An arithmetic valuative criterion for proper maps of tame algebraic stacks

The valuative criterion for proper maps of schemes has many applications in arithmetic, e.g. specializing $\mathbb{Q}_{p}$-points to $\mathbb{F}_{p}$-points. For algebraic stacks, the usual valuative criterion for proper maps is ill-suited for these kind of arguments, since it only gives a specialization point defined over an extension of the residue field, e.g. a $\mathbb{Q}_{p}$-point will specialize to an $\mathbb{F}_{p^{n}}$-point for some $n$. We give a new valuative criterion for proper maps of tame stacks which solves this problem and is well-suited for arithmetic applications. As a consequence, we prove that the Lang-Nishimura theorem holds for tame stacks.


Introduction
The well known and extremely useful valuative criterion for properness says, in particular, that if X → Y is a proper morphism of schemes, R is a DVR with quotient field K and residue field k, and we have a commutative diagram Spec K X

Spec R S
there exists a unique lifting Spec R → X of Spec R → Y extending Spec K → X.This has many arithmetic applications: most of them use that the statement above ensures the existence of a lifting Spec k → X of the composite Spec k ⊆ Spec R → Y .
If X and Y are algebraic stack, and X → Y is a proper morphism, then this fails, even in very simple examples, unless X → Y is representable.The correct general statement is that there exists a local extension of DVR R → R ′ , such that if we denote by K ′ the fraction field of R ′ , the composite Spec R ′ → Spec R → Y has a lifting Spec R ′ → X extending the composite Spec K ′ → Spec K → X (see for example [Sta21, Tag 0CLY]).For arithmetic applications this is problematic, because the extension R ⊆ R ′ will typically induce a nontrivial extension of residue fields, so it does not imply that Spec k → Y lifts to Spec k → X, as in the case of schemes.
When X and Y are Deligne-Mumford stacks over a field of characteristic 0 a substitute was found by the first author in [Bre21,Theorem 1].In this note we we extend this, in a somewhat more precise form, to positive and mixed characteristic.
The second author was partially supported by research funds from Scuola Normale Superiore, project SNS19 B VISTOLI, and by PRIN project "Derived and underived algebraic stacks and applications".The paper is based upon work partially supported by the Swedish Research Council under grant no.2016-06596 while the second author was in residence at Institut Mittag-Leffler in Djursholm.

1
In this context the correct generality is that of tame stacks, in the sense of [AOV08].Tame stacks are algebraic stacks with finite inertia, such that the automorphism group scheme of any object over a field is linearly reductive.In characteristic 0 they coincide with Deligne-Mumford stacks with finite inertia, but in positive and mixed characteristic there are Deligne-Mumford stacks with finite inertia that are not tame, and tame stacks that are not Deligne-Mumford.
In our version the role of the DVR R ′ above is played by a root stack n √ Spec R; this is not a scheme, but a tame stack with a map n √ Spec R → Spec R, which is an isomorphism above Spec K ⊆ Spec R (see the discussion at the beginning of §3).The statement of our main theorem 3.1 is that if X → Y is a proper morphism of tame algebraic stacks and we have a commutative diagram as above, there exists a unique positive integer n and a unique representable The key point for arithmetic applications is that the closed point Spec k → Spec R lifts to Spec k → n √ Spec R.This statement is much harder to prove in arbitrary characteristic than in characteristic 0.
Besides the original application to Grothendieck's section conjecture in [Bre21], this valuative criterion has been applied in [BVb] to give new proofs and stronger versions of the genericity theorem for essential dimension.
Recall that the Lang-Nishimura theorem states that the property of having a rational point is a birational invariant of smooth proper varieties.Another consequence of our version of the valuative criterion is that the Lang-Nishimura theorem generalizes to tame stacks, see 4.1.Our version of the Lang-Nishimura theorem has an immediate corollary, which we find surprising: if M is a smooth tame stack which is generically a scheme and M → M is a resolution of singularities of the coarse moduli space M → M , then a rational point of M (k) lifts to M if and only if it lifts to M .This gives a hint of the applications of the Lang-Nishimura theorem to fields of moduli, which will be the subject of the forthcoming papers [BVa, Bre].

