Last time we built étale morphisms — the algebraic replacement for local diffeomorphisms. The goal now is to generalize topological spaces and sheaves so that “étale covers” can play the role that open covers play classically. Unwinding what a sheaf actually needs, two ingredients stand out:
A Grothendieck topology is exactly enough extra structure on an abstract category $\mathcal{C}$ to say what a “covering family” is and to make sense of intersections — enough, in short, to define sheaves and their cohomology without ever mentioning points.
A Grothendieck topology on a category $\mathcal{C}$ is the data of, for every $X \in \operatorname{Ob}(\mathcal{C})$, a collection of sets of morphisms $\{X_\alpha \to X\}$, called covering families, such that:
A site is a category $\mathcal{C}$ equipped with a Grothendieck topology.
Let $X$ be a topological space. Take $\mathcal{C} = \mathrm{Open}(X)$: objects are open subsets $U \subset X$, and there is a unique morphism $U \to V$ iff $U \subset V$. A family $\{U_i \to U\}$ is a covering family iff $\bigcup_i U_i = U$. This recovers ordinary sheaf theory.
Let $X$ be a scheme. Take $\mathcal{C}$ to have objects the étale morphisms $U \to X$, with morphisms the $X$-morphisms $U_1 \to U_2$. A family $\{U_\alpha \to U\}$ is a covering family iff $\bigcup_\alpha \mathrm{im}(U_\alpha) = U$ and every $U_\alpha \to U$ is étale.
Take $\mathcal{C}$ to have objects all $X$-schemes (not just étale ones). A family $\{U_\alpha \to U\}$ is a covering family iff every $f_\alpha$ is étale and $\bigcup_\alpha \mathrm{im}(U_\alpha) = U$. The small site $X_{\mathrm{\acute{e}t}}$ sits inside $X_{\mathrm{Et}}$ as a full subcategory.
(“fidèlement plate et de présentation finie” — faithfully flat, finite presentation.) Take $\mathcal{C} = X_{\mathrm{fppf}}$ with objects all $X$-schemes $U \to X$ of finite presentation, and covering families those $\{U_\alpha \to U\}$ that are jointly faithfully flat. This topology is coarser to define than étale (no unramifiedness needed) but has more covers, and is essential for descent arguments.
The Nisnevich, crystalline, infinitesimal, and cdh sites are all variations on the same theme, tuned to different problems (motivic homotopy theory, $p$-adic Hodge theory, resolution of singularities). There is also the small Zariski site $X_{\mathrm{Zar}}$ of a scheme $X$: objects are open $U \subset X$, morphisms are open immersions, and $\{U_i \hookrightarrow U\}$ covers iff $\bigcup U_i = U$ — this is literally the classical topological site, now viewed inside the world of schemes.
For $X$ a complex analytic space, let $X_{\mathrm{an,\acute{e}t}}$ have objects complex analytic spaces $Y \to X$ such that, locally on $Y$, the map is an analytic isomorphism; morphisms and covers are the evident ones. One can check directly that $\mathrm{Sh}(X_{\mathrm{an,\acute{e}t}}) \simeq \mathrm{Sh}(X_{\mathrm{an}})$ — sheaves on this exotic site agree with ordinary sheaves on the underlying space. This is Exercise 1 in §8.
| Site | Objects | Covering families |
|---|---|---|
| $X_{\mathrm{Zar}}$ | Open $U \subset X$ | Open immersions, $\bigcup U_i = U$ |
| $X_{\mathrm{\acute{e}t}}$ (small) | Étale $U \to X$ | Étale, jointly surjective |
| $X_{\mathrm{Et}}$ (big) | All $X$-schemes | Étale, jointly surjective |
| $X_{\mathrm{fppf}}$ | Finite presentation $X$-schemes | Faithfully flat, finite presentation |
Let $\mathcal{C}$ be a category and $\mathcal{D}$ any category (e.g. sets, abelian groups). A $\mathcal{D}$-valued presheaf on $\mathcal{C}$ is a contravariant functor $F : \mathcal{C}^{\mathrm{op}} \to \mathcal{D}$. Note: this definition does not require a Grothendieck topology on $\mathcal{C}$ at all — presheaves make sense on any category. (For $X$ a topological space, a $\mathcal{D}$-valued presheaf on $X$ in the usual sense is exactly the same thing as a $\mathcal{D}$-valued presheaf on $\mathrm{Open}(X)$.)
A presheaf $F$ (valued in sets, say) on a site $\mathcal{C}$ is a sheaf if for every covering family $\{U_\alpha \to U\}$, the diagram
is an equalizer. Concretely, this says two things:
A morphism of presheaves (or of sheaves) $F \to G$ is simply a natural transformation of functors $\mathcal{C}^{\mathrm{op}} \to \mathcal{D}$.
