Grothendieck Topologies

In algebraic geometry, we define schemes by first defining affine schemes and then gluing them together. In the language of [Scheme Theory] §Faithfully Flat Descent, ⁋Definition 4, this amounts to descent along a Zariski open cover defined using open embeddings; as we saw in that post, descent is a formal unpacking of what is needed in this gluing process. What stands out most in this language is that a descent datum requires an isomorphism \(\Phi\) as part of its input: once the scheme is constructed, the images of two overlaps that were isomorphic to each other become literally the same intersection, which is why this gluing datum was hardly visible.

To glue local data together precisely in this way, we must first spell out separately which families of morphisms are admitted as coverings, and when compatible data over them descends to unique global data. The former was already generalized by the Grothendieck pretopology we examined in [Scheme Theory] §Faithfully Flat Descent, ⁋Definition 8, and the latter is expressed through the sheaf condition. In this post, the first of the series, we focus on this process and set up the language needed to define stacks later.

Sieves and covering sieves

As in [Scheme Theory] §Faithfully Flat Descent, ⁋Definition 8, we will declare a collection of morphisms with target \(U\) to be a covering of \(U\). To make such a definition, we inevitably have to explain what it means for one covering to be finer than another: given that \(V\rightarrow U\) is an element of a covering of \(U\), we must explain what it means for some \(W\rightarrow U\) to factor through it.

Definition 1 Given a category \(\mathcal{C}\) and an object \(U\in \mathcal{C}\), a sieve on \(U\) is a collection \(\mathcal{S}\) of morphisms satisfying the following conditions.

1. Every element of \(\mathcal{S}\) has target \(U\).
2. For any element \(f: V\rightarrow U\) of \(\mathcal{S}\) and any morphism \(g:W\rightarrow V\), the composite \(f\circ g: W\rightarrow U\) is also an element of \(\mathcal{S}\).

In other words, a sieve is a subfunctor of the representable presheaf \(h_U=\Hom_\mathcal{C}(-, U): \mathcal{C}^\op\rightarrow \Set\). Indeed, if \(\mathcal{S}\) is a sieve on \(U\), then assigning to each object \(V\) the subset

\[\mathcal{S}(V)=\{f: V \rightarrow U\mid f\in \mathcal{S}\}\subseteq \Hom_\mathcal{C}(V, U)\]

defines a subfunctor \(\mathcal{S}\) of \(h_U\), and when a morphism \(g:W\rightarrow V\) is given, \(\mathcal{S}(g)\) is given by

\[\mathcal{S}(g):\mathcal{S}(V)\rightarrow \mathcal{S}(W);\qquad f\mapsto f\circ g.\]

Conversely, any functor with this property (that is, any subfunctor of \(h_U\)) satisfies the sieve condition, so a sieve is exactly a subfunctor of \(h_U\). From this point of view, the maximal sieve is the largest of these, namely the sieve given by \(h_U\) itself, which is simply the collection of all morphisms into \(U\).

Meanwhile, given a family \(\{f_i: U_i \rightarrow U\}_{i\in I}\) of morphisms with target \(U\), there is a smallest sieve containing it, namely the collection of all composites of the \(f_i\) with arbitrary morphisms:

\[\langle f_i\rangle=\{f: V \rightarrow U\mid f=f_i\circ g\text{ for some }i\in I\text{ and }g: V \rightarrow U_i\}.\]

We call it the sieve generated by \(\{f_i\}\). We now define the pullback operation that pulls a sieve back to another object.

Definition 2 Given a morphism \(g: V \rightarrow U\) and a sieve \(\mathcal{S}\) on \(U\), the pullback sieve \(g^\ast \mathcal{S}\) on \(V\) is defined by

\[g^\ast \mathcal{S}=\{h: W \rightarrow V\mid g\circ h\in \mathcal{S}\}.\]

From the subfunctor viewpoint discussed above, the pullback sieve \(g^\ast \mathcal{S}\) is the preimage of the subfunctor \(\mathcal{S}\subseteq h_U\) along the natural transformation \(h_g: h_V \rightarrow h_U\), and that \(g^\ast \mathcal{S}\) is again a sieve on \(V\) follows immediately from the definition.

With these in hand, we can make the following definition.

