## 59.15 The fpqc topology

Before doing étale cohomology we study a bit the fpqc topology, since it works well for quasi-coherent sheaves.

Definition 59.15.1. Let $T$ be a scheme. An *fpqc covering* of $T$ is a family $\{ \varphi _ i : T_ i \to T\} _{i \in I}$ such that

each $\varphi _ i$ is a flat morphism and $\bigcup _{i\in I} \varphi _ i(T_ i) = T$, and

for each affine open $U \subset T$ there exists a finite set $K$, a map $\mathbf{i} : K \to I$ and affine opens $U_{\mathbf{i}(k)} \subset T_{\mathbf{i}(k)}$ such that $U = \bigcup _{k \in K} \varphi _{\mathbf{i}(k)}(U_{\mathbf{i}(k)})$.

Example 59.15.3. Examples of fpqc coverings.

Any Zariski open covering of $T$ is an fpqc covering.

A family $\{ \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)\} $ is an fpqc covering if and only if $A \to B$ is a faithfully flat ring map.

If $f: X \to Y$ is flat, surjective and quasi-compact, then $\{ f: X\to Y\} $ is an fpqc covering.

The morphism $\varphi : \coprod _{x \in \mathbf{A}^1_ k} \mathop{\mathrm{Spec}}(\mathcal{O}_{\mathbf{A}^1_ k, x}) \to \mathbf{A}^1_ k$, where $k$ is a field, is flat and surjective. It is not quasi-compact, and in fact the family $\{ \varphi \} $ is not an fpqc covering.

Write $\mathbf{A}^2_ k = \mathop{\mathrm{Spec}}(k[x, y])$. Denote $i_ x : D(x) \to \mathbf{A}^2_ k$ and $i_ y : D(y) \to \mathbf{A}^2_ k$ the standard opens. Then the families $\{ i_ x, i_ y, \mathop{\mathrm{Spec}}(k[[x, y]]) \to \mathbf{A}^2_ k\} $ and $\{ i_ x, i_ y, \mathop{\mathrm{Spec}}(\mathcal{O}_{\mathbf{A}^2_ k, 0}) \to \mathbf{A}^2_ k\} $ are fpqc coverings.

Lemma 59.15.4. The collection of fpqc coverings on the category of schemes satisfies the axioms of site.

**Proof.**
See Topologies, Lemma 34.9.7.
$\square$

It seems that this lemma allows us to define the fpqc site of the category of schemes. However, there is a set theoretical problem that comes up when considering the fpqc topology, see Topologies, Section 34.9. It comes from our requirement that sites are “small”, but that no small category of schemes can contain a cofinal system of fpqc coverings of a given nonempty scheme. Although this does not strictly speaking prevent us from defining “partial” fpqc sites, it does not seem prudent to do so. The work-around is to allow the notion of a sheaf for the fpqc topology (see below) but to prohibit considering the category of all fpqc sheaves.

Definition 59.15.5. Let $S$ be a scheme. The category of schemes over $S$ is denoted $\mathit{Sch}/S$. Consider a functor $\mathcal{F} : (\mathit{Sch}/S)^{opp} \to \textit{Sets}$, in other words a presheaf of sets. We say $\mathcal{F}$ *satisfies the sheaf property for the fpqc topology* if for every fpqc covering $\{ U_ i \to U\} _{i \in I}$ of schemes over $S$ the diagram (59.11.1.1) is an equalizer diagram.

We similarly say that $\mathcal{F}$ *satisfies the sheaf property for the Zariski topology* if for every open covering $U = \bigcup _{i \in I} U_ i$ the diagram (59.11.1.1) is an equalizer diagram. See Schemes, Definition 26.15.3. Clearly, this is equivalent to saying that for every scheme $T$ over $S$ the restriction of $\mathcal{F}$ to the opens of $T$ is a (usual) sheaf.

Lemma 59.15.6. Let $\mathcal{F}$ be a presheaf on $\mathit{Sch}/S$. Then $\mathcal{F}$ satisfies the sheaf property for the fpqc topology if and only if

$\mathcal{F}$ satisfies the sheaf property with respect to the Zariski topology, and

for every faithfully flat morphism $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ of affine schemes over $S$, the sheaf axiom holds for the covering $\{ \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)\} $. Namely, this means that

\[ \xymatrix{ \mathcal{F}(\mathop{\mathrm{Spec}}(A)) \ar[r] & \mathcal{F}(\mathop{\mathrm{Spec}}(B)) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(\mathop{\mathrm{Spec}}(B \otimes _ A B)) } \]

is an equalizer diagram.

