The Stacks project

34.10 The V topology

The V topology is weaker than all other topologies in this chapter. Roughly speaking it is generated by Zariski coverings and by quasi-compact morphisms satisfying a lifting property for specializations (Lemma 34.10.13). However, the procedure we will use to define V coverings is a bit different. We will first define standard V coverings of affines and then use these to define V coverings in general. Typographical point: in the literature sometimes “$v$-covering” is used instead of “V covering”.

Definition 34.10.1. Let $T$ be an affine scheme. A standard V covering is a finite family $\{ T_ j \to T\} _{j = 1, \ldots , m}$ with $T_ j$ affine such that for every morphism $g : \mathop{\mathrm{Spec}}(V) \to T$ where $V$ is a valuation ring, there is an extension $V \subset W$ of valuation rings (More on Algebra, Definition 15.112.1), an index $1 \leq j \leq m$, and a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[r] \ar[d] & T_ j \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r]^ g & T } \]

We first prove a few basic lemmas about this notion.

Proof. Let $\{ X_ i \to X\} _{i = 1, \ldots , n}$ be a standard fpqc covering (Definition 34.9.9). Let $g : \mathop{\mathrm{Spec}}(V) \to X$ be a morphism where $V$ is a valuation ring. Let $x \in X$ be the image of the closed point of $\mathop{\mathrm{Spec}}(V)$. Choose an $i$ and a point $x_ i \in X_ i$ mapping to $x$. Then $\mathop{\mathrm{Spec}}(V) \times _ X X_ i$ has a point $x'_ i$ mapping to the closed point of $\mathop{\mathrm{Spec}}(V)$. Since $\mathop{\mathrm{Spec}}(V) \times _ X X_ i \to \mathop{\mathrm{Spec}}(V)$ is flat we can find a specialization $x''_ i \leadsto x'_ i$ of points of $\mathop{\mathrm{Spec}}(V) \times _ X X_ i$ with $x''_ i$ mapping to the generic point of $\mathop{\mathrm{Spec}}(V)$, see Morphisms, Lemma 29.25.9. By Schemes, Lemma 26.20.4 we can choose a valuation ring $W$ and a morphism $h : \mathop{\mathrm{Spec}}(W) \to \mathop{\mathrm{Spec}}(V) \times _ X X_ i$ such that $h$ maps the generic point of $\mathop{\mathrm{Spec}}(W)$ to $x''_ i$ and the closed point of $\mathop{\mathrm{Spec}}(W)$ to $x'_ i$. We obtain a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[r] \ar[d] & X_ i \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & X } \]

where $V \to W$ is an extension of valuation rings. This proves the lemma. $\square$

Proof. Let $T$ be an affine scheme. Let $f : U \to T$ be a proper surjective morphism. Let $U = \bigcup _{j = 1, \ldots , m} U_ j$ be a finite affine open covering. We have to show that $\{ U_ j \to T\} $ is a standard V covering, see Definition 34.8.1. Let $g : \mathop{\mathrm{Spec}}(V) \to T$ be a morphism where $V$ is a valuation ring with fraction field $K$. Since $U \to T$ is surjective, we may choose a field extension $L/K$ and a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(L) \ar[rr] \ar[d] & & U \ar[d] \\ \mathop{\mathrm{Spec}}(K) \ar[r] & \mathop{\mathrm{Spec}}(V) \ar[r]^ g & T } \]

By Algebra, Lemma 10.49.2 we can choose a valuation ring $W \subset L$ dominating $V$. By the valuative criterion of properness (Morphisms, Lemma 29.41.1) we can then find the morphism $h$ in the commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(L) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(W) \ar[r]_ h \ar[d] & U \ar[d] \\ \mathop{\mathrm{Spec}}(K) \ar[r] & \mathop{\mathrm{Spec}}(V) \ar[r]^ g & X } \]

Since $\mathop{\mathrm{Spec}}(W)$ has a unique closed point, we see that $\mathop{\mathrm{Im}}(h)$ is contained in $U_ j$ for some $j$. Thus $h : \mathop{\mathrm{Spec}}(W) \to U_ j$ is the desired lift and we conclude $\{ U_ j \to T\} $ is a standard V covering. $\square$

Lemma 34.10.4. Let $\{ T_ j \to T\} _{j = 1, \ldots , m}$ be a standard V covering. Let $T' \to T$ be a morphism of affine schemes. Then $\{ T_ j \times _ T T' \to T'\} _{j = 1, \ldots , m}$ is a standard V covering.