Definition 3 A Grothendieck topology \(\tau\) on a category \(\mathcal{C}\) is an assignment to each object \(U\in \mathcal{C}\) of a collection \(\tau(U)\) of sieves on \(U\), whose elements are called covering sieves, subject to the following three conditions.

1. (Maximality) For every \(U\), the maximal sieve \(t_U\) belongs to \(\tau(U)\).
2. (Stability) If \(\mathcal{S}\in \tau(U)\) and \(g: V \rightarrow U\) is any morphism, then \(g^\ast \mathcal{S}\in \tau(V)\).
3. (Transitivity) If \(\mathcal{S}\in \tau(U)\) and \(\mathcal{R}\) is a sieve on \(U\) such that \(f^\ast \mathcal{R}\in \tau(V)\) for every \((f: V \rightarrow U)\in \mathcal{S}\), then \(\mathcal{R}\in \tau(U)\).

These three conditions are a precise abstraction of the properties that open covers of a topological space satisfy. Maximality says that \(U\) covers itself trivially; stability corresponds to the fact that when \(\{U_i\}\) covers \(U\) in a topological space, \(\{U_i\cap V\}\) covers \(V\). Transitivity, as its name suggests, says that a covering of a covering is again a covering.

Covering families and pretopologies

The notions of sieve and Grothendieck topology discussed above are completely general and apply to any category, but in the examples from algebraic geometry the pullback sieve in the second condition is realized by a better-behaved object, namely the fiber product. In a category with fiber products, we can make the following definition without the language of sieves.

Definition 4 A Grothendieck pretopology on a category \(\mathcal{C}\) with fiber products is an assignment to each object \(U\) of a collection of covering families \(\{f_i: U_i \rightarrow U\}_{i\in I}\) of morphisms with target \(U\), subject to the following three conditions.

1. (Isomorphism) If \(f: V \rightarrow U\) is an isomorphism, then the one-element family \(\{f: V \rightarrow U\}\) is a covering family.
2. (Base change) If \(\{f_i: U_i \rightarrow U\}\) is a covering family and \(g: V \rightarrow U\) is any morphism, then the family \(\{\pr_2: U_i\times_U V \rightarrow V\}\) obtained by base change is also a covering family.
3. (Transitivity) If \(\{f_i: U_i \rightarrow U\}\) is a covering family and, for each \(i\), \(\{g_{ij}: U_{ij} \rightarrow U_i\}_{j\in J_i}\) is a covering family, then the family \(\{f_i\circ g_{ij}: U_{ij} \rightarrow U\}_{i, j}\) obtained by composition is also a covering family.

In a category with fiber products, such manageable covering families carry enough information to recover a Grothendieck topology completely, without having to handle entire unwieldy sieves. Indeed, given a pretopology, one always obtains a unique Grothendieck topology by declaring as covering sieves those sieves that contain a sieve generated by a covering family; the following proposition guarantees this.

Proposition 5 Suppose a Grothendieck pretopology \(\Cov\) on \(\mathcal{C}\) is given. For each \(U\), declare that a sieve \(\mathcal{S}\) on \(U\) belongs to \(\tau(U)\) if there exists a covering family \(\{f_i: U_i \rightarrow U\}\in \Cov(U)\) such that \(\langle f_i\rangle\subseteq \mathcal{S}\). Then \(\tau\) is a Grothendieck topology on \(\mathcal{C}\).

Proof

We verify the three conditions of Definition 3 in turn.

