Proof. We identify N with M and treat $\phi $ as an endomorphism of M. Then M is also a module over $A[t]$ where $tx=\phi (x)$ for $x\in M$ . Since $\phi $ is surjective $tM=M$ and Nakayama’s lemma tells us that there is an element $a\in A[t]$ such that $(1-at)M=0$ . That is, $\phi $ has inverse given by $\phi ^{-1}(x)=ax$ .

Proof. Since $\varphi $ is surjective, it induces a surjection $\operatorname {\mathrm {Gr}}_n \varphi \colon I^n/I^{n+1}\to I^nS/I^{n+1}S$ of finitely generated $R/I$ -modules. By assumption, there is an abstract isomorphism $I^n/I^{n+1}\to I^nS/I^{n+1}S$ of $R/I$ -modules, so $\operatorname {\mathrm {Gr}}_n \varphi $ is an isomorphism by Lemma 2.6.

We have induced morphisms of exact sequences

and it follows that $\varphi _d\colon R/I^d\to S/I^dS$ is an isomorphism for every $d\geq 0$ by induction on d. In particular, $\ker \varphi \subseteq I^d$ for all $d\geq 0$ . But R is separated for the I-adic topology, so $\ker \varphi \subseteq \bigcap _{d\geq 0} I^d = (0)$ and the result follows.

Proof. We can replace f with $f_n$ . The first part is then [Reference Alper, Hall and RydhAHR19, Lem. 4.10]: The question is local and reduces to the affine case where it follows from Nakayama’s lemma. For the second part, we have seen that $f_n$ is a closed immersion and then it is adic if and only if $f_0$ is an isomorphism. The third part is also local and thus follows from Lemma 2.7.

Proof. The conditions are clearly necessary. Conversely, if the conditions of (1) (resp. (2)) hold, then $f_n$ is adic and a closed immersion (resp. an isomorphism) for every $n\geq 0$ by Proposition 2.8. Since $f_n$ is adic, we have that $f_n^{-1}(\mathscr {Y}_m)=\mathscr {X}_m$ for all $m\leq n$ . Since $\mathscr {Y}$ is coherently complete along $\mathscr {Y}_0$ , we obtain a closed substack $\mathscr {Z}\hookrightarrow \mathscr {Y}$ such that $\mathscr {Z}\times _{\mathscr {Y}} \mathscr {Y}_n=\mathscr {X}_n$ for all $n\geq 0$ . Under condition (2), we have that $\mathscr {Z}=\mathscr {Y}$ . Finally, since $(\mathscr {X},\mathscr {X}_0)$ is complete, we have by Tannaka duality a unique isomorphism $\mathscr {X}\to \mathscr {Z}$ over $\mathscr {Y}$ .

Proof. The lifting criterion in Definition 2.2 is equivalent to the same criterion for the map $f_n\colon \mathscr {Z}_n \to \mathscr {X}_n$ for $n \gg 0$ large enough that $\mathscr {Z}_n$ contains the image of $T'$ and $\mathscr {X}_n$ contains the image of $T"$ , so by our hypotheses we may assume that the map f is smooth. First, note that $T'$ has cohomological dimension $0$ because any quasi-coherent $\mathcal {O}_{T'}$ -module admits a finite filtration whose associated graded objects are pushforwards of objects in $\mathsf {QCoh}(T)$ . Also, because we may factor $T' \to T"$ into a sequence of square-zero extensions, it suffices to verify the lifting criterion in the case where $T' \to T"$ is a square-zero extension by some $M \in \mathsf {QCoh}(T')$ . In this case the obstruction to the existence of a dotted arrow is an element in the group $\operatorname {\mathrm {Ext}}^1_{T'}(\mathbb {L}_{\mathscr {Z}/\mathscr {X}}|_{T'},M)$ . Since f is smooth, $\mathbb {L}_{\mathscr {Z}/\mathscr {X}}$ is a perfect complex of Tor-amplitude $[0,1]$ . Hence, the $\operatorname {\mathrm {Ext}}$ group vanishes as $T'$ has cohomological dimension $0$ .

Proof. Consider the functor $F\colon \mathsf {Sch}_{/S}^{\mathrm {op}}\to \mathsf {Set}$ , where $F(U\to S)$ is the set of isomorphism classes of complexes of finitely presented quasi-coherent $\mathcal {O}_{U \times _S \mathscr {X}}$ -modules $\mathcal {E}_2\to \mathcal {E}_1\to \mathcal {O}_{U \times _S \mathscr {X}}$ such that $\mathcal {E}_1$ is locally free. This functor is locally of finite presentation.