Proof. Let $\mathop{\mathrm{Spec}}(V) \to T'$ be a morphism where $V$ is a valuation ring. By assumption we can find an extension of valuation rings $V \subset W$, an $i$, and a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[r] \ar[d] & T_ i \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & T } \]

By the universal property of fibre products we obtain a morphism $\mathop{\mathrm{Spec}}(W) \to T' \times _ T T_ i$ as desired. $\square$

Lemma 34.10.5. Let $T$ be an affine scheme. Let $\{ T_ j \to T\} _{j = 1, \ldots , m}$ be a standard V covering. Let $\{ T_{ji} \to T_ j\} _{i = 1, \ldots n_ j}$ be a standard V covering. Then $\{ T_{ji} \to T\} _{i, j}$ is a standard V covering.

Proof. This follows formally from the observation that if $V \subset W$ and $W \subset \Omega $ are extensions of valuation rings, then $V \subset \Omega $ is an extension of valuation rings. $\square$

Lemma 34.10.6. Let $T$ be an affine scheme. Let $\{ T_ j \to T\} _{j = 1, \ldots , m}$ be a family of morphisms with $T_ j$ affine for all $j$. The following are equivalent

  1. $\{ T_ j \to T\} _{j = 1, \ldots , m}$ is a standard V covering,

  2. there is a standard V covering which refines $\{ T_ j \to T\} _{j = 1, \ldots , m}$, and

  3. $\{ \coprod _{j = 1, \ldots , m} T_ j \to T\} $ is a standard V covering.

Proof. Omitted. Hints: This follows almost immediately from the definition. The only slightly interesting point is that a morphism from the spectrum of a local ring into $\coprod _{j = 1, \ldots , m} T_ j$ must factor through some $T_ j$. $\square$

Definition 34.10.7. Let $T$ be a scheme. A V covering of $T$ is a family of morphisms $\{ T_ i \to T\} _{i \in I}$ of schemes such that for every affine open $U \subset T$ there exists a standard V covering $\{ U_ j \to U\} _{j = 1, \ldots , m}$ refining the family $\{ T_ i \times _ T U \to U\} _{i \in I}$.

The V topology has the same set theoretical problems as the fpqc topology. Thus we refrain from defining V sites and we will not consider cohomology with respect to the V topology. On the other hand, given a $F : \mathit{Sch}^{opp} \to \textit{Sets}$ it does make sense to ask whether $F$ satisfies the sheaf property for the V topology, see below. Moreover, we can wonder about descent of object in the V topology, etc.

Lemma 34.10.8. Let $T$ be a scheme. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be a family of morphisms. The following are equivalent

  1. $\{ T_ i \to T\} _{i \in I}$ is a V covering,

  2. there is a V covering which refines $\{ T_ i \to T\} _{i \in I}$, and

  3. $\{ \coprod _{i \in I} T_ i \to T\} $ is a V covering.

Proof. Omitted. Hint: compare with the proof of Lemma 34.8.7. $\square$

Lemma 34.10.9. Let $T$ be a scheme.

  1. If $T' \to T$ is an isomorphism then $\{ T' \to T\} $ is a V covering of $T$.

  2. If $\{ T_ i \to T\} _{i\in I}$ is a V covering and for each $i$ we have a V covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a V covering.

  3. If $\{ T_ i \to T\} _{i\in I}$ is a V covering and $T' \to T$ is a morphism of schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is a V covering.

Proof. Assertion (1) is clear.