1. For maximality: since \(\id_U: U \rightarrow U\) is an isomorphism, \(\{\id_U\}\) is a covering family by the first condition of Definition 4, and the sieve it generates is the maximal sieve \(t_U\) itself. Hence \(t_U\in \tau(U)\).
2. For stability, if \(\mathcal{S}\in \tau(U)\) there is a covering family \(\{f_i: U_i \rightarrow U\}\) with \(\langle f_i\rangle\subseteq \mathcal{S}\). Given any \(g: V \rightarrow U\), the base change condition makes \(\{\pr_2: U_i\times_U V \rightarrow V\}\) a covering family, and it suffices to show that the sieve it generates is contained in \(g^\ast \mathcal{S}\). If \(h: W \rightarrow V\) factors through \(\pr_2: U_i\times_U V \rightarrow V\), then \(g\circ h\) factors through \(f_i\) via \(g\circ \pr_2=f_i\circ \pr_1\), so \(g\circ h\in \langle f_i\rangle\subseteq \mathcal{S}\), that is, \(h\in g^\ast \mathcal{S}\). Hence \(g^\ast \mathcal{S}\in \tau(V)\).
3. Finally, for transitivity, suppose \(\mathcal{S}\in \tau(U)\) and that a sieve \(\mathcal{R}\) on \(U\) satisfies \(f^\ast \mathcal{R}\in \tau(V)\) for every \((f: V \rightarrow U)\in \mathcal{S}\). Choose a covering family \(\{f_i: U_i \rightarrow U\}\) with \(\langle f_i\rangle\subseteq \mathcal{S}\); since each \(f_i\in \mathcal{S}\), we have \(f_i^\ast \mathcal{R}\in \tau(U_i)\), so there is a covering family \(\{g_{ij}: U_{ij} \rightarrow U_i\}\) with \(\langle g_{ij}\rangle\subseteq f_i^\ast \mathcal{R}\). That \(g_{ij}\in f_i^\ast \mathcal{R}\) means \(f_i\circ g_{ij}\in \mathcal{R}\), so every element of the covering family \(\{f_i\circ g_{ij}: U_{ij} \rightarrow U\}\) provided by the transitivity condition belongs to \(\mathcal{R}\). The sieve it generates is then contained in \(\mathcal{R}\), so \(\mathcal{R}\in \tau(U)\).

In other words, given a covering family we can form a covering sieve containing it, and in this way we can move freely between the two notions. One thing to keep in mind is that, unsurprisingly, two different Grothendieck pretopologies can generate the same Grothendieck topology; for instance, the coverings of an affine scheme by all principal opens and by all finite affine open covers give the same Zariski topology.

Definition 6 A pair \((\mathcal{C}, \tau)\) of a category \(\mathcal{C}\) and a Grothendieck topology \(\tau\) on it is called a site.

If a category \(\mathcal{C}\) has fiber products and is equipped with a Grothendieck pretopology, this data determines a Grothendieck topology on \(\mathcal{C}\) in a unique way, so in this case too we call such data a site. As always, making this fully rigorous requires appropriate set-theoretic size hypotheses (such as requiring \(\mathcal{C}\) to be a small category so that the products and limits appearing in the values of sheaves exist), but in this post we do not concern ourselves with such issues and simply assume that the necessary products and limits exist.

Examples of sites

The notion of a site did not appear out of thin air; it encompasses objects we already know. The most basic example is the site formed by the open sets of a topological space.

Example 7 (Topological spaces) Given a topological space \(X\) and the category \(\Open(X)\) of open sets of \(X\), one easily checks that in this category the intersection satisfies the universal property of the fiber product. Thus, declaring the covering families of each open set \(U\) to be the ordinary open covers of \(U\) (that is, families of inclusions \(\{U_i\hookrightarrow U\}\) of open sets satisfying \(\bigcup U_i=U\)) defines a pretopology. The meaning of each condition becomes clearer if we look at how the three conditions of Definition 4 translate in this example: the first says that \(U\) covers itself; the base change condition says that if \(\{U_i\}\) covers \(U\) and \(V\) is an open subset of \(U\), then \(\{U_i\cap V\}\) covers \(V\); and finally, transitivity, as explained above, says that a covering of a covering is again a covering.

Still, the examples of interest to us are the following topologies on schemes. For convenience, let us call a family \(\{f_i: U_i \rightarrow U\}\) of morphisms with common codomain jointly surjective if it satisfies \(\bigcup f_i(U_i)=U\).

Example 8 (Algebraic geometry) While treating descent, we already encountered various coverings defined on schemes. In this example we examine the sites they define. Fix a scheme \(X\).