**Proof.**
See Topologies, Lemma 34.9.13.
$\square$

An alternative way to think of a presheaf $\mathcal{F}$ on $\mathit{Sch}/S$ which satisfies the sheaf condition for the fpqc topology is as the following data:

for each $T/S$, a usual (i.e., Zariski) sheaf $\mathcal{F}_ T$ on $T_{Zar}$,

for every map $f : T' \to T$ over $S$, a restriction mapping $f^{-1}\mathcal{F}_ T \to \mathcal{F}_{T'}$

such that

the restriction mappings are functorial,

if $f : T' \to T$ is an open immersion then the restriction mapping $f^{-1}\mathcal{F}_ T \to \mathcal{F}_{T'}$ is an isomorphism, and

for every faithfully flat morphism $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ over $S$, the diagram

\[ \xymatrix{ \mathcal{F}_{\mathop{\mathrm{Spec}}(A)}(\mathop{\mathrm{Spec}}(A)) \ar[r] & \mathcal{F}_{\mathop{\mathrm{Spec}}(B)}(\mathop{\mathrm{Spec}}(B)) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}_{\mathop{\mathrm{Spec}}(B \otimes _ A B)}(\mathop{\mathrm{Spec}}(B \otimes _ A B)) } \]

is an equalizer.

Data (1) and (2) and conditions (a), (b) give the data of a presheaf on $\mathit{Sch}/S$ satisfying the sheaf condition for the Zariski topology. By Lemma 59.15.6 condition (c) then suffices to get the sheaf condition for the fpqc topology.

Example 59.15.7. Consider the presheaf

\[ \begin{matrix} \mathcal{F} :
& (\mathit{Sch}/S)^{opp}
& \longrightarrow
& \textit{Ab}
\\ & T/S
& \longmapsto
& \Gamma (T, \Omega _{T/S}).
\end{matrix} \]

The compatibility of differentials with localization implies that $\mathcal{F}$ is a sheaf on the Zariski site. However, it does not satisfy the sheaf condition for the fpqc topology. Namely, consider the case $S = \mathop{\mathrm{Spec}}(\mathbf{F}_ p)$ and the morphism

\[ \varphi : V = \mathop{\mathrm{Spec}}(\mathbf{F}_ p[v]) \to U = \mathop{\mathrm{Spec}}(\mathbf{F}_ p[u]) \]

given by mapping $u$ to $v^ p$. The family $\{ \varphi \} $ is an fpqc covering, yet the restriction mapping $\mathcal{F}(U) \to \mathcal{F}(V)$ sends the generator $\text{d}u$ to $\text{d}(v^ p) = 0$, so it is the zero map, and the diagram

\[ \xymatrix{ \mathcal{F}(U) \ar[r]^{0} & \mathcal{F}(V) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(V \times _ U V) } \]

is not an equalizer. We will see later that $\mathcal{F}$ does in fact give rise to a sheaf on the étale and smooth sites.

Lemma 59.15.8. Any representable presheaf on $\mathit{Sch}/S$ satisfies the sheaf condition for the fpqc topology.

**Proof.**
See Descent, Lemma 35.13.7.
$\square$

We will return to this later, since the proof of this fact uses descent for quasi-coherent sheaves, which we will discuss in the next section. A fancy way of expressing the lemma is to say that *the fpqc topology is weaker than the canonical topology*, or that the fpqc topology is *subcanonical*. In the setting of sites this is discussed in Sites, Section 7.12.

Example 59.15.10. Let $S$ be a scheme. Consider the additive group scheme $\mathbf{G}_{a, S} = \mathbf{A}^1_ S$ over $S$, see Groupoids, Example 39.5.3. The associated representable presheaf is given by

\[ h_{\mathbf{G}_{a, S}}(T) = \mathop{\mathrm{Mor}}\nolimits _ S(T, \mathbf{G}_{a, S}) = \Gamma (T, \mathcal{O}_ T). \]

By the above we now know that this is a presheaf of sets which satisfies the sheaf condition for the fpqc topology. On the other hand, it is clearly a presheaf of rings as well. Hence we can think of this as a functor

\[ \begin{matrix} \mathcal{O} :
& (\mathit{Sch}/S)^{opp}
& \longrightarrow
& \textit{Rings}
\\ & T/S
& \longmapsto
& \Gamma (T, \mathcal{O}_ T)
\end{matrix} \]

which satisfies the sheaf condition for the fpqc topology. Correspondingly there is a notion of $\mathcal{O}$-module, and so on and so forth.

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