Proof of (3). Let $U' \subset T'$ be an affine open subscheme. Since $U'$ is quasi-compact we can find a finite affine open covering $U' = U'_1 \cup \ldots \cup U'$ such that $U'_ j \to T$ maps into an affine open $U_ j \subset T$. Choose a standard V covering $\{ U_{jl} \to U_ j\} _{l = 1, \ldots , n_ j}$ refining $\{ T_ i \times _ T U_ j \to U_ j\} $. By Lemma 34.10.4 the base change $\{ U_{jl} \times _{U_ j} U'_ j \to U'_ j\} $ is a standard V covering. Note that $\{ U'_ j \to U'\} $ is a standard V covering (for example by Lemma 34.10.2). By Lemma 34.10.5 the family $\{ U_{jl} \times _{U_ j} U'_ j \to U'\} $ is a standard V covering. Since $\{ U_{jl} \times _{U_ j} U'_ j \to U'\} $ refines $\{ T_ i \times _ T U' \to U'\} $ we conclude.

Proof of (2). Let $U \subset T$ be affine open. First we pick a standard V covering $\{ U_ k \to U\} _{k = 1, \ldots , m}$ refining $\{ T_ i \times _ T U \to U\} $. Say the refinement is given by morphisms $U_ k \to T_{i_ k}$ over $T$. Then

\[ \{ T_{i_ kj} \times _{T_{i_ k}} U_ k \to U_ k\} _{j \in J_{i_ k}} \]

is a V covering by part (3). As $U_ k$ is affine, we can find a standard V covering $\{ U_{ka} \to U_ k\} _{a = 1, \ldots , b_ k}$ refining this family. Then we apply Lemma 34.10.5 to see that $\{ U_{ka} \to U\} $ is a standard V covering which refines $\{ T_{ij} \times _ T U \to U\} $. This finishes the proof. $\square$

Lemma 34.10.10. Any fpqc covering is a V covering. A fortiori, any fppf, syntomic, smooth, étale or Zariski covering is a V covering. Also, a ph covering is a V covering.

Proof. An fpqc covering can affine locally be refined by a standard fpqc covering, see Lemmas 34.9.8. A standard fpqc covering is a standard V covering, see Lemma 34.10.2. Hence the first statement follows from our definition of V covers in terms of standard V coverings. The conclusion for fppf, syntomic, smooth, étale or Zariski coverings follows as these are fpqc coverings, see Lemma 34.9.6.

The statement on ph coverings follows from Lemma 34.10.3 in the same manner. $\square$

Definition 34.10.11. Let $F$ be a contravariant functor on the category of schemes with values in sets. We say that $F$ satisfies the sheaf property for the V topology if it satisfies the sheaf property for any V covering (see Definition 34.9.12).

We try to avoid using the terminology “$F$ is a sheaf” in this situation since we are not defining a category of V sheaves as we explained above.

Lemma 34.10.12. Let $F$ be a contravariant functor on the category of schemes with values in sets. Then $F$ satisfies the sheaf property for the V topology if and only if it satisfies

  1. the sheaf property for every Zariski covering, and

  2. the sheaf property for any standard V covering.

Moreover, in the presence of (1) property (2) is equivalent to property

  1. the sheaf property for a standard V covering of the form $\{ V \to U\} $, i.e., consisting of a single arrow.

Proof. Assume (1) and (2) hold. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be a V covering. Let $s_ i \in F(T_ i)$ be a family of elements such that $s_ i$ and $s_ j$ map to the same element of $F(T_ i \times _ T T_ j)$. Let $W \subset T$ be the maximal open subset such that there exists a unique $s \in F(W)$ with $s|_{f_ i^{-1}(W)} = s_ i|_{f_ i^{-1}(W)}$ for all $i$. Such a maximal open exists because $F$ satisfies the sheaf property for Zariski coverings; in fact $W$ is the union of all opens with this property. Let $t \in T$. We will show $t \in W$. To do this we pick an affine open $t \in U \subset T$ and we will show there is a unique $s \in F(U)$ with $s|_{f_ i^{-1}(U)} = s_ i|_{f_ i^{-1}(U)}$ for all $i$.

