Let $\mathcal{C} = \mathbf{Sch}/S$ be the category of $S$-schemes. The Zariski pretopology declares that a family $\{U_i \to U\}_{i \in I}$ is a covering family if each $U_i \to U$ is an open immersion and $U = \bigcup_i \operatorname{im}(U_i)$.

For $U = \operatorname{Spec}(A)$, a Zariski cover corresponds to a collection of principal open sets $D(f_i)$ with $(f_1, \ldots, f_n) = A$, i.e., the $f_i$ generate the unit ideal.

This recovers the classical Zariski topology: the covering families are precisely the open covers.

The etale pretopology on $\mathbf{Sch}/S$ declares $\{U_i \to U\}$ to be a covering family if each $U_i \to U$ is etale and $\coprod_i U_i \to U$ is surjective.

Recall that a morphism $f: X \to Y$ of finite presentation is etale if it is flat and unramified, or equivalently if for every $y \in Y$, the fiber $X_y$ is a disjoint union of spectra of separable field extensions of $k(y)$.

For $U = \operatorname{Spec}(k)$ where $k$ is a field, etale covers are of the form $\{\operatorname{Spec}(L_i) \to \operatorname{Spec}(k)\}$ where each $L_i/k$ is a finite separable extension and $\prod L_i$ is a faithfully flat $k$-algebra.

The fppf pretopology (fidelement plat de presentation finie) declares $\{U_i \to U\}$ to be a covering family if each $U_i \to U$ is flat and locally of finite presentation, and $\coprod_i U_i \to U$ is surjective.

This is strictly finer than the etale pretopology: every etale cover is an fppf cover, but not conversely. For instance, $\operatorname{Spec}(k[\epsilon]/(\epsilon^p)) \to \operatorname{Spec}(k)$ over a field of characteristic $p$ is flat and finitely presented (hence an fppf cover) but not etale (it is purely inseparable).

The fppf topology is needed to study non-smooth group schemes and torsors correctly.

The fpqc pretopology (fidelement plat et quasi-compact) declares $\{U_i \to U\}$ to be a covering family if each $U_i \to U$ is flat and $\coprod_i U_i \to U$ is surjective and quasi-compact.

This is the finest of the standard algebraic topologies. Every fppf cover is an fpqc cover. Additionally, any faithfully flat morphism $V \to U$ that is quasi-compact gives a single-morphism cover $\{V \to U\}$.

The fpqc topology is the natural home for faithfully flat descent, but it is technically harder to work with because fpqc morphisms need not be of finite presentation.

Given a covering family $\{U_i \to U\}_{i \in I}$ (in some pretopology), the sieve generated by this family is

$S(V) = \{f: V \to U \mid f \text{ factors through some } U_i \to U\}.$

More precisely, $f \in S(V)$ if there exists $i \in I$ and a morphism $g: V \to U_i$ such that $f = (U_i \to U) \circ g$. This is indeed closed under precomposition: if $h: W \to V$ then $f \circ h$ factors through $U_i$ via $g \circ h$.

The passage from covering families to sieves is how a pretopology generates a full Grothendieck topology.

For any object $U$:

• The maximal sieve is $h_U$ itself: $S(V) = \operatorname{Hom}(V, U)$ for all $V$. Every morphism to $U$ belongs to this sieve.
• The empty sieve is $S(V) = \emptyset$ for all $V$. No morphism to $U$ belongs to this sieve.

If $\mathcal{C}$ has a terminal object $*$, then the maximal sieve on $*$ is the entire category (every object maps to $*$).

Let $X$ be a topological space and $\mathcal{C} = \operatorname{Open}(X)$ the category of open subsets (with inclusions). For an open set $U$, a sieve $S$ on $U$ consists of a collection of open subsets of $U$, closed under taking smaller opens. Define $S \in J(U)$ if and only if $\bigcup_{V \in S} V = U$.

This recovers the classical sheaf theory on $X$. The pretopology version says: a covering family of $U$ is an open cover $\{U_i \subseteq U\}$ with $\bigcup U_i = U$.

This shows that classical topology is a special case of Grothendieck topology.

The Nisnevich topology on $\mathbf{Sch}/k$ (for a field $k$) uses etale morphisms as covers, but with a stronger condition than surjectivity. A family $\{U_i \to U\}$ is a Nisnevich cover if each $U_i \to U$ is etale and for every point $x \in U$, there exists $i$ and a point $y \in U_i$ mapping to $x$ with $k(x) \xrightarrow{\sim} k(y)$ (an isomorphism on residue fields).