1. First, the small Zariski site \(X_\Zar\) of a scheme \(X\) is the site whose objects are the open subschemes of \(X\), whose morphisms are open embeddings, and whose covering families are the usual open covers. Since the topology of \(X\) itself is the Zariski topology anyway, this is essentially a special case of Example 7 (Topological spaces). To recast this in more formal language for the examples to come, \(X_\Zar\) is the category whose objects are open embeddings with target \(X\), and for each object \(U\rightarrow X\), the covering families of \(U\) are the families \(\{U_i\rightarrow U\}\) of \(X\)-morphisms that are jointly surjective open embeddings. We can also define the big Zariski site \((\Sch_{/X})_{\Zar}\): it is obtained from \(X_\Zar\) by enlarging only the objects, to arbitrary \(U\rightarrow X\), while the covering families of a fixed \(U\) are still the families \(\{U_i\rightarrow U\}\) of \(X\)-morphisms that are jointly surjective open embeddings.
2. Similarly, we define the small étale site \(X_\et\): its objects are the étale morphisms with target \(X\), and the covering families of each \(U\) are the families \(\{U_i\rightarrow U\}\) of \(X\)-morphisms that are jointly surjective and étale. The big étale site is defined analogously: it is obtained from the definition of the small étale site by replacing only the objects. Instead of étale morphisms with target \(X\), one takes arbitrary morphisms with target \(X\) (that is, arbitrary \(X\)-schemes).
3. One can similarly define the fppf site and the fpqc site, but one caveat is that the small versions do not work as one might expect. This is essentially because even when \(U, V\) are both flat over \(X\), a morphism \(U\rightarrow W\) need not be flat, so fiber products such as \(U\times_WV\) may fail to be flat over \(X\) and the language does not come out cleanly. Fortunately, the objects of interest to us are the big sites, where this concern does not arise in the first place, so no problem occurs. Thus, for fixed \(X\), the (big) fppf site \((\Sch_{/X})_\fppf\) of \(X\) is the site whose objects are \(X\)-schemes and whose covering families of each \(U\) are the families \(\{U_i\rightarrow U\}\) of \(X\)-morphisms that are jointly surjective and fppf.
4. For the (big) fpqc site \((\Sch_{/X})_\fpqc\) of \(X\), we likewise consider only the big version, but some care is needed in stating the condition on covering families. The reason is that, unlike for fppf morphisms, for fpqc coverings, as we saw in [Scheme Theory] §Faithfully Flat Descent, ⁋Definition 9, the quasi-compactness condition is imposed not on the individual morphisms but on the family itself. That is, \((\Sch_{/X})_\fpqc\) is the site whose objects are the morphisms \(U\rightarrow X\) and whose covering families of \(U\) are given by the fpqc coverings of [Scheme Theory] §Faithfully Flat Descent, ⁋Definition 9.

We now treat the notions of presheaf and sheaf defined over such sites.

Presheaves and sheaves on a site

A site is essentially a category equipped with additional topological information. We already know that a presheaf can be defined on any category, so the only new definition we need is the one that captures the sheaf condition using the Grothendieck topology.

Definition 9 A presheaf on a site \((\mathcal{C}, \tau)\) is a contravariant functor \(F:\mathcal{C}^\op \rightarrow \Set\), and a morphism of presheaves is a natural transformation. We write \(\PSh(\mathcal{C})\) for the functor category they form. Given a covering sieve \(\mathcal{S}\in \tau(U)\) on \(U\), a matching family of \(F\) over \(\mathcal{S}\) is a family \((x_f)_{f\in \mathcal{S}}\) consisting of an element \(x_f\in F(V)\) for each \((f: V \rightarrow U)\in \mathcal{S}\), such that for every \(g: W \rightarrow V\),

\[F(g)(x_f)=x_{f\circ g}.\]

An element \(x\in F(U)\) is an amalgamation of this matching family if \(F(f)(x)=x_f\) for every \((f: V \rightarrow U)\in \mathcal{S}\). We now say that a presheaf \(F\) is

1. a separated presheaf if every matching family over any covering sieve has at most one amalgamation, and
2. a sheaf if every matching family over any covering sieve has exactly one amalgamation.

We write \(\Sh(\mathcal{C}; \tau)\) for the full subcategory of \(\PSh(\mathcal{C})\) formed by the sheaves.

In other words, a matching family is local data given over a covering sieve, and an amalgamation is the global data obtained by gluing it. From this point of view it is intuitively clear that the two conditions above correspond exactly to the two sheaf axioms. As we saw earlier, in a category with fiber products this information is essentially given over covering families, in which case the sheaf condition can be spelled out as follows.