Notations and conventions
We will follow the conventions of [Knu71] and [LMB00]; so the diagonals of algebraic spaces and algebraic stacks will be separated and of finite type.In particular, every algebraic space will be decent, in the sense of [Sta21, Definition 03I8].
We will follow the terminology of [AOV08]: a tame stack is an algebraic stack X with finite inertia, such that its geometric points have linearly reductive automorphism group.This is equivalent to requiring that X is étale locally over its moduli space a quotient by a finite, linearly reductive group scheme [AOV08, Theorem 3.2].
More generally, a morphism f : X → Y of algebraic stacks is tame if the relative inertia group stack I X/Y → X, defined as in [Sta21, Section 050P], is finite and has linearly reductive geometric fibers.See [AOV11,§3].
Using [Sta21, Lemma 0CPK] one can easily prove the following.
Proposition 2.1.Let f : X → Y be a morphism of algebraic stacks.The following conditions are equivalent.
(2) If Z → Y is a morphism, and Z is a scheme, then Z × Y X is a tame stack.
(3) If Z → Y is a morphism, and Z a tame stack, then Z × Y X is also tame.
Furthermore, if X is tame, then the morphism f is also tame.

The valuative criterion
A basic example of tame stacks is root stacks (see [AGV08,Appendix B2]).We will need this in the following situation.Let R be a DVR with uniformizing parameter π and residue field k def = R/(π).If n is a positive integer, we will denote by n √ Spec R the n th root of the Cartier divisor Spec k ⊆ Spec R. It is a stack over Spec R, such that given a morphism φ : T → Spec R, the groupoid of liftings T → n √ Spec R is equivalent to the groupoid whose objects are triples (L, s, α), where L is an invertible sheaf on T , s ∈ L(T ) is a global section of L, and α is an isomorphism The following is our version of the valuative criterion for properness.
Theorem 3.1.Let f : X → Y be a tame, proper morphism of algebraic stacks, R a DVR with quotient field K. Suppose that we have a 2-commutative square Then there exists a unique positive integer n and a representable lifting n Furthermore, the lifting is unique up to a unique isomorphism.
Corollary 3.2.In the situation above, if k is the residue field of R, the composite As one would expect, these statements fail without the tameness hypothesis, even when Y is a scheme and X is a separated Deligne-Mumford stack.
Example 3.3.Let p be a prime, R a DVR whose fraction field K has characteristic 0 and contains a p-th root of 1, denoted by ζ p , while its residue field k has characteristic p and is not perfect.An example would be the localization of Z[ζ p ][t] at a prime ideal of height 1 containing p.
Choose an element a ∈ R * whose image in k is not a p-th power, and set Let X be the quotient stack [Spec R ′ /C p ]; this is a separated Deligne-Mumford stack, but it is not tame.Since (R ′ ) Cp = R the moduli space of X is Spec R, and we have we have a natural map X → Spec R, which is an isomorphism over Spec K ⊆ Spec R. Since k ′ /k is purely inseparable, then X k (k) is empty: such a krational point would correspond to a C p -torsor Spec A → Spec k with an equivariant morphism Spec A → Spec k ′ and thus an embedding of k ′ in the étale k-algebra A, which is clearly absurd.In particular, there is no map n √ Spec R → X for any n.
Definition 3.4.In the situation of Theorem 3.1, we call the integer n the loop index of the morphism Spec K → X at the place associated with R ⊆ K.If the loop index is 1, we say that Spec K → X is untangled.
Lemma 3.5.Let R ⊆ R ′ be an extension of DVRs with ramification index e, and let K ⊆ K ′ be the fraction fields of R and R ′ respectively.If X is a tame stack proper over R and Spec K → X is a morphism with loop index n, the composite Spec K ′ → Spec K → X has loop index equal to n/ gcd(n, e).