The Nisnevich topology sits strictly between Zariski and etale: $\text{Zariski} \subsetneq \text{Nisnevich} \subsetneq \text{etale} \subsetneq \text{fppf} \subsetneq \text{fpqc}$

It is the natural topology for algebraic K-theory and motivic homotopy theory (Morel--Voevodsky $\mathbb{A}^1$-homotopy theory).

The cdh topology is generated by Nisnevich covers together with abstract blowup squares: if $Z \hookrightarrow X$ is a closed immersion and $\pi: X' \to X$ is a proper morphism such that $\pi^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism, then $\{X' \to X, Z \to X\}$ is a cdh cover.

This topology is used in the study of algebraic K-theory of singular schemes, particularly in the work of Voevodsky on motivic cohomology.

Consider $X = \operatorname{Spec}(\mathbb{R})$. In the Zariski topology, $X$ is a single point with no nontrivial covers. But in the etale topology, the morphism $\operatorname{Spec}(\mathbb{C}) \to \operatorname{Spec}(\mathbb{R})$ is an etale cover (it is finite etale of degree 2).

The etale fundamental group $\pi_1^{\text{et}}(\operatorname{Spec}(\mathbb{R}))$ is $\operatorname{Gal}(\mathbb{C}/\mathbb{R}) \cong \mathbb{Z}/2\mathbb{Z}$, detecting the nontrivial extension $\mathbb{C}/\mathbb{R}$. The Zariski topology sees nothing.

Similarly, for $X = \operatorname{Spec}(\mathbb{F}_q)$, the etale fundamental group is $\hat{\mathbb{Z}}$ (the profinite completion of $\mathbb{Z}$), generated by the Frobenius. Etale cohomology with $\ell$-adic coefficients on varieties over $\mathbb{F}_q$ is the key to the Weil conjectures.

Over a field $k$ of characteristic $p > 0$, consider the group scheme $\alpha_p = \operatorname{Spec}(k[t]/(t^p))$ with the additive group structure. The map $x \mapsto x^p$ (Frobenius) $\mathbb{G}_a \to \mathbb{G}_a$ has kernel $\alpha_p$.

The sequence $0 \to \alpha_p \to \mathbb{G}_a \xrightarrow{F} \mathbb{G}_a \to 0$ is exact in the fppf topology but not in the etale topology, because $\alpha_p$ is not etale (it is infinitesimal).

To classify $\alpha_p$-torsors, one must use the fppf topology: $H^1_{\text{fppf}}(X, \alpha_p) \neq H^1_{\text{et}}(X, \alpha_p)$ in general.

The inclusion of topologies $\text{fppf} \subseteq \text{fpqc}$ is strict. For example, if $k$ is a field and $\overline{k}$ is its algebraic closure, the morphism $\operatorname{Spec}(\overline{k}) \to \operatorname{Spec}(k)$ is faithfully flat but not of finite presentation (when $[k^{\text{sep}}:k] = \infty$), so it is an fpqc cover but not an fppf cover.

However, for most sheaves arising in algebraic geometry (representable functors, quasi-coherent sheaves), the sheaf condition for fpqc and fppf agree: a presheaf satisfying the fppf sheaf condition automatically satisfies the fpqc condition as well (a deep result of Gabber and others).

On the category $\mathbf{Top}$ of topological spaces, the canonical topology consists of families $\{U_i \to U\}$ that are effective epimorphic: the diagram $\coprod_{i,j} U_i \times_U U_j \rightrightarrows \coprod_i U_i \to U$ is a coequalizer.

Open covers are effective epimorphic families, but so are some other families (e.g., surjective local homeomorphisms). The canonical topology on $\mathbf{Top}$ is thus finer than the standard open cover topology.

A topology $J$ on a category $\mathcal{C}$ is subcanonical if every representable presheaf is a $J$-sheaf. All the standard algebraic topologies (Zariski, Nisnevich, etale, fppf, fpqc) are subcanonical on the category of schemes.

That is, for any scheme $Y$, the functor $h_Y(T) = \operatorname{Hom}(T, Y)$ satisfies the sheaf condition for any Zariski/etale/fppf/fpqc cover. For the fpqc topology, this is the content of fpqc descent for morphisms: given an fpqc cover $\{U_i \to U\}$ and compatible morphisms $U_i \to Y$, there is a unique morphism $U \to Y$ restricting to each $U_i \to Y$.