Proposition 10 Suppose the Grothendieck topology \(\tau\) comes from a pretopology \(\Cov\). (Proposition 5) A presheaf \(F:\mathcal{C}^\op \rightarrow \Set\) is a sheaf if and only if, for every covering family \(\{f_i: U_i \rightarrow U\}\in \Cov(U)\), the following diagram

is an equalizer. Here \(e(x)=(F(f_i)(x))_i\), and the two morphisms \(p, q\) are the restrictions along the two projections \(\pr_1: U_i\times_U U_j \rightarrow U_i\) and \(\pr_2: U_i\times_U U_j \rightarrow U_j\), namely \(p((s_i)_i)=(F(\pr_1)(s_i))_{i, j}\) and \(q((s_i)_i)=(F(\pr_2)(s_j))_{i, j}\). \(F\) is separated if and only if \(e\) in the same diagram is injective.

Proof

It suffices to show that matching families over the generating covering sieve \(S=\langle f_i\rangle\) are in one-to-one correspondence with the data of the equalizer above.

Given a matching family \((x_f)_{f\in S}\) over \(S\), we obtain in particular the family \((s_i)\in \prod_i F(U_i)\) with \(s_i=x_{f_i}\in F(U_i)\) for each \(f_i\in S\). Composing the two projections \(\pr_1: U_i\times_U U_j \rightarrow U_i\) and \(\pr_2: U_i\times_U U_j \rightarrow U_j\) with \(f_i\) and \(f_j\) respectively yields one and the same morphism \(f_i\circ \pr_1=f_j\circ \pr_2\), which belongs to \(S\), so applying the matching condition with \(g=\pr_1\) and \(g=\pr_2\) gives

\[F(\pr_1)(s_i)=x_{f_i\circ \pr_1}=x_{f_j\circ \pr_2}=F(\pr_2)(s_j),\]

hence \(p((s_i))=q((s_i))\), that is, \((s_i)\) lies in the equalizer of \(p, q\).

Conversely, suppose \((s_i)\in \prod_i F(U_i)\) with \(p((s_i))=q((s_i))\) is given; we define a matching family over \(S\). If \(f\in S\), then \(f=f_i\circ g\) for some \(i\) and \(g: V \rightarrow U_i\), so we may set \(x_f=F(g)(s_i)\). To see this is well-defined, we must check that two expressions \(f=f_i\circ g=f_j\circ g'\) give the same value. This holds because, bundling the two morphisms into \((g, g'): V \rightarrow U_i\times_U U_j\) and pulling back the \((i, j)\)-component of \(p((s_i))=q((s_i))\) along \(F(g, g')\), we get \(F(g)(s_i)=F(g')(s_j)\). That this assignment satisfies the matching family condition, and that the two constructions are mutually inverse, can be checked directly from the definitions.

Therefore, the existence and uniqueness of an amalgamation \(x\in F(U)\) amounts exactly to \(e\) being a bijection onto the equalizer. The sheaf condition is the existence and uniqueness of amalgamations, so it is equivalent to \(e\) being a bijection onto the equalizer, that is, to the diagram above being an equalizer; the separated condition is uniqueness alone, so it is equivalent to \(e\) being injective.

Finally, a covering sieve \(\mathcal{S}'\) in \(\tau(U)\) need only contain a generating covering sieve \(\langle f_i\rangle\), not equal it, so we must check that the element \(x\) obtained by restricting a matching family over \(\mathcal{S}'\) to \(\langle f_i\rangle\) is an amalgamation for all of \(\mathcal{S}'\). For any \((f: V \rightarrow U)\in \mathcal{S}'\), take the base-changed covering family \(\{\pr_2: U_i\times_U V \rightarrow V\}\); since \(f\circ \pr_2=f_i\circ \pr_1\in \langle f_i\rangle\), we have \(F(\pr_2)(F(f)(x))=x_{f\circ \pr_2}=F(\pr_2)(x_f)\), and the injectivity of \(e\) for this covering family yields \(F(f)(x)=x_f\).