Proof.Write m def = n/ gcd(n, e); the statement follows from the fact that there is a natural representable morphism We spend the rest of this section proving Theorem 3.1.Given a DVR R and π ∈ R a uniformizing parameter, write Lemma 3.6.Let R be a DVR, m, n integers.A morphism m √ Spec R → n √ Spec R over R exists if and only if n|m, and in this case it is unique up to equivalence.
Proof.This follows from the fact that a section Spec R (m) → n √ Spec R exists if and only if n|m, and in this case it is unique up to equivalence.♠ Lemma 3.7.Let R be a DVR, D ⊂ Spec R the divisor corresponding to the closed point, m, n i , r i for i = 1, . . ., m positive integers, with n i ≥ 2 for every i.The fibered product is normal if and only if m = 1 and r 1 = 1.
Proof.Let V n,r be the scheme Spec R[t, s] s /(t n s − π r ), there is an action of G m on V n,r given by (λ, t, s) → (λt, λ −n s) The prime ideal p = (t 1 , . . ., t m , π) is the generic point of the special fiber and has height 1.Since V r,n → n √ Spec R, rD is smooth, then Y → X is smooth and hence X is normal if and only if Y is normal at p. Now consider the prime ideal p 0 = (t 1 , . . ., t m , π) ⊂ R[t 1 , s 1 , . . ., t m , s m ], p and p 0 have equal residue fields and there is a natural surjective linear map p 0 /p 2 0 → p/p 2 .We have that p 0 /p 2 0 has dimension m+1 generated by the classes of [t i ], [π].If m = 1 and r 1 = 1, then p 0 /p 2 0 has dimension 2 and [π] is in the kernel of p 0 /p 2 0 → p/p 2 , hence Y is normal at p.
On the other hand, assume that Y is normal at p, so that p/p 2 has dimension 1.Since n i ≥ 2 for every i, the kernel of p 0 /p 2 0 → p/p 2 is generated by the classes [π ri ] and hence has dimension 1 if r i = 1 for some i, and dimension 0 otherwise.Since p 0 /p 2 0 has dimension m + 1 and p/p 2 has dimension 1, this implies that m = 1 and r 1 = 1.♠ Lemma 3.8.Let A be a Dedekind domain with fraction field K, D ⊂ Spec A an effective, reduced divisor.Let f : X → n (Spec A, D) be a representable, proper morphism.Every generic section Spec K → X of f extends uniquely to a global section n (Spec A, D) → X.
Proof.Let Y ⊂ X be the schematic image of a generic section Spec K → X, we want to prove that Y → n (Spec A, D) is an isomorphism.Since the problem is local, we may assume that A = R is a DVR and D is either empty or the closed point.If D is empty, then n √ Spec R, D = Spec R and this is simply the valuative criterion of properness.Suppose that D is the closed point.Consider the flat morphism Spec R is the schematic image of the induced generic section Spec K (n) → X ′ .By the valuative criterion of properness, there is a section Spec R Lemma 3.9.Let R be a DVR, n, m positive integers.Assume that n is prime with the residue characteristic of R. Consider the µ n -torsor Spec R (n) → n √ Spec R.There exists a unique way of extending the action of µ n to m Spec R (n) , and the quotient [ m Spec R (n) /µ n ] is isomorphic to mn √ Spec R.
Proof.We have a natural action ρ : induced by the action on Spec R (n) .The action ρ gives a structure of µ n -torsor to the natural morphism η is 2-commutative, we want to show that ρ and η are equivalent.