This is a nontrivial theorem. It means we can "test" morphisms to $Y$ locally in any of these topologies.

Consider $U = \operatorname{Spec}(\mathbb{Z})$.

Zariski cover: $\{\operatorname{Spec}(\mathbb{Z}[1/p]) \to \operatorname{Spec}(\mathbb{Z})\}_{p \text{ prime}}$ together with $\operatorname{Spec}(\mathbb{Q}) \to \operatorname{Spec}(\mathbb{Z})$ do not form a finite Zariski cover. But $\{\operatorname{Spec}(\mathbb{Z}[1/6]), \operatorname{Spec}(\mathbb{Z}[1/10]), \operatorname{Spec}(\mathbb{Z}[1/15])\}$ is a Zariski cover since $\gcd(6, 10, 15) = 1$.

Etale cover: $\{\operatorname{Spec}(\mathbb{Z}[i]) \to \operatorname{Spec}(\mathbb{Z})\}$ is not an etale cover (the map is not etale at $p = 2$ since $2 = -i(1+i)^2$ is ramified). However, $\{\operatorname{Spec}(\mathbb{Z}[\sqrt{-1}, 1/2]) \to \operatorname{Spec}(\mathbb{Z}[1/2])\}$ is an etale cover.

fpqc cover: $\{\operatorname{Spec}(\mathbb{Z}_p) \to \operatorname{Spec}(\mathbb{Z})\}_{p \text{ prime}}$ is not even jointly surjective (missing the generic point). But $\operatorname{Spec}(\prod_p \mathbb{Z}_p \times \mathbb{Q}) \to \operatorname{Spec}(\mathbb{Z})$ is faithfully flat, hence an fpqc cover.

Let $K/k$ be a finite Galois extension with Galois group $G$. Then $\operatorname{Spec}(K) \to \operatorname{Spec}(k)$ is a finite etale cover, and the fiber product is

$\operatorname{Spec}(K) \times_{\operatorname{Spec}(k)} \operatorname{Spec}(K) \cong \operatorname{Spec}(K \otimes_k K) \cong \coprod_{g \in G} \operatorname{Spec}(K)$

where the isomorphism $K \otimes_k K \cong \prod_{g \in G} K$ sends $a \otimes b \mapsto (a \cdot g(b))_g$. This is the scheme-theoretic version of the fact that a Galois cover is "trivial over itself."

This example is fundamental: etale cohomology of $\operatorname{Spec}(k)$ with coefficients in a sheaf $\mathcal{F}$ computes Galois cohomology $H^i(G_k, \mathcal{F})$ where $G_k = \operatorname{Gal}(k^{\text{sep}}/k)$.

For $X = \operatorname{Spec}(k)$, the small etale site $X_{\text{et}}$ has objects $\operatorname{Spec}(L)$ where $L/k$ is a finite separable extension. A covering family of $\operatorname{Spec}(L)$ is a finite family $\{\operatorname{Spec}(L_i)\}$ with $\prod L_i$ faithfully flat over $L$.

The category of sheaves on $X_{\text{et}}$ is equivalent to the category of continuous discrete $G_k$-sets, where $G_k = \operatorname{Gal}(k^{\text{sep}}/k)$ is the absolute Galois group. This fundamental equivalence connects etale cohomology to Galois cohomology.

The identity functor $\operatorname{id}: \mathbf{Sch}/S \to \mathbf{Sch}/S$ is continuous from the etale site to the Zariski site (every Zariski cover is an etale cover, so etale sheaves restrict to Zariski sheaves). This gives a comparison morphism of sites $\epsilon: X_{\text{et}} \to X_{\text{Zar}}$.

For any etale sheaf $\mathcal{F}$, we get a natural map

$H^i_{\text{Zar}}(X, \epsilon_*\mathcal{F}) \to H^i_{\text{et}}(X, \mathcal{F})$

This is generally not an isomorphism. For example, $H^1_{\text{Zar}}(\operatorname{Spec}(k), \mathbb{G}_m) = 0$ but $H^1_{\text{et}}(\operatorname{Spec}(k), \mathbb{G}_m) = 0$ as well (Hilbert's Theorem 90). However, $H^2_{\text{et}}(\operatorname{Spec}(k), \mathbb{G}_m) = \operatorname{Br}(k)$ (the Brauer group), which is invisible to Zariski cohomology.