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54.15 The fpqc topology

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

Definition 54.15.1. Let $T$ be a scheme. An fpqc covering of $T$ is a family $\{ \varphi _ i : T_ i \to T\} _{i \in I}$ such that

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

  2. 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 $U = \bigcup _{k \in K} \varphi _{\mathbf{i}(k)}(U_{\mathbf{i}(k)})$.

Remark 54.15.2. The first condition corresponds to fp, which stands for fidèlement plat, faithfully flat in french, and the second to qc, quasi-compact. The second part of the first condition is unnecessary when the second condition holds.

Example 54.15.3. Examples of fpqc coverings.

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

  2. 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.

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

  4. 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.

  5. Write $\mathbf{A}^2_ k = \mathop{\mathrm{Spec}}(k[x, y])$. Denote $i_ x : D(x) \to \mathbf{A}^2_ k$ and $i_ y : D(y) \hookrightarrow \mathbf{A}^2_ k$ 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 54.15.4. The collection of fpqc coverings on the category of schemes satisfies the axioms of site.

Proof. See Topologies, Lemma 33.9.7. $\square$

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 33.9. 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 54.15.5. 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 ( 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 ( is an equalizer diagram. See Schemes, Definition 25.15.3. 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 54.15.6. 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

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

  2. 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 33.9.13. $\square$

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:

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

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

such that

  1. the restriction mappings are functorial,

  2. 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

  3. 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 54.15.6 condition (c) then suffices to get the sheaf condition for the fpqc topology.

Example 54.15.7. 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 54.15.8. Any representable presheaf on $\mathit{Sch}/S$ satisfies the sheaf condition for the fpqc topology.

Proof. See Descent, Lemma 34.10.7. $\square$

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.

Remark 54.15.9. The fpqc is the finest topology that we will see. Hence any presheaf satisfying the sheaf condition for the fpqc topology will be a sheaf in the subsequent sites (étale, smooth, etc). In particular representable presheaves will be sheaves on the étale site of a scheme for example.

Example 54.15.10. Let $S$ be a scheme. Consider the additive group scheme $\mathbf{G}_{a, S} = \mathbf{A}^1_ S$ over $S$, see Groupoids, Example 38.5.3. The associated representable presheaf is given by

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

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.

Comments (2)

Comment #152 by on

In the paragraph after tag 54.15.4 (it doesn't belong to a specific tag itself) the reference for the set theoretical issues with the fpqc topology is the section of the fpqc topology itself. This postmodern self-reference might induce set theoretical issues by its mere existence, so maybe (given the title that is used) tag 7.47 is meant?

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