The stack Isom(ρ, η) has a proper, representable morphism and for every connected component of m Spec R (n) × R µ n there is a generic section.By Lemma 3.8, these generic sections extend to global sections, hence η ≃ ρ. ♠ Corollary 3.10.Let R be a DVR, n, m positive integers.Assume that n is prime with the residue characteristic of R. Let X → n √ Spec R be a morphism, and assume that the base change of Lemma 3.11.Let R ′ /R be a local, quasi-finite étale extension of DVRs and X a tame stack over R, X Proof.Let K be the residue field of R, clearly we have that then A is a product of Dedekind domains with a finite number of closed points.Let D ⊂ Spec A be the effective, reduced divisor of all closed points and → X be the composite, and consider the two projections p 1 , p 2 : S → n √ Spec R ′ .Since X R ′ is separated, then X is separated, too, and hence Isom(p * 1 φ, p * 2 φ) is an algebraic stack with a proper, representable morphism to S. There is a generic section S K → Isom(p * 1 φ, p * 2 φ) which extends to a global section thanks to Lemma 3.8, this gives descent data for a morphism f : n √ Spec R → X (the cocycle condition can be checked on the generic point, where it is obvious).Since the base change to R ′ of f is an isomorphism, we have that f is an isomorphism, too.♠ Proposition 3.12.Let X be a normal, tame stack of finite type over a DVR R, and assume that there is a generic section Spec K → X which is an open, schemetheoretically dense embedding.Then X ≃ n √ Spec R for some n.
Proof.Since X is of finite type over R, there exists a DVR R 0 ⊂ R which is the localization of a Z-algebra of finite type and a stack X 0 /R 0 such that X ≃ X 0,R .Furthermore, we may assume that the uniformizing parameter of R 0 maps to a uniformizing parameter of R, so that n √ Spec R 0 × R0 Spec R ≃ n √ Spec R. Up to replacing R, X with R 0 , X 0 we may assume that R is Nagata.Let k be the residue field of R and p its characteristic.
By [AOV08, Theorem 3.2], there exists a DVR R ′ quasi-finite and étale over R and a finite, flat, linearly reductive group scheme G/R ′ with an action on a scheme is constant and tame.We may furthermore assume that the degree of ∆ is a power of p. Thanks to Lemma 3.11, we may assume R ′ = R.
Case 1. X is tame and Deligne-Mumford.Since ∆ k is connected and X is Deligne-Mumford and generically a scheme, the action of ∆ is free (because otherwise X would have ramified inertia), hence up to replacing U with U/∆ we may assume that G is constant and tame.Since X is normal and G is constant and tame, then U is normal, too.If u ∈ U is a geometric point, the stabilizer G u acts faithfully on the tangent space, hence the automorphism groups of the points of X are cyclic and tame.By [Ryd11, Lemma 8.5] and Lemma 3.7, since X is normal we have we have that Y 0 is Deligne-Mumford and there is a natural birational morphism X → Y 0 whose relative inertia is diagonalizable.Let Y → Y 0 be the normalization, it is finite over Y 0 since R is Nagata and since X is normal the morphism X → Y 0 lifts to a morphism X → Y .
By case 1, there exists an n prime with p and an isomorphism Y ≃ n √ Spec R.
Consider the morphism Spec R (n) → n √ Spec R, it is a µ n -torsor and hence finite étale since n is prime with p, it follows that the base change X × n √ Spec R Spec R (n) is normal with diagonalizable inertia.By [Ryd11,Lemma 8.5] and Lemma 3.7, we have n) for some integer m, hence X ≃ mn √ Spec R thanks to Corollary 3.10.♠ Proof of Theorem 3.1.By base change, we may assume that Y = Spec R and that X is a tame stack proper over R. With an argument similar to the one in the proof of Proposition 3.12, we may reduce to the case in which R is Nagata.By [Ryd11, Theorem B], we may assume that Spec K → X is an open, scheme theoretically dense embedding.Since R is Nagata, the normalization X is finite and representable over X.By Proposition 3.12 we have X = n √ Spec R, hence an extension exists.If m is another integer with a representable extension m √ Spec R → X, it factors through X = n √ Spec R since m √ Spec R is normal by Lemma 3.7.We conclude the proof of Theorem 3.1 by Lemma 3.6.♠

The Lang-Nishimura theorem
Here is our version of the Lang-Nishimura theorem for tame stacks.
Theorem 4.1.Let S be a scheme and X Y a rational map of algebraic stacks over S, with X locally noetherian and integral and Y tame and proper over S. Let k be a field, s : Spec k → S a morphism.Assume that s lifts to a regular point Spec k → X; then it also lifts to a morphism Spec k → Y .
