On the Zariski site of $X = \operatorname{Spec}(A)$, the structure sheaf $\mathcal{O}_X$ assigns $\mathcal{O}_X(D(f)) = A_f$ to each principal open set. Given a Zariski cover $\{D(f_i)\}$ with $(f_1, \ldots, f_n) = A$, the sheaf condition requires the sequence

$0 \to A \to \prod_i A_{f_i} \rightrightarrows \prod_{i,j} A_{f_i f_j}$

to be exact. This is equivalent to saying $A \to \prod A_{f_i}$ is injective (separation) and any compatible family of elements in $\prod A_{f_i}$ comes from a unique element of $A$ (gluing).

This is a theorem in commutative algebra: the sequence above is exact whenever $(f_1, \ldots, f_n)$ generates the unit ideal.

The presheaf $\mathbb{G}_m: U \mapsto \Gamma(U, \mathcal{O}_U)^\times$ is a sheaf on the etale site $X_{\text{et}}$. Given an etale cover $\{U_i \to U\}$, we need to verify:

$\Gamma(U, \mathcal{O}_U)^\times \to \prod_i \Gamma(U_i, \mathcal{O}_{U_i})^\times \rightrightarrows \prod_{i,j} \Gamma(U_i \times_U U_j, \mathcal{O}_{U_i \times_U U_j})^\times$

is an equalizer. This follows from the fact that the structure sheaf $\mathcal{O}$ is an etale sheaf (by fpqc descent for quasi-coherent sheaves) and that $(-)^\times$ preserves the equalizer diagram when $\mathcal{O}$ is a sheaf of rings.

The first cohomology group $H^1_{\text{et}}(X, \mathbb{G}_m) = \operatorname{Pic}(X)$ classifies line bundles on $X$, just as in the Zariski case, by Hilbert's Theorem 90.

For an abelian group $A$, the constant sheaf $\underline{A}$ on $X_{\text{et}}$ assigns to each connected etale $U \to X$ the group $A$, and more generally to a possibly disconnected $U$ the group $A^{\pi_0(U)}$ (functions from connected components to $A$).

On $X = \operatorname{Spec}(k)$, the global sections of the constant sheaf $\underline{A}$ are $\underline{A}(\operatorname{Spec}(k)) = A$. But the cohomology detects Galois information:

$H^n_{\text{et}}(\operatorname{Spec}(k), \underline{A}) = H^n(\operatorname{Gal}(k^{\text{sep}}/k), A)$

where the right side is group cohomology of the absolute Galois group with trivial action on $A$.

On $X_{\text{et}}$ where $n$ is invertible on $X$, the sheaf $\mu_n$ (defined by $\mu_n(U) = \{f \in \mathcal{O}(U)^\times : f^n = 1\}$) sits in an exact sequence of etale sheaves:

$1 \to \mu_n \to \mathbb{G}_m \xrightarrow{(\cdot)^n} \mathbb{G}_m \to 1$

This is the Kummer sequence. It is exact in the etale topology: the surjectivity of $(\cdot)^n$ means that every unit locally (in the etale topology) has an $n$-th root. This fails in the Zariski topology (not every unit on a Zariski open has a Zariski-local $n$-th root).

The long exact sequence in cohomology gives

$\cdots \to H^0(X, \mathbb{G}_m) \xrightarrow{n} H^0(X, \mathbb{G}_m) \to H^1(X, \mu_n) \to \operatorname{Pic}(X) \xrightarrow{n} \operatorname{Pic}(X) \to \cdots$

On $\mathbb{R}$ with the usual topology (viewed as a site), the presheaf $\mathcal{B}(U) = \{f: U \to \mathbb{R} \mid f \text{ continuous and bounded}\}$ is separated: if two bounded continuous functions agree on every open set of a cover, they are equal. But gluing fails: locally bounded functions may not be globally bounded.

In the etale-site context, the presheaf $U \mapsto \operatorname{Pic}(U)$ on the big Zariski site is separated (an isomorphism of line bundles that is trivial locally is trivial globally) but not necessarily a sheaf.

The sheafification process can be done in two steps:

1. Start with a presheaf $F$.
2. Apply the "plus construction" once to get $F^+$, which is always a separated presheaf.
3. Apply the plus construction again to get $F^{++}$, which is a sheaf.

For a separated presheaf, a single application of the plus construction already produces a sheaf.