Fix a site $\mathcal{C}$. The category $\mathrm{Sh}(\mathcal{C}, \mathbf{Ab})$ of abelian-group-valued sheaves on $\mathcal{C}$ is an abelian category with enough injectives. Kernels of maps of sheaves are computed exactly as for presheaves (kernel is again a sheaf automatically); but be warned — cokernels are subtler: the presheaf cokernel need not already be a sheaf, and one must sheafify. This subtlety is worth sitting with; it is the reason $\mathbb{G}_m \xrightarrow{n} \mathbb{G}_m$ behaves so differently on different sites (see §6 below).
The general colimit construction is Exercise 2 in §8.
Any representable functor $\operatorname{Hom}_{\mathcal{C}}(-, X)$ is a sheaf on $X_{\mathrm{\acute{e}t}}$. Moreover, the same functor, restricted appropriately, is a sheaf on the finer site $X_{\mathrm{fppf}}$.
This single theorem manufactures most of the sheaves we care about. Four examples:
The functor $U \mapsto \{f \in \mathcal{O}(U)^\times : f^n = 1\}$ is representable by $\operatorname{Spec}\,\mathbb{Z}[t]/(t^n-1)$, so it is a sheaf on $X_{\mathrm{\acute{e}t}}$.
Represented by $n$ disjoint copies of $X$ (i.e. $\coprod_{\mathbb{Z}/n\mathbb{Z}} X$). We have $\underline{\mathbb{Z}/n\mathbb{Z}}(U) = \operatorname{Hom}(U,\, \mathbb{Z}/n\mathbb{Z}) = $ locally constant functions $U \to \mathbb{Z}/n\mathbb{Z}$.
$U \mapsto \mathcal{O}(U)^\times$, represented by $\mathbb{G}_{m} = \operatorname{Spec}\,\mathbb{Z}[t^{\pm1}]$. This is the fundamental example for Kummer theory.
$U \mapsto \operatorname{Hom}(U, \mathbb{P}^1)$ is representable (by $\mathbb{P}^1$ itself), hence a sheaf on $X_{\mathrm{\acute{e}t}}$ — and indeed on $X_{\mathrm{fppf}}$.
We would like to define $H^i_{\mathrm{\acute{e}t}}(X,\, \underline{\mathbb{Z}/n\mathbb{Z}})$ using $\underline{\mathbb{Z}/n\mathbb{Z}}$ as above. Two things need to be checked first: (1) that $\underline{\mathbb{Z}/n\mathbb{Z}}$ really is a sheaf on $X_{\mathrm{\acute{e}t}}$ (it is, by the theorem above); and (2) that the category of abelian-group-valued sheaves on $X_{\mathrm{\acute{e}t}}$ has enough injectives, so that right derived functors make sense.
Let $F$ be an abelian-group-valued sheaf on $X_{\mathrm{\acute{e}t}}$. Global sections $\Gamma(X_{\mathrm{\acute{e}t}}, -) = F \mapsto F(X)$ is a left-exact functor $\mathrm{Sh}(X_{\mathrm{\acute{e}t}}, \mathbf{Ab}) \to \mathbf{Ab}$. Since the source category has enough injectives, we define
the $i$-th right derived functor of global sections, computed via an injective resolution $F \to I^\bullet$ in $\mathrm{Sh}(X_{\mathrm{\acute{e}t}}, \mathbf{Ab})$.
Once cohomology is defined this way, it is automatically functorial, comes with long exact sequences from short exact sequences of sheaves, and specializes correctly: for $X$ a variety over $\mathbb{C}$, $H^i_{\mathrm{\acute{e}t}}(X, \underline{\mathbb{Z}/n\mathbb{Z}})$ recovers ordinary singular cohomology $H^i_{\mathrm{sing}}(X(\mathbb{C}), \mathbb{Z}/n\mathbb{Z})$. None of this is visible from the Zariski topology alone, which is why the extra covers matter.
Here is the first real surprise of the étale topology, and it is worth dwelling on because it is exactly the phenomenon that makes étale cohomology see more than Zariski cohomology.
The $n$-th power map of sheaves on $X_{\mathrm{\acute{e}t}}$,
is not in general an epimorphism of sheaves on $X_{\mathrm{Zar}}$, but is an epimorphism of sheaves on $X_{\mathrm{\acute{e}t}}$ once $n$ is invertible on $X$. On $X_{\mathrm{fppf}}$, it is an epimorphism for every $n \geq 1$, with no invertibility hypothesis; this is Exercise 3 in §8.
Take $X = \operatorname{Spec}\,\mathbb{R}$ and $n = 2$. Then $\mathbb{G}_m(X) = \mathbb{R}^\times$, and the map $\mathbb{R}^\times \to \mathbb{R}^\times$, $x \mapsto x^2$, is not surjective: $-1$ has no square root in $\mathbb{R}$. Since $\operatorname{Spec}\,\mathbb{R}$ has no nontrivial Zariski covers (it's a single point), surjectivity on global sections is the only thing being asked, and it fails.