Let $\widehat {S}$ be the completion of S along $S_0$ . Then $T \to \widehat {S}$ is a closed immersion by Proposition 2.9, because $(T,T_0)$ is complete, $f_0$ is an isomorphism and $f_1$ is a closed immersion. Now let

$$\begin{align*}\mathcal{O}_{\widehat{S}}^{\oplus n} \to \mathcal{O}_{\widehat{S}} \twoheadrightarrow \mathcal{O}_{T} \end{align*}$$

be a presentation of the structure sheaf of $T \hookrightarrow \widehat {S}$ . Pulling back to $\widehat {S} \times _S \mathscr {X}$ we get a resolution

$$\begin{align*}\ker(\beta) \xrightarrow{\alpha} \mathcal{O}_{\widehat{S} \times_S \mathscr{X}}^{\oplus n} \xrightarrow{\beta} \mathcal{O}_{\widehat{S} \times_S \mathscr{X}} \twoheadrightarrow \mathcal{O}_{T \times_S \mathscr{X}}. \end{align*}$$

We regard the pair $(\alpha ,\beta )$ as an element of $F(\widehat {S})$ . Note that by increasing N if necessary, we may assume that both $\alpha $ and $\beta $ satisfy the Artin–Rees condition $\mathrm {(AR)_{N}}$ of [Reference Alper, Hall and RydhAHR20, Def. A.15] with respect to $\mathscr {Z}_0$ .

Let $(S^h,S_0)$ denote the henselization of the pair $(S,S_0)$ . By Artin approximation over henselian pairs [Reference Alper, Hall and RydhAHR19, Thm. 3.4], one can find a class in $F(S^h)$ which restricts to the same class as $(\alpha ,\beta )$ in $F(S_N)$ . Then because $S^h$ is constructed as an inverse limit of étale neighborhoods of $S_0$ , we lift this class in $F(S^h)$ to a class $(\alpha ',\beta ') \in F(S')$ for some étale map $S' \to S$ lying under $S^h$ such that $S' \times _S S_0 \simeq S_0$ .

We now let $\mathscr {W} \hookrightarrow S' \times _S \mathscr {X}$ be the closed substack defined by $\operatorname {\mathrm {im}}(\beta ') \subset \mathcal {O}_{S' \times _S \mathscr {X}}$ . By construction, we have

$$\begin{align*}\mathcal{O}_{\mathscr{W}} \otimes_{\mathcal{O}_{S' \times_S \mathscr{X}}} \mathcal{O}_{S^{\prime}_N} \simeq \operatorname{\mathrm{coker}}(\beta'|_{S^{\prime}_N}) \simeq \mathcal{O}_{T_N \times_S \mathscr{X}} \end{align*}$$

as $\mathcal {O}_{S' \times _S \mathscr {X}}$ -algebras, which is the second condition of the lemma.

Now, consider $(\alpha ,\beta ) \in F(\widehat {S})$ and the restriction of $(\alpha ',\beta ')$ to $F(\widehat {S})$ . Both complexes are isomorphic after tensoring with $\mathcal {O}_{S_N}$ , and by hypothesis the complex defined by $(\alpha ,\beta )$ is exact and satisfies the Artin–Rees criterion $\mathrm {(AR)_{N}}$ , so the refined Artin–Rees theorem [Reference Alper, Hall and RydhAHR20, Thm. A.16] implies that

$$\begin{align*}\operatorname{\mathrm{Gr}}_{\mathcal{I}_{\mathscr{Z}}} \mathcal{O}_{\mathscr{Z}} \cong \operatorname{\mathrm{Gr}}_{\mathcal{I}_{\mathscr{Z}}} (\operatorname{\mathrm{coker}}(\beta)) \cong \operatorname{\mathrm{Gr}}_{\mathcal{I}_{\mathscr{W}}} ( \operatorname{\mathrm{coker}}(\beta')) \cong \operatorname{\mathrm{Gr}}_{\mathcal{I}_{\mathscr{W}}} \mathcal{O}_{\mathscr{W}}.\\[-36pt] \end{align*}$$

Proof. It suffices to prove the claims after base change to the completion of T, so we may assume that T is complete along $T_0$ . Now, write

$$ \begin{align*}T = \varprojlim_\lambda T_\lambda,\end{align*} $$

where $T_\lambda $ is a cofiltered system of affine S-schemes of finite type. For $\lambda $ sufficiently large, $T_1 \to T_\lambda $ is a closed immersion. Increasing $\lambda $ if necessary, standard limit methods give us an algebraic stack $\mathscr {Z}_\lambda $ of finite presentation over $T_\lambda $ fitting into a commutative diagram

(1)

It now suffices to replace S with $T_\lambda $ , and $\mathscr {X}$ with $\mathscr {Z}_\lambda $ , and to find a stack over $\mathscr {Z}_\lambda $ meeting the conditions of the theorem. We may therefore assume that $\mathscr {X}$ is algebraic and of finite presentation over S, and that $T_1 \to S$ is a closed immersion, in which case the theorem follows immediately from Lemma 2.11 with $S_0$ as the image of $T_0$ .

