphism of degree $n^2$. Its kernel is $E[n] \cong (\mathbb{Z}/n\mathbb{Z})^2$ (over $\bar{k}$).

So $\{[n]: E \to E\}$ is an etale cover. The induced map on etale fundamental groups gives the full Tate module: $\pi_1^{\text{et}}(E) \cong \hat{\mathbb{Z}}^2 \times (\text{Galois part})$.

The etale covers of $E$ are classified by the etale fundamental group, which encodes both the geometric covers and the Galois action.

For $\mathbb{A}^n_k$ over an algebraically closed field $k$ of characteristic $0$, the etale fundamental group is trivial: $\pi_1^{\text{et}}(\mathbb{A}^n_k) = 1$. Every finite etale cover is trivial.

In characteristic $p > 0$, this fails dramatically: the Artin-Schreier cover $\operatorname{Spec}(k[x,y]/(y^p - y - x)) \to \mathbb{A}^1_k$ is a connected etale $\mathbb{Z}/p\mathbb{Z}$-cover. The etale fundamental group of $\mathbb{A}^1_{\overline{\mathbb{F}_p}}$ is huge (it surjects onto every finite group generated by at most one element at each prime-to-$p$ part, by Abhyankar's conjecture, proved by Raynaud and Harbater).

The morphism $\operatorname{Spec}(A[t]) \to \operatorname{Spec}(A)$ is flat (and faithfully flat) for any ring $A$, since $A[t]$ is a free $A$-module. The morphism $\operatorname{Spec}(A[t]/(t^n)) \to \operatorname{Spec}(A)$ is flat (and faithfully flat) since $A[t]/(t^n)$ is free of rank $n$ as an $A$-module.

More generally, any morphism corresponding to a flat ring map $A \to B$ gives a flat morphism of schemes. Flatness is the scheme-theoretic analogue of "having fibers of constant dimension" (though this is a rough heuristic).

Over $\mathbb{F}_p$, the Frobenius morphism $F: \mathbb{A}^1 \to \mathbb{A}^1$ given by $x \mapsto x^p$ corresponds to the ring map $\mathbb{F}_p[t] \to \mathbb{F}_p[t]$ sending $t \mapsto t^p$. This makes $\mathbb{F}_p[t]$ a free $\mathbb{F}_p[t]$-module of rank $p$ (via the basis $1, t, t^2, \ldots, t^{p-1}$).

So the Frobenius is finite flat of degree $p$, hence an fppf cover $\{F: \mathbb{A}^1 \to \mathbb{A}^1\}$. However, it is not etale: it is totally inseparable (the fiber over any closed point is a single point with multiplicity $p$).

This is a key example showing what fppf detects that etale does not: inseparable maps and group schemes like $\alpha_p = \ker(F: \mathbb{G}_a \to \mathbb{G}_a)$.

Over a field $k$ of characteristic $p$, the group scheme $\mu_p = \operatorname{Spec}(k[t]/(t^p - 1)) = \operatorname{Spec}(k[t]/((t-1)^p))$ is not reduced (and not etale over $k$). The morphism $[p]: \mathbb{G}_m \to \mathbb{G}_m$ given by $t \mapsto t^p$ has kernel $\mu_p$.

The sequence $1 \to \mu_p \to \mathbb{G}_m \xrightarrow{[p]} \mathbb{G}_m \to 1$ is exact in the fppf topology but not in the etale topology. Over a field $k$, $H^1_{\text{fppf}}(k, \mu_p)$ classifies $\mu_p$-torsors, which correspond to degree-$p$ purely inseparable extensions and Brauer-Severi varieties in certain cases. The etale cohomology $H^1_{\text{et}}(k, \mu_p)$ gives $k^\times/(k^\times)^p$ only when $\mu_p$ is etale (i.e., when $p$ is invertible in $k$, which contradicts $\operatorname{char}(k) = p$).

The Brauer group $\operatorname{Br}(X)$ can be defined as $H^2_{\text{et}}(X, \mathbb{G}_m)$ (etale cohomology) or as the group of Azumaya algebras modulo Brauer equivalence.