Thus a sheaf on a site, defined in this way, is formally identical to [Topology] §Sheaves, ⁋Definition 1, except that overlaps are generalized to fiber products \(U_i\times_U U_j\). For topological spaces, every morphism in a covering is a monomorphism, so \(U_i\times_U U_i=U_i\); but for étale or fpqc coverings, \(U_i\times_U U_i\) can be larger than \(U_i\), which is what makes descent over non-injective coverings possible.

Sheafification and the plus construction

Over topological spaces, we characterized sheafification as the left adjoint of the forgetful functor and constructed it directly as the sheaf of compatible germs. ([Topology] §Sheaves, ⁋Definition 5) A general site has no points or stalks the way a topological space does, so the construction using germs is unavailable in general; instead one uses the plus construction, which takes a colimit over all coverings.

Definition 11 Suppose a presheaf \(F:\mathcal{C}^\op \rightarrow \Set\) and an object \(U\) are given. Equip the collection \(\operatorname{Cov}_\tau(U)\) of covering sieves on \(U\) with the ordering

\[\mathcal{S}\preceq\mathcal{S}'\quad\iff\quad\mathcal{S}'\subseteq\mathcal{S}.\]

Then, since the intersection of two covering sieves gives a common refinement, \(\operatorname{Cov}_\tau(U)\) is a filtered preorder. An inclusion \(\mathcal{S}'\subseteq\mathcal{S}\) induces a restriction of matching families

\[\operatorname{Match}(\mathcal{S},F)\rightarrow\operatorname{Match}(\mathcal{S}',F),\]

and the colimit of this filtered diagram

\[F^+(U)=\varinjlim_{\mathcal{S}\in\operatorname{Cov}_\tau(U)}\operatorname{Match}(\mathcal{S},F)\]

is called the plus construction of \(F\). Pulling covering sieves and matching families back along a morphism \(V\rightarrow U\) yields a restriction \(F^+(U)\rightarrow F^+(V)\), so \(U\mapsto F^+(U)\) forms a presheaf. Moreover, since a matching family over the maximal sieve is nothing but an element of \(F(U)\), there is a natural morphism \(F\rightarrow F^+\).

Intuitively, an element of \(F^+(U)\) is local data given compatibly over some covering, with two such data identified whenever they become equal after passing to a common refinement (this is precisely the meaning of the colimit), and in this way the failure of uniqueness of amalgamations is corrected. In the language of pretopologies, \(F^+(U)\) is the filtered colimit, over coverings \(\{U_i \rightarrow U\}\), of the 0th Čech cohomology

\[\check{H}^0(\{U_i \rightarrow U\}, F)=\operatorname{eq}\Big(\prod_i F(U_i)\rightrightarrows \prod_{i, j}F(U_i\times_U U_j)\Big).\]

The following theorem records the key properties of this construction.

Theorem 12 For any presheaf \(F\), the following hold.

1. \(F^+\) is a separated presheaf.
2. If \(F\) is separated, then \(F^+\) is a sheaf. Hence \(F^{++}=(F^+)^+\) is always a sheaf.
3. The assignment \(a(F)=F^{++}\) is left adjoint to the forgetful functor \(\iota:\Sh(\mathcal{C}; \tau)\hookrightarrow \PSh(\mathcal{C})\), with the natural morphism \(F \rightarrow F^{++}\) as its unit. That is, for any sheaf \(G\),

\[\Hom_{\Sh}(F^{++}, G)\cong \Hom_{\PSh}(F, G).\]

Moreover, \(a\) preserves finite limits.

The proof of this theorem is somewhat technical, so we take it as a standard property of the plus construction. The essential point is that, thanks to Theorem 12, the sheafification adjunction treated in [Topology] §Sheaves, ⁋Definition 5 extends to arbitrary sites. The fact that the left adjoint \(a\) preserves finite limits is especially important: it implies that the category of sheaves enjoys the same kind of good structure as the category of sheaves on a topological space.

The category of sheaves \(\Sh(\mathcal{C}; \tau)\) on a site \((\mathcal{C}, \tau)\) is called a Grothendieck topos. Topos theory, however, is a vast subject in its own right, so we will develop only as much of it as is needed in the context of stacks later.