We can find a standard V covering $\{ U_ j \to U\} _{j = 1, \ldots , n}$ refining $\{ U \times _ T T_ i \to U\} $, say by morphisms $h_ j : U_ j \to T_{i_ j}$. By (2) we obtain a unique element $s \in F(U)$ such that $s|_{U_ j} = F(h_ j)(s_{i_ j})$. Note that for any scheme $V \to U$ over $U$ there is a unique section $s_ V \in F(V)$ which restricts to $F(h_ j \circ \text{pr}_2)(s_{i_ j})$ on $V \times _ U U_ j$ for $j = 1, \ldots , n$. Namely, this is true if $V$ is affine by (2) as $\{ V \times _ U U_ j \to V\} $ is a standard V covering (Lemma 34.10.4) and in general this follows from (1) and the affine case by choosing an affine open covering of $V$. In particular, $s_ V = s|_ V$. Now, taking $V = U \times _ T T_ i$ and using that $s_{i_ j}|_{T_{i_ j} \times _ T T_ i} = s_ i|_{T_{i_ j} \times _ T T_ i}$ we conclude that $s|_{U \times _ T T_ i} = s_ V = s_ i|_{U \times _ T T_ i}$ which is what we had to show.

Proof of the equivalence of (2) and (2') in the presence of (1). Suppose $\{ T_ i \to T\} _{i = 1, \ldots , n}$ is a standard V covering, then $\coprod _{i = 1, \ldots , n} T_ i \to T$ is a morphism of affine schemes which is clearly also a standard V covering. In the presence of (1) we have $F(\coprod T_ i) = \prod F(T_ i)$ and similarly $F((\coprod T_ i) \times _ T (\coprod T_ i)) = \prod F(T_ i \times _ T T_{i'})$. Thus the sheaf condition for $\{ T_ i \to T\} $ and $\{ \coprod T_ i \to T\} $ is the same. $\square$

The following lemma shows that being a V covering is related to the possibility of lifting specializations.

Lemma 34.10.13. Let $X \to Y$ be a quasi-compact morphism of schemes. The following are equivalent

  1. $\{ X \to Y\} $ is a V covering,

  2. for any valuation ring $V$ and morphism $g : \mathop{\mathrm{Spec}}(V) \to Y$ there exists an extension of valuation rings $V \subset W$ and a commutative diagram

    \[ \xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & Y } \]
  3. for any morphism $Z \to Y$ and specialization $z' \leadsto z$ of points in $Z$, there is a specialization $w' \leadsto w$ of points in $Z \times _ Y X$ mapping to $z' \leadsto z$.

Proof. Assume (1) and let $g : \mathop{\mathrm{Spec}}(V) \to Y$ be as in (2). Since $V$ is a local ring there is an affine open $U \subset Y$ such that $g$ factors through $U$. By Definition 34.10.7 we can find a standard V covering $\{ U_ j \to U\} $ refining $\{ X \times _ Y U \to U\} $. By Definition 34.10.1 we can find a $j$, an extension of valuation rings $V \subset W$ and a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[r] \ar[d] & U_ j \ar[d] \ar@{..>}[r] & X \ar[ld] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & Y } \]

We have the dotted arrow making the diagram commute by the refinement property of the covering and we see that (2) holds.

Assume (2) and let $Z \to Y$ and $z' \leadsto z$ be as in (3). By Schemes, Lemma 26.20.4 we can find a valuation ring $V$ and a morphism $\mathop{\mathrm{Spec}}(V) \to Z$ such that the closed point of $\mathop{\mathrm{Spec}}(V)$ maps to $z$ and the generic point of $\mathop{\mathrm{Spec}}(V)$ maps to $z'$. By (2) we can find an extension of valuation rings $V \subset W$ and a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[rr] \ar[d] & & X \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & Z \ar[r] & Y } \]

The generic and closed points of $\mathop{\mathrm{Spec}}(W)$ map to points $w' \leadsto w$ in $Z \times _ Y X$ via the induced morphism $\mathop{\mathrm{Spec}}(W) \to Z \times _ Y X$. This shows that (3) holds.