A map of sheaves is an epimorphism iff it is locally surjective: surjective after passing to a cover. So given $f \in \mathcal{O}(U)^\times$, we need an étale cover $\{V \to U\}$ on which $f$ acquires an $n$-th root. Take
with structure map $V \to U$. Since $f \in \mathcal{O}(U)^\times$ and $n$ is invertible on $X$, the polynomial $t^n - f$ has derivative $nt^{n-1}$, which is a unit modulo $(t^n - f)$ (because $t$ is a unit there, $t \cdot t^{n-1} = f \cdot(\text{unit})$, so $t^{n-1}$ is a unit, so $nt^{n-1}$ is a unit) — exactly the standard-étale criterion reviewed in Meeting 2. So $V \to U$ is étale, and by construction the pullback of $f$ to $V$ is $t^n$, an $n$-th power. Thus $[n]$ is surjective étale-locally, i.e. an epimorphism of sheaves on $X_{\mathrm{\acute{e}t}}$.
“Epimorphism” in a sheaf category never means “surjective on global sections” — it means surjective after passing to a cover. Enlarging the site (Zariski $\to$ étale $\to$ fppf) enlarges the supply of covers, and hence enlarges the supply of epimorphisms. This is precisely why the Kummer sequence $$1 \to \mu_n \to \mathbb{G}_m \xrightarrow{[n]} \mathbb{G}_m \to 1$$ is exact as sheaves on $X_{\mathrm{\acute{e}t}}$ (with $n$ invertible on $X$) but simply false as a statement about $\mathcal{O}(X)^\times$ on the nose. This is the engine behind Kummer theory and the definition of étale cohomology with $\mu_n$-coefficients.
Let $\mathcal{T}, \mathcal{T}'$ be sites. A continuous map of sites $f : \mathcal{T} \to \mathcal{T}'$ is a functor $u : \mathcal{T}' \to \mathcal{T}$ (note the direction reversal, exactly as for continuous maps and pullback of opens) such that:
If $f: X \to Y$ is a continuous map of topological spaces, define $u : \mathrm{Open}(Y) \to \mathrm{Open}(X)$ by $u(V) = f^{-1}(V)$. Exercise 4 in §8 asks you to check directly that $u$ preserves finite intersections (fiber products in $\mathrm{Open}$) and sends covers to covers. It is therefore a continuous map of sites $X_{\mathrm{Zar}} \to Y_{\mathrm{Zar}}$ in the sense above — recovering the ordinary pushforward/pullback formalism for sheaves.
Consider the inclusion of the closed point $i : \operatorname{Spec}\,\mathbb{F}_p \hookrightarrow \operatorname{Spec}\,\mathbb{Z}$. On the étale sites, this induces a map
identifying decomposition/inertia data at $p$ with the absolute Galois group of the residue field. This is the germ of the entire theory of Frobenius elements and unramified representations — a preview of why the étale fundamental group, defined via automorphisms of fiber functors on $X_{\mathrm{\acute{e}t}}$, will recover $\operatorname{Gal}(\overline{k}/k)$ when $X = \operatorname{Spec}\,k$.
Next time: morphisms of sites in greater depth and the first part of fppf descent, following Litt’s Lecture 4.
These are the exercises explicitly marked in Litt’s Lecture 3. They follow the order in which the relevant ideas appeared above: first sites, then sheaves, then local surjectivity in the fppf topology, and finally maps of sites.
Let $X$ be a complex analytic space. Let $X_{\mathrm{an,\acute{e}t}}$ be the site whose objects are local analytic isomorphisms $Y \to X$, with the evident coverings. Prove that
Thus passing from ordinary open subsets to local analytic isomorphisms does not change the resulting category of sheaves. Hint: ordinary open subsets form a basis for this site; every local analytic isomorphism is locally an open embedding, so compare restriction and gluing on this basis.
Let $\mathcal{C}$ be a site. Prove that the category
of abelian-group-valued sheaves on $\mathcal{C}$ admits all small colimits. Hint: first form the colimit objectwise in the category of presheaves. The result need not be a sheaf, so apply sheafification and use its universal property to verify the universal property of the colimit in $\mathrm{Sh}(\mathcal{C},\mathbf{Ab})$.
Let $n \geq 1$. Show that the $n$-th power map
is an epimorphism of sheaves on $X_{\mathrm{fppf}}$. Unlike the étale argument in §6, no invertibility hypothesis on $n$ is required.
Hint: for $f \in \mathcal{O}(U)^\times$, consider
Because $t^n-f$ is monic, $V\to U$ is finite locally free of rank $n$; check that it is surjective, hence an fppf cover. On $V$, the pullback of $f$ is $t^n$. Compare this with the standard-étale calculation reviewed in Meeting 2.
Let $f:X\to Y$ be a continuous map of topological spaces. Consider
Check directly that this defines a continuous map of sites in the convention used in these notes. What to check: