Root Stack Valuative Criterion

• The root stack valuative criterion is a framework where a morphism from a trait’s generic point extends over a root stack, encoding ramification via added stabilizer rather than residue field extension.
• It supports tame proper morphisms by ensuring a unique, representable lift characterized by a loop index that measures the residual stacky ramification.
• The criterion extends to Artin stacks with good moduli spaces and reductive gerbes, providing existence results in settings like birational geometry and even wild characteristic cases.

Searching arXiv for the most relevant papers on root-stack-based valuative criteria, tame stacks, and related extensions. Search 1: "root stack valuative criterion good moduli spaces" A root stack valuative criterion is a valuative extension principle in which a morphism from the generic point of a trait extends not necessarily over $\operatorname{Spec} R$ itself, and not merely after replacing $R$ by a ramified DVR extension, but after replacing the trait by a root stack $\sqrt[n]{\operatorname{Spec} R}$ along its closed point. In current work this idea appears in several distinct but closely related settings: proper tame morphisms of algebraic stacks, Artin stacks admitting good moduli spaces, gerbes banded by reductive groups, and birational extension problems for rational maps to tame stacks. Across these settings, the root stack is the device that records the necessary stacky structure at the special fiber while preserving the original fraction field and residue field (Bresciani et al., 2022).

1. Rooted traits as valuative test objects

Let $R$ be a DVR with fraction field $K$, residue field $k$, and uniformizer $\pi$. The $n$-th root stack of the trait is described concretely by

$$\sqrt[n]{\operatorname{Spec} R}\simeq [\operatorname{Spec} R[t]/(t^n-\pi)/\mu_n].$$

Equivalently, a lifting $T\to \sqrt[n]{\operatorname{Spec} R}$ of a map $R$0 is given by a triple $R$1, where $R$2 is an invertible sheaf on $R$3, $R$4, and $R$5 satisfies $R$6 (Bresciani et al., 2022).

This object is an isomorphism over the generic point $R$7, but its reduced closed fiber is noncanonically $R$8. In particular, the closed immersion $R$9 lifts to a morphism $\sqrt[n]{\operatorname{Spec} R}$0. That feature is the essential arithmetic advantage over the usual valuative criterion for algebraic stacks: one encodes ramification by adding stabilizer at the closed point rather than by enlarging the residue field (Bresciani et al., 2022).

The same local model is used in the good-moduli-space setting, where the paper adopts the convention

$\sqrt[n]{\operatorname{Spec} R}$1

Its geometric meaning is explicit: over the generic point the root stack is just $\sqrt[n]{\operatorname{Spec} R}$2, while over the closed point one obtains a stacky point with stabilizer $\sqrt[n]{\operatorname{Spec} R}$3 (Bejleri et al., 11 Jul 2025).

2. Proper tame morphisms and the arithmetic criterion

For tame proper morphisms of algebraic stacks, the root stack valuative criterion is stated as a replacement for the usual stack-valuative criterion. Given a $\sqrt[n]{\operatorname{Spec} R}$4-commutative square

$\sqrt[n]{\operatorname{Spec} R}$5

with $\sqrt[n]{\operatorname{Spec} R}$6 a tame, proper morphism, there exists a unique positive integer $\sqrt[n]{\operatorname{Spec} R}$7 and a representable lifting

$\sqrt[n]{\operatorname{Spec} R}$8

making the diagram $\sqrt[n]{\operatorname{Spec} R}$9-commutative; moreover, the lifting is unique up to a unique isomorphism (Bresciani et al., 2022).

This result is stronger for arithmetic purposes than the usual criterion for proper algebraic stacks. The standard criterion only gives extension after a local extension of DVRs $R$0, which may force a residue field extension $R$1. By contrast, the rooted-trait version immediately yields a $R$2-point of $R$3: if $R$4 is the residue field of $R$5, then the composite $R$6 has a lifting $R$7 (Bresciani et al., 2022).

The theorem also introduces the loop index: the unique integer $R$8 is called the loop index of the morphism $R$9 at the place associated with $K$0. If the loop index is $K$1, the generic morphism is called untangled. Morphisms between rooted traits are rigid: a map

$K$2

exists if and only if $K$3, and then it is unique up to equivalence. Under an extension of DVRs of ramification index $K$4, the loop index changes by

$K$5

These formulas make the root index a precise measure of the residual stacky ramification carried by the generic point (Bresciani et al., 2022).

Tameness is essential. The paper gives counterexamples showing that the result fails without tameness, even when the target is a scheme and the source is a separated Deligne–Mumford stack; it also shows that the corresponding Lang–Nishimura statement fails in the non-tame setting (Bresciani et al., 2022).