Finally, if $\mathscr {Z}$ were fundamental, meaning $\mathscr {Z}$ admits an affine map $f\colon \mathscr {Z} \to B\mathrm {GL}_{n,\mathbb {Z}}$ for some n, then in this case one can simultaneously approximate both the map f and the map $\mathscr {Z} \to \mathscr {X}$ by replacing $\mathscr {X}$ with $\mathscr {X} \times _S (B\mathrm {GL}_{n,S})$ in the argument above. The map $\mathscr {Z} \to B \mathrm {GL}_{n,S}$ is affine, so [Reference RydhRyd15, Thm. C] guarantees that we can arrange for $\mathscr {Z}_\lambda $ in (1) to be affine over $B \mathrm {GL}_{n,S}$ as well. The stack $\mathscr {W}$ constructed in Lemma 2.11 will be affine over $B\mathrm {GL}_{n,S}$ as well, hence fundamental.

Proof of Theorem 2.3

Let T be the good moduli space of $\mathscr {Z}$ and $T_0$ the good moduli space of $\mathscr {Z}_0$ . Choose an $N \geq 1$ . Then $T_0\to S$ is of finite type, so Proposition 2.12 produces a stack $\mathscr {W}$ satisfying the first two conditions of the theorem along with a map $\xi \colon \mathscr {W} \to \mathscr {X}$ and an isomorphism $\psi _N\colon \mathscr {W}_N \cong \mathscr {Z}_N$ over $\mathscr {X}$ .

We would like to extend the isomorphism $\psi _N$ to a compatible sequence of isomorphisms $\psi _n\colon \mathscr {W}_n \to \mathscr {Z}_n$ over $\mathscr {X}$ for all $n\geq N$ . Extending the map $\psi _n$ to $\psi _{n+1}$ is equivalent to finding a dotted arrow such that the diagram

is $2$ -commutative. It is possible to do this for all $n \geq N$ because by hypothesis the map $\eta $ is formally versal at $\mathscr {W}_0 = \mathscr {Z}_0$ (see Definition 2.2). The resulting sequence of maps $\psi _n\colon \mathscr {W}_n \to \mathscr {Z}_n$ and the induced map $\widehat {\psi }\colon \widehat {\mathscr {W}} \to \mathscr {Z}$ are isomorphisms by Proposition 2.9 and part (3) of Proposition 2.12. If we define $\varphi $ to be the inverse of $\psi $ followed by the canonical map $\widehat {\mathscr {W}} \to \mathscr {W}$ , then by construction we have a compatible sequence of $2$ -isomorphisms $\xi \circ \varphi |_{\mathscr {Z}_{n}} \cong \eta |_{\mathscr {Z}_{n}}$ for all $n\geq 1$ .

If $\mathscr {X}$ is an algebraic stack with quasi-separated diagonal, then the stack $I := \operatorname {\mathrm {Isom}}_{\mathscr {Z}} (\xi \circ \varphi , \eta )$ is a quasi-separated algebraic space, locally of finite type over $\mathscr {Z}$ . The $2$ -isomorphisms $\xi \circ \varphi |_{\mathscr {Z}_n} \cong \eta |_{\mathscr {Z}_n}$ give a compatible sequence of sections $\sigma _n$ of $I \to \mathscr {Z}$ over $\mathscr {Z}_n$ for all $n \geq 1$ . The image of all of the $\sigma _n$ lie in some quasi-compact open substack $I' \subset I$ , so we may replace I with $I'$ . Then Tannaka duality implies that there is a unique section $\sigma \colon \mathscr {Z} \to I' \subset I$ of $I \to \mathscr {Z}$ , which corresponds to a $2$ -isomorphism $\xi \circ \varphi \simeq \eta $ satisfying the conditions of the theorem.

Proof of Theorem 1.3(1)—smooth/étale case

Step 1: An effective formally versal solution. Since $\mathscr {W}_0$ is quasi-compact, we may assume that $\mathscr {X}$ is quasi-compact after replacing $\mathscr {X}$ with a quasi-compact open substack containing the image of $\mathscr {W}_0$ . Since $\mathscr {W}_0$ is linearly fundamental, we can apply [Reference Alper, Hall and RydhAHR19, Thm. 1.11] to obtain a Cartesian square

where $\widehat {f} \colon \widehat {\mathscr {W}} \to \mathscr {X}$ is a flat morphism and $\widehat {\mathscr {W}}$ is a linearly fundamental stack coherently complete along $\mathscr {W}_0$ . Since $\mathscr {W}_n\to \mathscr {X}_n$ is smooth, $\widehat {f} \colon \widehat {\mathscr {W}}\to \mathscr {X}$ is formally versal at $\mathscr {W}_0$ (Lemma 2.10).

Step 2: Algebraization. We now apply algebraization for linearly fundamental pairs (Theorem 2.3) to the pair $(\widehat {\mathscr {W}}, \mathscr {W}_0)$ and morphism $\widehat {f} \colon \widehat {\mathscr {W}}\to \mathscr {X}$ to obtain a fundamental pair $(\mathscr {W}, \mathscr {W}_0)$ with $\mathscr {W}$ of finite type over S and a morphism $f \colon \mathscr {W} \to \mathscr {X}$ smooth (resp. étale) along $\mathscr {W}_0$ such that $\widehat {\mathscr {W}}$ is isomorphic over $\mathscr {X}$ to the coherent completion of $\mathscr {W}$ along $\mathscr {W}_0$ . After replacing $\mathscr {W}$ with an open neighborhood, we may arrange that f is smooth (resp. étale) and $\mathscr {W}$ is fundamental. Indeed, if $\mathscr {U} \subset \mathscr {W}$ is an open neighborhood of $\mathscr {W}_0$ such that $f|_{\mathscr {U}}$ is smooth (resp. étale) and if $\pi \colon \mathscr {W}\to W$ denotes the adequate moduli space, then we replace $\mathscr {W}$ with the inverse image of any affine open subscheme of $W\smallsetminus \pi (\mathscr {W}\smallsetminus \mathscr {U}))$ containing $\pi (\mathscr {W}_0)$ .

Proof of Theorem 1.3(2)—syntomic case

Step 1: We may assume that $f_0 \colon \mathscr {W}_0 \to \mathscr {X}_0$ is affine. We may assume that $\mathscr {X}$ is quasi-compact. Since $\mathscr {W}_0$ is fundamental, there is an affine morphism $\mathscr {W}_0 \to B \mathrm {GL}_n$ for some n. Since $\mathscr {X}_0$ has affine diagonal (as it has the resolution property), the induced morphism $\mathscr {W}_0 \to \mathscr {X}_0 \times B \mathrm {GL}_n$ is affine. Since $B \mathrm {GL}_n$ is smooth with smooth diagonal, we may replace $(\mathscr {X}, \mathscr {X}_0)$ with $(\mathscr {X} \times B \mathrm {GL}_n, \mathscr {X}_0 \times B \mathrm {GL}_n)$ .

Step 2: There is a factorization $f_0 \colon \mathscr {W}_0 \hookrightarrow \mathscr {Y}_0 \to \mathscr {X}_0$ , where $\mathscr {W}_0 \hookrightarrow \mathscr {Y}_0$ is a regular closed immersion, $\mathscr {Y}_0 \to \mathscr {X}_0$ is smooth and affine, and $\mathscr {Y}_0$ is linearly fundamental. Since $\mathscr {X}_0$ has the resolution property and $(f_0)_* \mathcal {O}_{\mathscr {W}_0}$ is a finite type $\mathcal {O}_{\mathscr {X}_0}$ -algebra, there exist a vector bundle $\mathcal {E}_0$ on $\mathscr {X}_0$ and a surjection $\operatorname {\mathrm {Sym}}(\mathcal {E}_0)\twoheadrightarrow (f_0)_*\mathcal {O}_{\mathscr {W}_0}$ . Setting $\mathscr {Y}_0 = \mathbb {V}(\mathcal {E}_0)=\operatorname {\mathrm {Spec}}(\operatorname {\mathrm {Sym}}\mathcal {E}_0)$ yields a factorization such that $\mathscr {W}_0 \hookrightarrow \mathscr {Y}_0$ is a regular closed immersion and $\mathscr {Y}_0 \to \mathscr {X}_0$ is smooth and affine. To arrange that $\mathscr {Y}_0$ is linearly fundamental, we apply the étale case of the local structure theorem (Theorem 1.3 (1)) to the closed immersion $\mathscr {W}_0\hookrightarrow \mathscr {Y}_0$ to extend the isomorphism $\mathscr {W}_0 \stackrel {\sim }{\to } \mathscr {W}_0$ to an étale morphism $\mathscr {Y}^{\prime }_0 \to \mathscr {Y}_0$ with $\mathscr {Y}^{\prime }_0$ fundamental. Since $\mathscr {W}_0$ satisfies (PC) or (N), or $\mathscr {Y}_0$ satisfies (FC), there is an open neighborhood of $\mathscr {W}_0\hookrightarrow \mathscr {Y}^{\prime }_0$ that is linearly fundamental [Reference Alper, Hall and RydhAHR19, Prop. 7.20].

Step 3: Apply the smooth version of the local structure theorem. Since $\mathscr {Y}_0$ is linearly fundamental, we may apply the smooth case of the local structure theorem (Theorem 1.3 (1)) to the closed immersion $\mathscr {X}_0 \hookrightarrow \mathscr {X}$ and smooth morphism $\mathscr {Y}_0 \to \mathscr {X}_0$ to obtain a commutative diagram

with a Cartesian square such that $\mathscr {Y} \to \mathscr {X}$ is smooth and $\mathscr {Y}$ is fundamental.

Step 4: Lift the closed substack $\mathscr {W}_0 \hookrightarrow \mathscr {Y}_0$ to a closed substack $\mathscr {W} \hookrightarrow \mathscr {Y}$ syntomic over $\mathscr {X}$ . Let $\mathcal {I}_0$ be the ideal sheaf defining $\mathscr {W}_0 \hookrightarrow \mathscr {Y}_0$ , and consider the conormal bundle $\mathcal {N}_0:=\mathcal {I}_0/\mathcal {I}_0^2$ . After replacing $\mathscr {Y}$ with an étale fundamental neighborhood of $\mathscr {W}_0\hookrightarrow \mathscr {Y}$ , we may extend the conormal bundle $\mathcal {N}_0$ to a vector bundle $\mathcal {N}$ on $\mathscr {Y}$ ; this follows from applying [Reference Alper, Hall and RydhAHR19, Prop. 7.18(4)] to the fundamental pair $(\mathscr {Y}, \mathscr {W}_0)$ .

We claim that after replacing $\mathscr {Y}$ with a fundamental étale neighborhood of $\mathscr {W}_0$ the canonical homomorphism $\mathcal {N}\twoheadrightarrow \mathcal {N}_0 \hookrightarrow \mathcal {O}_{\mathscr {Y}_0}/\mathcal {I}_0^2$ extends to a diagram

If $\mathscr {Y}$ is linearly fundamental, this is immediate as the functor $\operatorname {\mathrm {Hom}}_{\mathcal {O}_{\mathscr {Y}}}(\mathcal {N},-) = \Gamma (\mathscr {Y}, \mathcal {N}^{\vee } \otimes -)$ is exact. In general, let $\mathscr {W}_1 \hookrightarrow \mathscr {Y}_0$ be the closed substack defined by $\mathcal {I}_0^2$ . Then we have a morphism $\mathscr {W}_1\to \mathbb {V}(\mathcal {N})$ over $\mathscr {Y}$ which by [Reference Alper, Hall and RydhAHR19, Prop. 7.18(1)] extends to a section $\mathscr {Y}\to \mathbb {V}(\mathcal {N})$ after replacing $\mathscr {Y}$ with a fundamental étale neighborhood of $\mathscr {W}_0$ . This gives the requested map $\mathcal {N}\to \mathcal {O}_{\mathscr {Y}}$ .

Let $\mathscr {W}$ be the closed substack defined by the image of $\mathcal {N}\to \mathcal {O}_{\mathscr {Y}}$ . By construction, $\mathscr {W}$ contains the closed substack $\mathscr {W}_0$ . We claim that $\mathscr {W}_0\hookrightarrow \mathscr {W}\times _{\mathscr {X}} \mathscr {X}_0$ is an isomorphism and $\mathscr {W} \to \mathscr {X}$ is syntomic in an open neighborhood of $\mathscr {W}_0$ . This establishes the theorem as we may shrink further to arrange that $\mathscr {W} \to \mathscr {X}$ is syntomic in a fundamental open neighborhood of $\mathscr {W}_0$ . These claims can be verified smooth-locally on $\mathscr {X}$ and $\mathscr {Y}$ , so we may assume that $\mathscr {X}$ and $\mathscr {Y}$ are affine schemes and $\mathcal {N}$ is a trivial vector bundle. By construction, $\mathcal {N}\mathcal {O}_{\mathscr {Y}_0}+\mathcal {I}_0^2=\mathcal {I}_0$ , so it follows that $\mathscr {W}_0\hookrightarrow \mathscr {W}\times _{\mathscr {Y}} \mathscr {Y}_0$ is an isomorphism in an open neighborhood of $\mathscr {W}_0$ by Nakayama’s lemma.

Let $f_1,\dots ,f_n\in \mathcal {O}_{\mathscr {Y}}$ be the image of a basis of $\mathcal {N}$ . We claim that $f_1,\dots ,f_n$ is a regular sequence in a neighborhood of $\mathscr {W}_0$ and that $\mathscr {W}\to \mathscr {X}$ is flat in a neighborhood of $\mathscr {W}_0$ . By [Reference GrothendieckEGA_IV, Thm. 11.3.8 (c) $\Rightarrow$ (b’)], it is enough to prove that the images of $f_1,\dots ,f_n$ in $\mathcal {O}_{\mathscr {Y}_x,w}$ is a regular sequence for every $w\in |\mathscr {W}_0|$ with image $x\in |\mathscr {X}_0|$ , which follows by construction.

Proof. Case (1) is precisely Theorem 1.3. For cases (2)–(3), after replacing $\mathscr {X}$ with a quasi-compact open substack we can assume that $\mathscr {X}$ is quasi-compact. If $\mathscr {X}_0$ satisfies (FC), we can assume that $\mathscr {X}$ is also (FC). Indeed, let S be the spectrum of $\mathbb {Z}$ localized in the characteristics of $\mathscr {X}_0$ , and replace $\mathscr {X}$ , $\mathscr {X}_0$ and $\mathscr {W}_0$ with their base changes along $S\to \operatorname {\mathrm {Spec}} \mathbb {Z}$ . Once the theorem is established in this case, we can use standard limit methods to replace S with an open subscheme of $\operatorname {\mathrm {Spec}} \mathbb {Z}$ . If instead $\mathscr {X}_0$ satisfies (PC) or (N), we let $S=\operatorname {\mathrm {Spec}} \mathbb {Z}$ .

By [Reference RydhRyd23], we can write $\mathscr {X}_0\hookrightarrow \mathscr {X}$ as a limit of finitely presented closed immersions $\mathscr {X}_{0,\lambda }\hookrightarrow \mathscr {X}$ with transition maps that are closed immersions. For sufficiently large $\lambda $ , we can extend $f_0\colon \mathscr {W}_0\to \mathscr {X}_0$ to a map $f_{0,\lambda }\colon \mathscr {W}_{0,\lambda }\to \mathscr {X}_{0,\lambda }$ of finite presentation. For sufficiently large $\lambda $ , we have that $f_{0,\lambda }$ is smooth/étale/syntomic. If $\mathscr {X}_0$ has the resolution property, then so does $\mathscr {X}_{0,\lambda }$ for sufficiently large $\lambda $ . This follows from the Totaro–Gross characterization of the resolution property as having a quasi-affine morphism to $B\mathrm {GL}_N$ for some N [Reference TotaroTot04, Reference GrossGro17] and [Reference RydhRyd15, Thm. C]. For sufficiently large $\lambda $ , we also have that $\mathscr {W}_{0,\lambda }$ is linearly fundamental [Reference Alper, Hall and RydhAHR19, Thm. 7.3] using that either $\mathscr {W}_0$ is (PC) or (N), or $\mathscr {X}$ is (FC). After replacing $\mathscr {W}_0$ and $\mathscr {X}_0$ with $\mathscr {W}_{0,\lambda }$ and $\mathscr {X}_{0,\lambda }$ we may thus assume that $\mathscr {X}_0\hookrightarrow \mathscr {X}$ is of finite presentation.

Using [Reference RydhRyd23], we may further write $\mathscr {X}\to S$ as a limit of algebraic stacks $\mathscr {X}_{\lambda }\to S$ of finite presentation. For sufficiently large $\lambda $ , we can descend the finitely presented maps $f_0\colon \mathscr {W}_0\to \mathscr {X}_0$ and $i\colon \mathscr {X}_0 \hookrightarrow \mathscr {X}$ to finitely presented maps $f_{0,\lambda }\colon \mathscr {W}_{0,\lambda }\to \mathscr {X}_{0,\lambda }$ and $i_\lambda \colon \mathscr {X}_{0,\lambda }\hookrightarrow \mathscr {X}_{\lambda }$ . For sufficiently large $\lambda $ , we have that $\mathscr {X}_\lambda $ has affine stabilizers [Reference Hall and RydhHR15, Thm. 2.8] and, as before, that $f_{0,\lambda }$ is smooth/étale/syntomic, that $i_\lambda $ is a closed immersion, that $\mathscr {W}_{0,\lambda }$ is linearly fundamental and that $\mathscr {X}_{0,\lambda }$ has the resolution property. We are now in the situation of Theorem 1.3.

