The arc-topology and friends

Section 20 The arc-topology and friends

Reference.

We describe an exotic Grothendieck topology on the category of schemes, the arc-topology, and its close relatives, the h-topology and v-topology. This will be useful in the study of the étale comparison map (Section 22).

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Subsection 20.1 Grothendieck topologies

In Section 11, we introduced indiscrete Grothendieck topologies as a shortcut to getting to the construction of prismatic cohomology. Since we will be discussing various Grothendieck topologies on the category of schemes, we must say a bit more now.

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Definition 20.1.1.

A Grothendieck topology on a category

C

consists of a collection of (set-indexed) families of morphisms

{U_{i} \to U}_{i \in I}

with a single target

U,

the coverings, subject to the following restrictions.

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Any isomorphism, viewed as a singleton family, is a covering.
If ${U_{i} \to U}_{i \in I}$ is a covering and, for each $i,$ ${V_{i j} \to U_{i}}_{j \in J_{i}}$ is a covering, then the composition ${V_{i j} \to U}_{i \in I, j \in J_{i}}$ is a covering. (In short, a covering of the terms in a covering gives a covering.)
If ${U_{i} \to U}_{i \in I}$ is a covering and $V \to U$ is any morphism of $C,$ then the fiber products $U_{i} \times_{U} V$ exist for all $i \in I$ and ${U_{i} \times_{U} V \to V}_{i \in I}$ is a covering. (In short, the restriction of a covering is a covering.)

A category equipped with a Grothendieck topology is called a site.

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A presheaf on a site valued in

Set

is a contravariant functor

F : C \to Set .

A sheaf is a presheaf such that for every covering

{U_{i} \to U}_{i \in I},

F (U)

is the limit of the diagram

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\prod_{i \in I} F (U_{i}) ⇉ \prod_{(i_{0}, i_{1}) \in I \times I} F (U_{i_{0}} \times_{U} U_{i_{1}}) .

The category of sheaves of sets on the site is called the topos associated to the site; it is in many ways a more canonical object, in that there are usually many different ways to construct families of coverings (or even underlying categories) that give rise to equivalent topoi. In particular, one can “sheafify” the definition of a morphism of sites to obtain morphisms of topoi, some of which do not arise from morphisms of the underlying sites. We will not dwell on this too much here, but see [117], tag 00X9.

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Subsection 20.2 Valuation rings

Definition 20.2.1.

A valuation ring is a local integral domain

V

which, as a subring of its fraction field

K,

is maximal with respect to local inclusions of local rings. In this case, the group

Γ = K^{\times} / V^{\times}

(the value group of

A

) is totally ordered with the nonnegative elements being

(V ∖ {0}) / V^{\times} .

See [117], tag 00I8 for more on valuation rings.

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We say that

V

is eudoxian if its value group satisfies the equivalent conditions of Lemma 20.2.2.

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We define an arc to be a scheme of the form

Spec (V)

where

V

is a eudoxian valuation ring. For example, a scheme which is the spectrum of a discrete valuation ring (sometimes called a trait or a dash) is an arc. (This terminology is hinted at in [21] but not actually introduced there.)

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Lemma 20.2.2.

For

Γ

a totally ordered abelian group, the following statements are equivalent.

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For any two elements $α, β \in Γ$ with $α, β > 0,$ there exists a positive integer $n$ such that $n α > β .$
The group $Γ$ admits an order-preserving isomorphism with a subgroup of the additive group $R .$

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Proof.

It is obvious that (2) implies (1). Conversely, if (1) holds and

Γ

is nontrivial (as otherwise there is nothing to check), we can fix a single

α \in Γ

and define a function

f : Γ \to R

by the formula

f (β) = sup {\frac{r}{s} : r, s \in Z, s > 0, s β > r α}

(condition (1) guaranteeing that the set in question is bounded above). We leave it to the reader to verify that this indeed gives an injective order-preserving homomorphism (Exercise 20.4.1).

Remark 20.2.3.

A typical example of a totally ordered abelian group not satisfying the conditions of Lemma 20.2.2 is the group

R \times R

with the lexicographic ordering.

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Remark 20.2.4.

A eudoxian valuation ring is microbial in the sense of Huber [67], but not conversely; the latter requires that there be a “leading term” while still having intermediate specializations. An example of a totally ordered abelian group that is not microbial is the infinite direct sum

\oplus_{m \in Z} R

with the lexicographic ordering.

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Corollary 20.2.5.

For

V

a valuation ring,

Spec (V)

is an arc if and only if it contains at most two points (the generic point and the special point, which coincide if and only if

V

is a field).

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Proof.

Exercise (see Exercise 20.4.2).

Remark 20.2.6.

Condition (1) in Lemma 20.2.2 is commonly called the archimedean property of a totally ordered group. We prefer the adjective eudoxian both for historical accuracy and to avoid creating confusion with the use of the term nonarchimedean in reference to an associated absolute value of a eudoxian valuation.

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Remark 20.2.7.

Recall (Definition 19.4.1) that a ring

R

is said to be absolutely integrally closed (or AIC) if every monic polynomial over

R

has a root in

R .

When

R = V

is a valuation ring, this is equivalent to requiring that its fraction field is algebraically closed. In particular, any (eudoxian) valuation ring can be embedded in an AIC (eudoxian) valuation ring.

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Subsection 20.3 The arc-topology

Definition 20.3.1.