Subcanonical topologies and representable presheaves

In the post [Scheme Theory] §Functor of Points, we decided to regard a (representable) functor \(\Sch^\op\rightarrow \Set\) itself as a scheme. As we extend this viewpoint to sheaves defined on a site, we first check whether the representable functor of each scheme is a sheaf on that site.

Definition 13 A site \((\mathcal{C}, \tau)\) is subcanonical if, for every object \(X\in \mathcal{C}\), the representable presheaf \(h_X=\Hom_\mathcal{C}(-, X)\) is a \(\tau\)-sheaf. Among the subcanonical topologies there exists a finest one, called the canonical topology. In other words, a topology is subcanonical if and only if it is coarser than or equal to the canonical topology.

The subcanonical condition is stated concretely through Proposition 10: that \(h_X\) is a sheaf means that, for each covering family \(\{U_i \rightarrow U\}\), a morphism from \(U\) to \(X\) glues uniquely from morphisms given compatibly over the covering; that is,

\[\Hom(U, X) \rightarrow \prod_i \Hom(U_i, X)\rightrightarrows \prod_{i, j}\Hom(U_i\times_U U_j, X)\]

is an equalizer. This is nothing other than the descent condition for morphisms. For the Zariski or étale topology this holds by the familiar fact that morphisms are determined locally over a covering, but for coverings as coarse as fpqc it is by no means trivial; its validity is a form of faithfully flat descent.

Theorem 14 The fpqc topology on \(\Sch\) (or \(\Sch_{/S}\)) is subcanonical. That is, for any scheme \(X\), the functor of points \(h_X=\Hom_{\Sch}(-, X)\) is an fpqc sheaf. Consequently, every representable presheaf is also a sheaf in the Zariski, étale, and fppf topologies.

The functor of points \(h_X:\Sch^\op \rightarrow \Set\) of a scheme \(X\) is a representable presheaf, and one can check directly, by gluing morphisms over an open covering, that \(h_X\) is a sheaf on the big Zariski site. The point of Theorem 14 is that the same \(h_X\) satisfies the sheaf condition even for the more general fpqc coverings, so that morphisms given compatibly descend uniquely. The proof proceeds as follows. An fpqc covering can be refined, over affine opens, to a single faithfully flat affine covering \(\Spec B\rightarrow\Spec A\). If the target \(X=\Spec R\) is also affine, the Amitsur equalizer \(A\rightarrow B\rightrightarrows B\otimes_A B\) of [Scheme Theory] §Faithfully Flat Descent, ⁋Lemma 3 descends a morphism \(\Spec B\rightarrow X\) to a unique morphism \(\Spec A\rightarrow X\). For a general \(X\), one uses the fact that the above faithfully flat affine morphism is universally submersive to descend the underlying continuous map, and then glues the local morphisms obtained by applying the same argument over the affine opens of \(X\).

Having reduced the problem to the affine case, let us verify the computation concretely for \(\mathbb{G}_a=\Spec \mathbb{Z}[\x]=\mathbb{A}^1\). In [Scheme Theory] §Functor of Points, ⁋Proposition 1 we already saw that the corresponding functor of points assigns to each scheme \(T\) the additive group of global sections

\[\mathbb{G}_a(T)=\Hom_{\Sch}(T, \mathbb{A}^1)=\Gamma(T, \mathcal{O}_T).\]

The sheaf condition for a single fpqc covering \(\{\Spec B \rightarrow \Spec A\}\) therefore says that

\[A \rightarrow B \rightrightarrows B\otimes_A B\]

is an equalizer. The two morphisms are \(b\mapsto b\otimes 1\) and \(b\mapsto 1\otimes b\), and by the Amitsur equalizer above, \(A=\{b\in B\mid b\otimes 1=1\otimes b\}\). In other words, an element of \(\mathbb{G}_a(\Spec A)=A\) is recovered uniquely from elements given compatibly over the covering.

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References

[FGA] B. Fantechi, L. Göttsche, L. Illusie, S. Kleiman, N. Nitsure, A. Vistoli, Fundamental algebraic geometry: Grothendieck’s FGA explained. Mathematical Surveys and Monographs. American Mathematical Society, 2005.
[Stacks] The Stacks Project Authors, The Stacks Project. Available online.