Proof of Theorem 1.12

The implications (1) $\implies $ (2) $\implies $ (3) and (4) $\implies $ (5) are trivial. The implication (3) $\implies $ (5) is [Reference Hall, Neeman and RydhHNR19, Thm. 1.1] since every closed point has positive characteristic. It remains to prove that (5) implies (1) and (4). To this end, let x be a closed point of $\mathscr {X}$ , which we view as morphism $x\colon \operatorname {\mathrm {Spec}} l \to \mathscr {X}$ , where l is an algebraically closed field. Let $i_x \colon \mathcal {G}_x \hookrightarrow \mathscr {X}$ be the closed immersion of the residual gerbe of x. Then there is a field $\kappa (x)$ such that $\mathcal {G}_x \to \operatorname {\mathrm {Spec}} \kappa (x)$ is a coarse moduli space. Certainly, $\kappa (x) \subseteq l$ . After taking a finite extension $\kappa (x) \subseteq k \subseteq l$ , $(\mathcal {G}_x)_k \simeq BH$ , for some group scheme H over k. After passing to an additional finite extension of k, there is a subgroup scheme $H' \hookrightarrow H$ such that $H^{\prime }_l \simeq G^0_{\mathrm {red}}$ . By assumption, $G^0_{\mathrm {red}}$ is a torus, so $H'$ is of multiplicative type. Set $\mathscr {W}_0^x = BH'$ , $\mathscr {X}_0^x = \mathcal {G}_x$ and let $f_0^x \colon \mathscr {W}_0^x \to \mathscr {X}_0^x$ be the induced morphism. We claim that $f_0^x$ is syntomic. Indeed, $f_0^x$ is the composition $BH' \to BH \to \mathcal {G}_x$ . Now $BH \to \mathcal {G}_x$ is the base change of $\operatorname {\mathrm {Spec}} k \to \operatorname {\mathrm {Spec}} \kappa (x)$ , which is syntomic. Also, $BH' \to BH$ is fppf-locally the morphism $H/H' \to \operatorname {\mathrm {Spec}} k$ . Since $H \to \operatorname {\mathrm {Spec}} k$ is syntomic (Lemma 5.2) and $H \to H/H'$ is fppf, $H/H' \to \operatorname {\mathrm {Spec}} k$ is syntomic. By descent, $BH' \to BH$ is syntomic and so $f_0^x$ is too.

We now apply Theorem 5.1 (2) (b) to $f_0^x$ : This results in a syntomic morphism $f^x\colon \mathscr {W}^x \to \mathscr {X}$ such that $\mathscr {W}^x$ is fundamental and $f^x|_{\mathscr {X}_0^x} \simeq f_0^x$ . Since $\mathscr {W}^x$ is fundamental and $f_0^x$ is finite, we may shrink $\mathscr {W}^x$ so that $f^x$ is quasi-finite [Reference Alper, Hall and RydhAHR20, Lem. 3.1]. Additionally, since $\mathscr {X}$ has affine diagonal, we may further shrink $\mathscr {W}^x$ so that $f^x$ is affine [Reference Alper, Hall and RydhAHR19, Prop. 5.7]. By [Reference Alper, Hall and RydhAHR19, Prop. 6.7], after passing to a strictly étale neighborhood of $\mathscr {W}_0^x$ , we may further shrink $\mathscr {W}^x$ so that it is nicely fundamental.

Since $\mathscr {X}$ is quasi-compact and the $f^x$ are all open morphisms, there is a finite set of closed points $x_1$ , $\dots $ , $x_m$ of $\mathscr {X}$ such that the induced morphism $f\colon \mathscr {W} = \amalg _{i=1}^m \mathscr {W}^{x_i} \xrightarrow {\amalg f^{x_i}} \mathscr {X}$ is affine, quasi-finite, syntomic and faithfully flat. But $\mathscr {W}$ is nicely fundamental, so it is $\aleph _0$ -crisp [Reference Hall and RydhHR17, Ex. 8.6]. By [Reference Hall and RydhHR17, Thm. C], $\mathscr {X}$ is $\aleph _0$ -crisp. This proves (5) $\implies $ (1). Since $\mathscr {W}\to \mathscr {X}$ is quasi-finite and surjective and the reduced identity components of the stabilizers of $\mathscr {W}$ are tori, so are those of $\mathscr {X}$ . This proves (5) $\implies $ (4).

