61.10 The V topology and the pro-h topology

The V topology was introduced in Topologies, Section 34.10. The h topology was introduced in More on Flatness, Section 38.34. A kind of intermediate topology, namely the ph topology, was introduced in Topologies, Section 34.8.

Given a topology $\tau$ on a suitable category $\mathcal{C}$ of schemes, we can introduce a “pro-$\tau$ topology” on $\mathcal{C}$ as follows. Recall that for $X$ in $\mathcal{C}$ we use $h_ X$ to denote the representable presheaf associated to $X$. Let us temporarily say a morphism $X \to Y$ of $\mathcal{C}$ is a $\tau$-cover1 if the $\tau$-sheafification of $h_ X \to h_ Y$ is surjective. Then we can define the pro-$\tau$ topology as the coarsest topology such that

1. the pro-$\tau$ topology is finer than the $\tau$ topology, and

2. $X \to Y$ is a pro-$\tau$-cover if $Y$ is affine and $X = \mathop{\mathrm{lim}}\nolimits X_\lambda$ is a directed limit of affine schemes $X_\lambda$ over $Y$ such that $h_{X_\lambda } \to h_ Y$ is a $\tau$-cover for all $\lambda$.

We use this pedantic formulation because we do not want to specify a choice of pro-$\tau$ coverings: for different $\tau$ different choices of collections of coverings are suitable. For example, in Section 61.12 we will see that in order to define the pro-étale topology looking at families of weakly étale morphisms with some finiteness property works well. More generally, the proposed construction given in this paragraph is meant mainly to motivate the results in this section and we will never implicitly define a pro-$\tau$ topology using this method.

The following lemma tells us that the pro-V topology is equal to the V topology.

Lemma 61.10.1. Let $Y$ be an affine scheme. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of affine schemes over $Y$. The following are equivalent

1. $\{ X \to Y\}$ is a standard V covering (Topologies, Definition 34.10.1), and

2. $\{ X_ i \to Y\}$ is a standard V covering for all $i$.

Proof. A singleton $\{ X \to Y\}$ is a standard V covering if and only if given a morphism $g : \mathop{\mathrm{Spec}}(V) \to Y$ there is an extension of valuation rings $V \subset W$ and a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r]^ g & Y }$

Thus (1) $\Rightarrow$ (2) is immediate from the definition. Conversely, assume (2) and let $g : \mathop{\mathrm{Spec}}(V) \to Y$ as above be given. Write $\mathop{\mathrm{Spec}}(V) \times _ Y X_ i = \mathop{\mathrm{Spec}}(A_ i)$. Since $\{ X_ i \to Y\}$ is a standard V covering, we may choose a valuation ring $W_ i$ and a ring map $A_ i \to W_ i$ such that the composition $V \to A_ i \to W_ i$ is an extension of valuation rings. In particular, the quotient $A'_ i$ of $A_ i$ by its $V$-torsion is a faitfhully flat $V$-algebra. Flatness by More on Algebra, Lemma 15.22.10 and surjectivity on spectra because $A_ i \to W_ i$ factors through $A'_ i$. Thus

$A = \mathop{\mathrm{colim}}\nolimits A'_ i$

is a faithfully flat $V$-algebra (Algebra, Lemma 10.39.20). Since $\{ \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(V)\}$ is a standard fpqc cover, it is a standard V cover (Topologies, Lemma 34.10.2) and hence we can choose $\mathop{\mathrm{Spec}}(W) \to \mathop{\mathrm{Spec}}(A)$ such that $V \to W$ is an extension of valuation rings. Since we can compose with the morphism $\mathop{\mathrm{Spec}}(A) \to X = \mathop{\mathrm{Spec}}(\mathop{\mathrm{colim}}\nolimits A_ i)$ the proof is complete. $\square$

The following lemma tells us that the pro-h topology is equal to the pro-ph topology is equal to the V topology.

Lemma 61.10.2. Let $X \to Y$ be a morphism of affine schemes. The following are equivalent

1. $\{ X \to Y\}$ is a standard V covering (Topologies, Definition 34.10.1),

2. $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a directed limit of affine schemes over $Y$ such that $\{ X_ i \to Y\}$ is a ph covering for each $i$, and

3. $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a directed limit of affine schemes over $Y$ such that $\{ X_ i \to Y\}$ is an h covering for each $i$.