3. Good moduli spaces and reductive gerbes

A second major version of the root stack valuative criterion concerns Artin stacks with good moduli spaces. If $K$6 is a good moduli space morphism, where $K$7 is an Artin stack with affine diagonal and of finite type over a locally Noetherian base, then any compatible generic-point diagram over a DVR admits an extension after replacing $K$8 by some root stack $K$9. If the generic point maps to the closed point of $k$0, one can further arrange that the closed point of $k$1 maps to the closed point of $k$2 (Bejleri et al., 11 Jul 2025).

This is an existence theorem rather than a uniqueness theorem. It is therefore closer to semistable reduction than to separatedness. Its significance is that it extends root-stack-based valuative extension from tame Deligne–Mumford settings to Artin stacks that may have positive-dimensional stabilizers, provided they admit good moduli spaces (Bejleri et al., 11 Jul 2025).

The paper also proves a gerbe version. If

$k$3

is a gerbe for a reductive group scheme $k$4, and either $k$5 is special, or the orders of the Weyl groups of its fibers are coprime to the residue characteristic, or $k$6 fits into an exact sequence

$k$7

with the Weyl groups of the fibers of $k$8 prime to the residue characteristic and $k$9 special, then any $\pi$0-point of $\pi$1 extends after passing to a further root stack over $\pi$2 (Bejleri et al., 11 Jul 2025).

Setting	Extension object	Conclusion
Tame proper morphism	$\pi$3	Representable lift; unique $\pi$4; unique up to unique isomorphism
Good moduli space map	$\pi$5	Existence of a lift; closed-point control in the polystable case
Gerbe for reductive group	Further root stack over $\pi$6	Existence under special or tame-Weyl hypotheses

The proof strategy is structured. The good-moduli-space theorem is reduced to the case of a polystable generic point by a Kempf/Hilbert–Mumford type degeneration, then to a gerbe by canonical reduction of stabilizers, and finally to extension problems for torsors and gerbes over root stacks. In this way the rooted trait becomes the universal local carrier of the automorphism data needed by the limit object (Bejleri et al., 11 Jul 2025).

The residue-characteristic assumptions are genuine. The paper gives a counterexample for $\pi$7 in residue characteristic $\pi$8, showing that without the tame Weyl group condition a $\pi$9-point need not extend to any root stack of $n$0 (Bejleri et al., 11 Jul 2025).

4. Root stacks as the output of birational extension

In birational geometry of tame stacks, root stacks appear not as objects satisfying an independent valuative criterion, but as the canonical stacky modifications produced by a valuative argument. For rational maps from a regular surface to a proper tame stack, the relevant valuative input is Bresciani–Vistoli’s valuative criterion for proper tame morphisms, as recalled in the paper: for a DVR $n$1 and a generic morphism $n$2, there exists a representable morphism

$n$3

with $n$4 minimal, where the root stack is taken along the closed-point divisor of $n$5 (Jeon, 17 Jun 2025).

The birational extension theorem then globalizes these codimension-one rooted extensions. After resolving the boundary to a simple normal crossings divisor, the source of the extended map is a global root stack

$n$6

where $n$7 is proper birational, $n$8 is an SNC divisor, and $n$9 is the unique minimal tuple. Thus the codimension-one root orders extracted from valuative data become the global stack structure along the resolved boundary (Jeon, 17 Jun 2025).

A local higher-dimensional analogue is formulated in Lemma 4.4 of that paper. If $$\sqrt[n]{\operatorname{Spec} R}\simeq [\operatorname{Spec} R[t]/(t^n-\pi)/\mu_n].$$0 is a regular local ring of dimension $$\sqrt[n]{\operatorname{Spec} R}\simeq [\operatorname{Spec} R[t]/(t^n-\pi)/\mu_n].$$1 with parameters $$\sqrt[n]{\operatorname{Spec} R}\simeq [\operatorname{Spec} R[t]/(t^n-\pi)/\mu_n].$$2 cutting out divisors $$\sqrt[n]{\operatorname{Spec} R}\simeq [\operatorname{Spec} R[t]/(t^n-\pi)/\mu_n].$$3, and a map is given away from $$\sqrt[n]{\operatorname{Spec} R}\simeq [\operatorname{Spec} R[t]/(t^n-\pi)/\mu_n].$$4, then there exists a tuple of minimal positive integers $$\sqrt[n]{\operatorname{Spec} R}\simeq [\operatorname{Spec} R[t]/(t^n-\pi)/\mu_n].$$5 and a morphism

$$\sqrt[n]{\operatorname{Spec} R}\simeq [\operatorname{Spec} R[t]/(t^n-\pi)/\mu_n].$$6

making the extension diagram commute; the lift is unique up to unique isomorphism (Jeon, 17 Jun 2025).