Assume (3) holds and let $U \subset Y$ be an affine open. Choose a finite affine open covering $U \times _ Y X = \bigcup _{j = 1, \ldots , m} U_ j$. This is possible as $X \to Y$ is quasi-compact. We claim that $\{ U_ j \to U\} $ is a standard V covering. The claim implies (1) is true and finishes the proof of the lemma. In order to prove the claim, let $V$ be a valuation ring and let $g : \mathop{\mathrm{Spec}}(V) \to U$ be a morphism. By (3) we find a specialization $w' \leadsto w$ of points of

\[ T = \mathop{\mathrm{Spec}}(V) \times _ X Y = \mathop{\mathrm{Spec}}(V) \times _ U (U \times _ X Y) \]

such that $w'$ maps to the generic point of $\mathop{\mathrm{Spec}}(V)$ and $w$ maps to the closed point of $\mathop{\mathrm{Spec}}(V)$. By Schemes, Lemma 26.20.4 we can find a valuation ring $W$ and a morphism $\mathop{\mathrm{Spec}}(W) \to T$ such that the generic point of $\mathop{\mathrm{Spec}}(W)$ maps to $w'$ and the closed point of $\mathop{\mathrm{Spec}}(W)$ maps to $w$. The composition $\mathop{\mathrm{Spec}}(W) \to T \to \mathop{\mathrm{Spec}}(V)$ corresponds to an inclusion $V \subset W$ which presents $W$ as an extension of the valuation ring $V$. Since $T = \bigcup \mathop{\mathrm{Spec}}(V) \times _ U U_ j$ is an open covering, we see that $\mathop{\mathrm{Spec}}(W) \to T$ factors through $\mathop{\mathrm{Spec}}(V) \times _ U U_ j$ for some $j$. Thus we obtain a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[d] \ar[r] & U_ j \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & U } \]

and the proof of the claim is complete. $\square$

A V covering gives a universally submersive family of maps. The converse of this lemma is false, see Examples, Section 108.71.

Lemma 34.10.14. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be a V covering. Then

\[ \coprod \nolimits _{i \in I} f_ i : \coprod \nolimits _{i \in I} X_ i \longrightarrow X \]

is a universally submersive morphism of schemes (Morphisms, Definition 29.24.1).

Proof. We will use without further mention that the base change of a V covering is a V covering (Lemma 34.10.9). In particular it suffices to show that the morphism is submersive. Being submersive is clearly Zariski local on the base. Thus we may assume $X$ is affine. Then $\{ X_ i \to X\} $ can be refined by a standard V covering $\{ Y_ j \to X\} $. If we can show that $\coprod Y_ j \to X$ is submersive, then since there is a factorization $\coprod Y_ j \to \coprod X_ i \to X$ we conclude that $\coprod X_ i \to X$ is submersive. Set $Y = \coprod Y_ j$ and consider the morphism of affines $f : Y \to X$. By Lemma 34.10.13 we know that we can lift any specialization $x' \leadsto x$ in $X$ to some specialization $y' \leadsto y$ in $Y$. Thus if $T \subset X$ is a subset such that $f^{-1}(T)$ is closed in $Y$, then $T \subset X$ is closed under specialization. Since $f^{-1}(T) \subset Y$ with the reduced induced closed subscheme structure is an affine scheme, we conclude that $T \subset X$ is closed by Algebra, Lemma 10.40.5. Hence $f$ is submersive. $\square$


Comments (2)

Comment #5415 by on

Is the v-topology subcanonical? I can't find a statement saying yea or nay. I did find that the h-topology is not subcanonical (https://stacks.math.columbia.edu/tag/0EV1)

Comment #5419 by Laurent Moret-Bailly on

Beginning of section: by "weaker", don't you mean "stronger"?


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