Proof. Proof of (2) $\Rightarrow$ (1). Recall that a V covering given by a single arrow between affines is a standard V covering, see Topologies, Definition 34.10.7 and Lemma 34.10.6. Recall that any ph covering is a V covering, see Topologies, Lemma 34.10.10. Hence if $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as in (2), then $\{ X_ i \to Y\}$ is a standard V covering for each $i$. Thus by Lemma 61.10.1 we see that (1) is true.

Proof of (3) $\Rightarrow$ (2). This is clear because an h covering is always a ph covering, see More on Flatness, Definition 38.34.2.

Proof of (1) $\Rightarrow$ (3). This is the interesting direction, but the interesting content in this proof is hidden in More on Flatness, Lemma 38.34.1. Write $X = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(R)$. We can write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ with $A_ i$ of finite presentation over $R$, see Algebra, Lemma 10.127.2. Set $X_ i = \mathop{\mathrm{Spec}}(A_ i)$. Then $\{ X_ i \to Y\}$ is a standard V covering for all $i$ by (1) and Topologies, Lemma 34.10.6. Hence $\{ X_ i \to Y\}$ is an h covering by More on Flatness, Definition 38.34.2. This finishes the proof. $\square$

The following lemma tells us, roughly speaking, that an h sheaf which is limit preserving satisfies the sheaf condition for V coverings. Please also compare with Remark 61.10.4.

Lemma 61.10.3. Let $S$ be a scheme. Let $F$ be a contravariant functor defined on the category of all schemes over $S$. If

1. $F$ satisfies the sheaf property for the h topology, and

2. $F$ is limit preserving (Limits, Remark 32.6.2),

then $F$ satisfies the sheaf property for the V topology.

Proof. We will prove this by verifying (1) and (2') of Topologies, Lemma 34.10.12. The sheaf property for Zariski coverings follows from the fact that $F$ has the sheaf property for all h coverings. Finally, suppose that $X \to Y$ is a morphism of affine schemes over $S$ such that $\{ X \to Y\}$ is a V covering. By Lemma 61.10.2 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as a directed limit of affine schemes over $Y$ such that $\{ X_ i \to Y\}$ is an h covering for each $i$. We obtain

\begin{align*} & \text{Equalizer}( \xymatrix{ F(X) \ar@<1ex>[r] \ar@<-1ex>[r] & F(X \times _ Y X) } ) \\ & = \text{Equalizer}( \xymatrix{ \mathop{\mathrm{colim}}\nolimits F(X_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathop{\mathrm{colim}}\nolimits F(X_ i \times _ Y X_ i) } ) \\ & = \mathop{\mathrm{colim}}\nolimits \text{Equalizer}( \xymatrix{ F(X_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & F(X_ i \times _ Y X_ i) } ) \\ & = \mathop{\mathrm{colim}}\nolimits F(Y) = F(Y) \end{align*}

which is what we wanted to show. The first equality because $F$ is limit preserving and $X = \mathop{\mathrm{lim}}\nolimits X_ i$ and $X \times _ Y X = \mathop{\mathrm{lim}}\nolimits X_ i \times _ Y X_ i$. The second equality because filtered colimits are exact. The third equality because $F$ satisfies the sheaf property for h coverings. $\square$

Remark 61.10.4. Let $S$ be a scheme contained in a big site $\mathit{Sch}_ h$. Let $F$ be a sheaf of sets on $(\mathit{Sch}/S)_ h$ such that $F(T) = \mathop{\mathrm{colim}}\nolimits F(T_ i)$ whenever $T = \mathop{\mathrm{lim}}\nolimits T_ i$ is a directed limit of affine schemes in $(\mathit{Sch}/S)_ h$. In this situation $F$ extends uniquely to a contravariant functor $F'$ on the category of all schemes over $S$ such that (a) $F'$ satisfies the sheaf property for the h topology and (b) $F'$ is limit preserving. See More on Flatness, Lemma 38.35.4. In this situation Lemma 61.10.3 tells us that $F'$ satisfies the sheaf property for the V topology.

[1] This should not be confused with the notion of a covering. For example if $\tau = {\acute{e}tale}$, any morphism $X \to Y$ which has a section is a $\tau$-covering. But our definition of étale coverings $\{ V_ i \to Y\} _{i \in I}$ forces each $V_ i \to Y$ to be étale.

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