The etale and fppf $H^2$ agree for $\mathbb{G}_m$: $H^2_{\text{et}}(X, \mathbb{G}_m) = H^2_{\text{fppf}}(X, \mathbb{G}_m)$. This is because $\mathbb{G}_m$ is smooth, so the comparison theorem applies.

For a field $k$, $\operatorname{Br}(k)$ classifies central simple algebras over $k$ up to Morita equivalence. For example, $\operatorname{Br}(\mathbb{R}) = \mathbb{Z}/2\mathbb{Z}$ (generated by the quaternions $\mathbb{H}$), $\operatorname{Br}(\mathbb{F}_q) = 0$ (Wedderburn), and $\operatorname{Br}(\mathbb{Q}_p) = \mathbb{Q}/\mathbb{Z}$ (local class field theory).

For any scheme $X$, the Picard group $\operatorname{Pic}(X) = H^1_{\text{et}}(X, \mathbb{G}_m) = H^1_{\text{Zar}}(X, \mathbb{G}_m) = H^1_{\text{fppf}}(X, \mathbb{G}_m)$. All three topologies give the same answer.

This equality for $H^1$ is essentially Hilbert's Theorem 90: for the etale case, it says $H^1_{\text{et}}(\operatorname{Spec}(L/k), \mathbb{G}_m) = 0$ for any Galois extension $L/k$. For the fppf case, it generalizes to say that every $\mathbb{G}_m$-torsor in the fppf topology is already locally trivial in the Zariski topology.

However, for higher cohomology, the topologies may differ. For example, $H^3_{\text{et}}$ and $H^3_{\text{fppf}}$ can differ for non-smooth coefficient sheaves.

Consider the etale cover $\{f: \operatorname{Spec}(k[t, t^{-1}]) \to \operatorname{Spec}(k[t^2, t^{-2}])\}$ given by the inclusion $k[t^2, t^{-2}] \hookrightarrow k[t, t^{-1}]$ (adjoin a square root of $t^2$). This is a degree-2 etale cover of $\mathbb{G}_m$ by $\mathbb{G}_m$.

The fiber product is $\operatorname{Spec}(k[t, t^{-1}]) \times_{\operatorname{Spec}(k[t^2, t^{-2}])} \operatorname{Spec}(k[t, t^{-1}]) = \operatorname{Spec}(k[t, t^{-1}] \otimes_{k[t^2, t^{-2}]} k[t, t^{-1}])$. The tensor product $k[t, t^{-1}] \otimes_{k[s, s^{-1}]} k[t, t^{-1}]$ (where $s = t^2$) is $k[t, t^{-1}, u, u^{-1}]/(t^2 - u^2) = k[t, t^{-1}, u, u^{-1}]/((t-u)(t+u))$, which splits as $k[t, t^{-1}] \times k[t, t^{-1}]$ (identifying $u = t$ or $u = -t$).

This is characteristic of a Galois cover: the fiber product over itself splits as copies indexed by the Galois group $\mathbb{Z}/2\mathbb{Z}$.

Let $G = \mu_2 = \operatorname{Spec}(k[t]/(t^2-1))$ over a field $k$ with $\operatorname{char}(k) \neq 2$. A $\mu_2$-torsor over $\operatorname{Spec}(k)$ is a scheme $P$ with a $\mu_2$-action that is locally trivial. These are classified by $H^1_{\text{fppf}}(k, \mu_2) = H^1_{\text{et}}(k, \mu_2) = k^\times / (k^\times)^2$.

Concretely, each class $a \in k^\times / (k^\times)^2$ gives the torsor $P_a = \operatorname{Spec}(k[x]/(x^2 - a))$ with $\mu_2$-action $t \cdot x = tx$ (where $t^2 = 1$). The torsor $P_a$ is trivial (isomorphic to $\mu_2$ itself) if and only if $a$ is a square in $k$.

For $k = \mathbb{Q}$, $H^1(\mathbb{Q}, \mu_2) = \mathbb{Q}^\times / (\mathbb{Q}^\times)^2$, which is infinite (generated by $-1$ and the primes $p$).