As per [21] (and an as yet unavailable sequel to [106]), we say that a morphism

f : Y \to X

of schemes is an arc-covering if for any morphism

Spec (V) \to X

from an arc into

X,

there exists a commuting diagram as in Figure 20.3.2 in which

Spec (W) \to Spec (V)

is a faithfully flat morphism of arcs. (We do not require the map

V \to W

to be integral.)

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Lemma 20.3.3.

Let

f : Y \to X

be a morphism of schemes.

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If $f$ is faithfully flat, then it is an arc-covering.
If $f$ is proper and surjective, then it is an arc-covering.

Moreover, in both cases

f

is also a v-covering (see Remark 20.3.7).

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Proof.

For (1), we first lift the closed point of

Spec (V),

and then lift generizations. For (2), we first lift the generic point of

Spec (V),

and then apply the valuative criterion for properness. In both cases, the condition that

V

is eudoxian plays no role. (Compare [106], Remark 2.5 or [24], Example 2.3.)

Example 20.3.4.

Let

X = Spec R

where

R \in Ring

is noetherian. Then the map

R \to \prod_{m} R_{m}^{\land},

where

m

runs over the product of all maximal ideals of

R,

is a faithfully flat morphism and hence an arc-covering.

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Example 20.3.5.

Let

X = Spec (k [x, y])

be the affine plane over a field

k,

let

\tilde{f} : \tilde{Y} \to X

be the blowup at the origin, let

Y

be the complement in

\tilde{Y}

of a single closed point in the exceptional locus, and let

f : Y \to X

be the induced morphism. Then

f : Y \to X

is surjective but not an arc-covering: we can choose an arc whose special point maps to the origin in

X

and whose generic point maps to the direction corresponding to the missing point in the exceptional locus, and such an arc will not lift to

Y .

(Again, compare [106], Remark 2.5 or [24], Example 2.3.)

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Definition 20.3.6.

The arc-topology on the category of schemes is the Grothendieck topology in which a family

{f_{i} : Y_{i} \to X}_{i \in I}

of morphisms is considered to be a covering if for any open affine

V \subseteq X,

there exists a map

t : K \to I

of sets with

K

finite and some affine opens

U_{k} \subseteq f_{t (k)}^{- 1} (V)

for each

k \in K

such that the induced map

⊔_{k} U_{k} \to V

is an arc-covering.

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Remark 20.3.7.

Lemma 20.3.3 shows that the arc-topology includes many more coverings than the flat topology. This leads to some potentially confusing behavior: for instance, the structure presheaf

X \mapsto Γ (X, O)

is not a sheaf for the arc-topology, because its sheafification does not include nilpotents (the inclusion of the reduced closed subscheme is an arc-covering). This can (and generally should) be circumvented by working with derived categories.

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In any case, there is plenty of precedent for considering topologies of this nature. For example, Voevodsky [123] considered the h-topology, generated by étale coverings and proper surjective morphisms. A more recent variant is the universally subtrusive topology of Rydh [106], which is defined similarly to the arc-topology except that the lifting property is required for all valuation rings, not just eudoxian ones; following [24], this is now commonly called the v-topology. For morphisms of finite type between noetherian schemes, the h-topology, the v-topology, and the arc-topology all coincide, but not otherwise; see Remark 20.3.11 for a minimal counterexample and [21], section 1.1 for more discussion.

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We describe two fundamental examples of coverings.

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Example 20.3.8.

Let

A

be a ring. Let

(A \to V_{i})_{i \in I}

be a set of isomorphism class representatives of

A

-algebras which are AIC valuation rings of cardinality at most

max {ℵ_{0}, # A}

and put

B = \prod_{i \in I} V_{i} .

The map

A \to B

is a v-covering: any morphism

f : A \to V

to a valuation ring factors through the intersection

Frac (f (A)) \cap V

within

Frac (V),

and hence through some

V_{i} .

(Compare [21], Proposition 3.30.)

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Remark 20.3.9.

In Example 20.3.8, the connected components of the ring

V

are indexed by the set

I .

However, if

I

is infinite, then the spectrum of

V

is much larger than the set of kernels of projections

V \to V_{i} :

it also includes maximal ideals corresponding to ultraproducts of the

V_{i} .

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Example 20.3.10.

Let V be a valuation ring and let

p

be a prime ideal of

V .

Then

V \to V_{p} \times V / p

is an arc-covering, but not a v-covering unless

p

is zero or the maximal ideal. (See [21], Corollary 2.9.)

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Remark 20.3.11.

One can modify Example 20.3.10 to obtain a finitely presented morphism, as follows. Let

V

be a valuation ring which is not eudoxian. Let

p

be a prime ideal which is neither zero nor the maximal ideal (see Corollary 20.2.5). Then for any

f \in V ∖ p,

V \to V_{f} \times V / f

is an arc-covering but not a v-covering. (Compare [21], Example 1.3.)

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We record some variations on Example 20.3.8.

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Remark 20.3.12.

In Example 20.3.8, let

J

be the subset of

i \in I

for which

V_{i}

is eudoxian. Then

C = \prod_{j \in J} V_{j}

is an arc-covering, but not in general a v-covering as per Example 20.3.10.

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This remains true if we replace each

V_{j}

with a larger valuation ring. In particular, we can ensure that each factor is not a field, and even complete with respect to its valuation.

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Remark 20.3.13.

It is possible to characterize arc-coverings of qcqs schemes in purely topological terms: they are precisely the universal spectral submersions ([21], Proposition 2.19). See Exercise 20.4.3 for a related observation.

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