Consider the presheaf $F(U) = \{s \in \mathcal{O}(U) : s^2 - t = 0\}$ on the affine line $X = \mathbb{A}^1_k = \operatorname{Spec}(k[t])$ over a field $k$ with $\operatorname{char}(k) \neq 2$.

As a Zariski presheaf: $F(U)$ is the set of regular functions $s$ on $U$ with $s^2 = t$. For a Zariski open $U \subseteq \mathbb{A}^1$, the function $\sqrt{t}$ is not regular (it has a branch point at $0$). So $F(U)$ is either empty or consists of two elements $\pm\sqrt{t}$ (the latter only on opens not containing $0$).

As an etale presheaf: The etale cover $\operatorname{Spec}(k[t,s]/(s^2-t)) \to \mathbb{A}^1$ provides a "local square root" of $t$ in the etale topology. On this cover, $F$ has a section, and the etale sheafification is nontrivial.

This illustrates how the etale topology can "see" algebraic data (square roots, Galois extensions) invisible to the Zariski topology.

Over $\mathbb{F}_p$, the Artin-Schreier sequence $0 \to \mathbb{F}_p \to \mathbb{G}_a \xrightarrow{F - 1} \mathbb{G}_a \to 0$ (where $F$ is the Frobenius $x \mapsto x^p$) is exact in the etale topology. The map $F - 1$ is surjective in the etale topology because for any $a$ in a ring $A$, the equation $x^p - x = a$ is separable (its derivative $-1$ is a unit), so it has a root after a finite etale extension.

The resulting cohomology $H^1_{\text{et}}(\operatorname{Spec}(k), \mathbb{F}_p)$ classifies Artin-Schreier extensions of $k$.

For a scheme $X$, a sheaf $\mathcal{F}$ on the big etale site $(\mathbf{Sch}/X)_{\text{et}}$ restricts to a sheaf on the small etale site $X_{\text{et}}$ via the inclusion functor. Conversely, a sheaf on $X_{\text{et}}$ can be extended to the big site (not uniquely in general).

However, for computing cohomology, the two sites give the same answer: $H^i(X_{\text{et}}, \mathcal{F}|_{X_{\text{et}}}) \cong H^i((\mathbf{Sch}/X)_{\text{et}}, \mathcal{F})$. This is because the inclusion $X_{\text{et}} \hookrightarrow (\mathbf{Sch}/X)_{\text{et}}$ is a morphism of sites that induces an equivalence on the relevant derived categories.

For $X = \operatorname{Spec}(k)$ with $k$ a field:

Zariski sheaves: $\operatorname{Sh}(X_{\text{Zar}})$ is equivalent to $\mathbf{Set}$ (a single point has no nontrivial open covers).

Etale sheaves: $\operatorname{Sh}(X_{\text{et}})$ is equivalent to the category of continuous discrete $G_k$-sets, where $G_k = \operatorname{Gal}(k^{\text{sep}}/k)$. The abelian category $\operatorname{Ab}(X_{\text{et}})$ is equivalent to the category of continuous discrete $G_k$-modules.

fppf sheaves: $\operatorname{Sh}(X_{\text{fppf}})$ is strictly larger. It contains sheaves represented by non-smooth group schemes like $\alpha_p$ and $\mu_p$ in characteristic $p$.

This shows concretely how finer topologies give richer categories of sheaves.

The category $\operatorname{Ab}(X_{\text{et}})$ of abelian etale sheaves on a scheme $X$ has:

• Kernels: computed as the presheaf kernel (which is automatically a sheaf).
• Cokernels: the sheafification of the presheaf cokernel.
• Images: the sheafification of the presheaf image.
• Exact sequences: a sequence $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ is exact if and only if it is exact on all stalks $\mathcal{F}_{\bar{x}} \to \mathcal{G}_{\bar{x}} \to \mathcal{H}_{\bar{x}}$ at all geometric points $\bar{x} \to X$.

The existence of enough injectives ensures that the derived functor cohomology $H^i(X_{\text{et}}, \mathcal{F}) = R^i\Gamma(X_{\text{et}}, \mathcal{F})$ is well-defined.

For the etale site of a scheme $X$, the points correspond to geometric points $\bar{x}: \operatorname{Spec}(\Omega) \to X$ where $\Omega$ is a separably closed field. The stalk at $\bar{x}$ is

$\mathcal{F}_{\bar{x}} = \varinjlim_{(U, u)} \mathcal{F}(U)$

where the colimit runs over etale neighborhoods $(U, u)$ of $\bar{x}$, i.e., pairs of an etale $U \to X$ and a lift $u: \operatorname{Spec}(\Omega) \to U$ of $\bar{x}$.