The logic is explicitly codimension-sensitive. The paper treats codimension one by rooted valuative extension and codimension at least two by purity on regular tame root stacks. Root stacks are therefore not merely convenient notation: they are the mechanism by which the codimension-one stacky obstruction is packaged in birationally meaningful form (Jeon, 17 Jun 2025).

5. Foundations, non-results, and wild analogues

The basic algebraic definition of a root stack is moduli-theoretic. For a scheme $$\sqrt[n]{\operatorname{Spec} R}\simeq [\operatorname{Spec} R[t]/(t^n-\pi)/\mu_n].$$7, an invertible sheaf $$\sqrt[n]{\operatorname{Spec} R}\simeq [\operatorname{Spec} R[t]/(t^n-\pi)/\mu_n].$$8, a section $$\sqrt[n]{\operatorname{Spec} R}\simeq [\operatorname{Spec} R[t]/(t^n-\pi)/\mu_n].$$9, and an integer $T\to \sqrt[n]{\operatorname{Spec} R}$0, the root stack $T\to \sqrt[n]{\operatorname{Spec} R}$1 is the category whose objects over a scheme $T\to \sqrt[n]{\operatorname{Spec} R}$2 are quadruples

$T\to \sqrt[n]{\operatorname{Spec} R}$3

where $T\to \sqrt[n]{\operatorname{Spec} R}$4 is an invertible sheaf on $T\to \sqrt[n]{\operatorname{Spec} R}$5, $T\to \sqrt[n]{\operatorname{Spec} R}$6, and $T\to \sqrt[n]{\operatorname{Spec} R}$7 satisfies $T\to \sqrt[n]{\operatorname{Spec} R}$8. In the affine trivialized case,

$T\to \sqrt[n]{\operatorname{Spec} R}$9

These formulas furnish the local Kummer model used throughout later valuative arguments, but the paper itself does not state or prove an explicit valuative criterion for root stacks (Biswas et al., 2012).

The topological and functorial side is developed in the study of the Kato–Nakayama space as a transcendental root stack. There the finite root stack $R$00 is described by a lifting problem for the Deligne–Faltings structure, and the infinite root stack $R$01 parametrizes compatible systems of all rational roots. The paper provides a detailed functor-of-points framework and the comparison morphism

$R$02

but it explicitly does not state a valuative criterion in the sense of extension from valuation rings or punctured disks (Talpo et al., 2016).

In characteristic $R$03, ordinary Kummer root stacks no longer capture cyclic stabilizers of order $R$04. The replacement is the Artin–Schreier root stack $R$05, built from triples $R$06 and a normalized pullback along the universal Artin–Schreier cover

$R$07

Its local charts are not Kummer equations $R$08, but Artin–Schreier equations

$R$09

The paper proves that every stacky curve with a point of stabilizer order $R$10 is locally an Artin–Schreier root stack, and that cyclic $R$11-covers factor étale-locally through such stacks. It does not, however, formulate a standalone valuative criterion for Artin–Schreier root stacks; rather, it supplies the wild local structure theory that a genuine criterion would have to incorporate (Kobin, 2019).

6. Conceptual profile and limitations

The literature does not present a single universal theorem called “the” root stack valuative criterion. Instead, the phrase designates a family of results sharing one structural principle: when an honest trait is too small to carry the limiting object, the correct replacement is a rooted trait with the same generic field and the same residue field, but with controlled stabilizer along the closed point (Bresciani et al., 2022).

Within that family, uniqueness varies sharply. For tame proper morphisms, the rooted extension has a unique minimal index and is unique up to unique isomorphism. For good moduli space maps, one generally has existence without uniqueness. In birational applications, the criterion does not characterize root stacks intrinsically; rather, it produces them as the canonical codimension-one output of the extension process (Bejleri et al., 11 Jul 2025).

The limits of the theory are also precise. Older foundational work on root stacks provides the moduli-theoretic and quotient-stack descriptions needed for valuative formulations, but no independent criterion. Wild characteristic-$R$12 analogues require Artin–Schreier equations, ramification jumps, and normalization, showing that the simple Kummer picture is not stable beyond tame or linearly reductive situations (Biswas et al., 2012). A plausible implication is that any fully general root-stack valuative theory must be sensitive to the distinct degeneration mechanisms present in the geometry under study, rather than to stabilizer order alone.