In the standard version of the Lang-Nishimura theorem (see for example [Poo17, Theorem 3.6.11]),which is a standard tool in arithmetic geometry, X and Y are schemes, and S = Spec k.In the applications that we have in mind, the additional flexibility of having a base scheme is important.
Proof.According to [LMB00, Théorème 6.3] we can find a smooth morphism U → X with a lifting Spec k → U of Spec k → X; hence we can replace X by U , and assume that X is scheme.Furthermore, if x denotes the image of Spec k → X and k(x) its residue field, we have a factorization Spec k → Spec k(x) → X, and we may assume k = k(x).If x has height 0, then Spec k dominates X, and the composite Spec k → X Y is well defined.Otherwise, call U ⊆ X the open subset where f is defined.By [BVb, Lemma 4.3] there exists a DVR R with residue field k = k(x) and a morphism Spec R → X that maps the generic point Spec K of Spec R into U .Thus we get a morphism Spec K → U , and we apply Corollary 3. Example 4.2.Let X → Spec R be the stack constructed in 3.3, it is a non-tame regular Deligne-Mumford stack.Let k be the residue field of R.There is a rational map Spec R X and Spec R has a k rational point, but X has no k-rational points.As a consequence of Theorem 4.1, we can decide whether a residual gerbe of a tame stack is neutral or not by looking at a resolution of singularities of the coarse moduli space.We find this rather surprising.
Corollary 4.4.Let X be a locally noetherian, regular and integral tame stack with coarse moduli space X → M , and M → M a proper birational morphism, with M integral and regular.Assume that there is a lifting Spec k(M ) → X of the generic point Spec k(M ) → M .
If k is a field and m : Spec k → M a morphism, then m lifts to a morphism Spec k → X if and only if it lifts to a morphism Spec k → M .♠ So, for example, if M is regular all morphisms Spec k → M lift to X, and all residual gerbes are neutral.
desired morphism Spec k → Y .♠ The Lang-Nishimura theorem fails for non-tame separated stacks.Let us give two examples, one in mixed characteristic, the other in positive characteristic.

Example 4. 3 .
Let C 0 be a smooth, projective curve of positive genus over a finite field F of characteristic p with C 0 (F ) = ∅.Let a be an indeterminate, writek def = F (a) and C def = C 0,k ; since C 0 has positive genus C(k) = C 0 (F ) is finite.Let f ∈ k(C)be a rational function such that each rational point is a zero of f (this can be easily found using Riemann-Roch).Consider the ramified cover D → C given by the equation t p − f p−1 t = a ; in other words, D is the smooth projective curve associated with the field extensionk(C)[t]/(t p − f p−1 t − a).Let c ∈ C(k) be a rational point, and write R c def = O C,c [t]/(t p − f p−1 t − a), it is a normal domain: if R c is the normalization, both R c ⊗ k(C) → R c ⊗ k(C) and R c ⊗ k → R c ⊗ k are isomorphisms for degree reasons since R c ⊗ k = k[t]/(t p − a) is a field of degree p over k.Hence, R c is a DVR with residue field k ′ def = k[t]/(t p − a).It follows that D has no k-rational points.The cyclic group C p acts on D by t → t + f , the field extension k(D)/k(C) is a cyclic Galois cover and C is the quotient scheme D/C p .Let X be the quotient stack [D/C p ], there is a natural birational morphism X → C = D/C p .A rational point Spec k → X corresponds to a C p -torsor Spec A → Spec k with an equivariant morphism Spec A → D: since the fibers of D → C over rational points are isomorphic to Spec k ′ , a rational point of X gives an embedding of k ′ in an étale algebra A, which is clearly absurd.It follows that X is a proper Deligne-Mumford stack over k with X(k) = ∅ and a birational map C X.
for its fraction field and k ′ = k( p the cyclic group of order p generated by ζ p ∈ K * .The extension K ′ /K is Galois with cyclic Galois group C p acting by p √ a → ζ p p √ a.The action of C p on K ′ naturally extends to R ′ .