The etale site has enough points (this is a theorem of Grothendieck). So a morphism of etale sheaves is an isomorphism if and only if it induces isomorphisms on all stalks at geometric points.

The fppf site also has enough points, but the points are more complicated: they correspond to maps from spectra of strictly henselian local rings (for the etale case) or from spectra of henselian local rings with algebraically closed residue field (for the fppf case, roughly).

In practice, one rarely computes fppf stalks directly. Instead, one uses the comparison between fppf and etale cohomology: for a smooth group scheme $G$, the etale and fppf cohomology agree: $H^i_{\text{et}}(X, G) \cong H^i_{\text{fppf}}(X, G)$.

For sheaves $\mathcal{F}, \mathcal{G}$ on $(\mathcal{C}, J)$, the sheaf hom (or internal hom) $\underline{\operatorname{Hom}}(\mathcal{F}, \mathcal{G})$ is the sheaf defined by

$\underline{\operatorname{Hom}}(\mathcal{F}, \mathcal{G})(U) = \operatorname{Hom}_{\operatorname{Sh}(\mathcal{C}/U)}(\mathcal{F}|_U, \mathcal{G}|_U)$

where $\mathcal{F}|_U$ denotes the restriction to the "localized" site over $U$.

This gives the sheaf category a closed monoidal structure (if we also have a suitable tensor product). For sheaves of $\mathcal{O}$-modules on a ringed site, $\underline{\operatorname{Hom}}_{\mathcal{O}}(\mathcal{F}, \mathcal{G})$ is the sheaf of $\mathcal{O}$-module homomorphisms.

Let $G$ be a sheaf of groups on $(\mathcal{C}, J)$. A $G$-torsor is a sheaf $P$ with a $G$-action $G \times P \to P$ such that:

1. $P$ is locally nonempty: there exists a covering $\{U_i \to U\}$ such that $P(U_i) \neq \emptyset$.
2. The action is simply transitive: $G \times P \to P \times P$ given by $(g, p) \mapsto (gp, p)$ is an isomorphism.

Isomorphism classes of $G$-torsors form the pointed set $H^1((\mathcal{C}, J), G)$ (nonabelian first cohomology). For $G = \mathbb{G}_m$, this gives $\operatorname{Pic}$ (line bundles); for $G = GL_n$, this gives isomorphism classes of rank-$n$ vector bundles.

In general, $\check{H}^n$ and $H^n$ (derived functor cohomology) differ. However:

• $\check{H}^0(U, \mathcal{F}) = H^0(U, \mathcal{F}) = \mathcal{F}(U)$ always.
• $\check{H}^1(U, \mathcal{F}) = H^1(U, \mathcal{F})$ always (for abelian sheaves on a site).
• For $n \geq 2$, Cech and derived functor cohomology agree when the site has "enough acyclic covers" (Leray's theorem / Cartan's theorem).

On the etale site of a Noetherian scheme, Cech cohomology agrees with derived functor cohomology for all abelian sheaves. This is a key technical result that makes etale cohomology computable.

For a topological space $X$, the classical category of sheaves $\operatorname{Sh}(X)$ equals $\operatorname{Sh}(\operatorname{Open}(X), J_{\text{open}})$ where $J_{\text{open}}$ is the open cover topology.

For a scheme $X$ with Zariski topology:

• $\operatorname{Sh}(X_{\text{Zar}}) =$ classical Zariski sheaves on $X$.
• $\operatorname{Sh}(X_{\text{et}}) \supsetneq \operatorname{Sh}(X_{\text{Zar}})$ strictly, as etale sheaves encode Galois-theoretic information.
• $\operatorname{Sh}(X_{\text{fppf}}) \supsetneq \operatorname{Sh}(X_{\text{et}})$ strictly, as fppf sheaves can detect inseparable phenomena.

The comparison morphism $\epsilon: X_{\text{et}} \to X_{\text{Zar}}$ gives a pushforward $\epsilon_*: \operatorname{Sh}(X_{\text{et}}) \to \operatorname{Sh}(X_{\text{Zar}})$. For a quasi-coherent sheaf $\mathcal{F}$ on $X$, the etale and Zariski cohomology agree: $H^i_{\text{et}}(X, \mathcal{F}) = H^i_{\text{Zar}}(X, \mathcal{F})$ (a theorem of Grothendieck).