Proof. Since $G \to S$ is flat and locally of finite presentation, we reduce immediately to the situation where S is the spectrum of an algebraically closed field k [SP, Tags 01UF & 069N] and we must show that $G \to \operatorname {\mathrm {Spec}} k$ is a local complete intersection morphism. Let $G^0 \subseteq G$ be the connected component of the identity, which is a normal, irreducible and quasi-compact flat closed subgroup scheme of G [SP, Tag 0B7R]. Since G is locally of finite type, $G^0 \subseteq G$ is even open and closed. Hence, the quotient $G/G^0$ is étale. Thus, we may replace G with $G^0$ and assume that G is connected and of finite type. If the characteristic of k is $0$ , then $G \to S$ is smooth and we are done (Cartier’s theorem [SP, Tag 047N]). In general, there is an extension of groups $1 \to G_{\mathrm {ant}} \to G \to G_{\mathrm {aff}} \to 1$ , where $G_{\mathrm {ant}}$ is anti-affine (i.e., $\Gamma (G_{\mathrm {ant}},\mathcal {O}_{G_{\mathrm {ant}}}) \simeq k$ ) and $G_{\mathrm {aff}}$ is affine [Reference BrionBri09, (0.2)]. Then $G_{\mathrm {ant}}$ is smooth, so it suffices to prove the claim when G is affine. In this case, $G \hookrightarrow \mathrm {GL}_n$ for some $n>0$ . The cover $\mathrm {GL}_n \to \mathrm {GL}_n/G$ is faithfully flat and of finite presentation. Since $\mathrm {GL}_n$ is smooth over $\operatorname {\mathrm {Spec}} k$ , $\mathrm {GL}_n/G$ is also smooth over $\operatorname {\mathrm {Spec}} k$ —this follows immediately from the descent of regularity under faithfully flat extensions of local rings [Reference GrothendieckEGA_IV, 0 ${}_{\mathrm{IV}}$ .17.3.3]. Hence, $\mathrm {GL}_n \to \mathrm {GL}_n/G$ is syntomic [SP, Tags 069M & 069K]. The fiber of the syntomic morphism $\mathrm {GL}_n \to \mathrm {GL}_n/G$ over a k-point is $G \to \operatorname {\mathrm {Spec}} k$ , which is consequently a local complete intersection morphism.

Proof. As a first preliminary step, we can as before assume that $\mathscr {X}$ is quasi-compact and, if $\mathscr {X}_0$ satisfies (FC), that $\mathscr {X}$ satisfies (FC) by base changing along $S \to \operatorname {\mathrm {Spec}} \mathbb {Z}$ where S is the spectrum of $\mathbb {Z}$ localized in the characteristics of $\mathscr {X}_0$ .

By assumption, $\mathscr {X}_0 = \cap _\lambda \mathscr {X}_\lambda $ is an intersection of a cofiltered system of immersions $\mathscr {X}_\lambda \hookrightarrow \mathscr {X}$ with eventually affine inclusions $\mathscr {X}_\mu \hookrightarrow \mathscr {X}_{\lambda }$ . Pick $\alpha $ sufficiently large such that $\mathscr {X}_\lambda \hookrightarrow \mathscr {X}_{\alpha }$ is affine for all $\lambda \geq \alpha $ and pick a quasi-compact open neighborhood $\mathscr {U}$ of $\mathscr {X}_0$ in $\mathscr {X}_\alpha $ . Then $\mathscr {X}_0 = \cap _{\lambda \geq \alpha } (\mathscr {X}_\lambda \cap \mathscr {U})$ , so we may assume that all the $\mathscr {X}_\lambda $ are quasi-compact and that all the $\mathscr {X}_\mu \hookrightarrow \mathscr {X}_\lambda $ are affine.

By standard limit methods, the morphism $f_0 \colon \mathscr {W}_0 \to \mathscr {X}_0$ descends to a morphism $f_\alpha \colon \mathscr {W}_\alpha \to \mathscr {X}_\alpha $ , which is étale, smooth or syntomic if $f_0$ is so. If $\lambda \geq \alpha $ , set $\mathscr {W}_{\lambda } = \mathscr {W}_\alpha \times _{\mathscr {X}_\alpha } \mathscr {X}_{\lambda }$ . Then $\mathscr {W}_0 = \cap _{\lambda \geq \alpha } \mathscr {W}_\lambda $ . Now, either $\mathscr {X}$ satisfies (FC) (by the initial reduction) or $\mathscr {W}_0$ satisfies (PC) or (N). Hence, $\mathscr {W}_\beta $ is linearly fundamental for some $\beta \gg \alpha $ [Reference Alper, Hall and RydhAHR19, Thm. 7.3]. After replacing $\mathscr {X}$ with an open neighborhood of $\mathscr {X}_\beta $ , we may assume that $\mathscr {X}_\beta \hookrightarrow \mathscr {X}$ is a closed immersion. We may now apply Theorem 5.1 (2) or (3) to $f_\beta $ and the result follows.

Proof of Theorem 1.5

Arguing as in the proof of Theorem 5.8, we may assume that $\mathscr {W}_0$ satisfies (PC) or (N) or $\mathscr {W}$ satisfies (FC). We may further assume that there is a factorization of $\mathscr {W}_0 \hookrightarrow \mathscr {W} \to \mathscr {X}$ through an immersion $\mathscr {W}_\beta \hookrightarrow \mathscr {W}$ such that $\mathscr {W}_{\beta }$ is fundamental and $\mathscr {W}_\beta \hookrightarrow \mathscr {W} \to \mathscr {X}$ is representable [Reference RydhRyd15, Thm. C]. We now factor $\mathscr {W}_\beta \hookrightarrow \mathscr {W}$ as $\mathscr {W}_\beta \hookrightarrow \mathscr {Z} \subseteq \mathscr {W}$ , where $\mathscr {W}_\beta \hookrightarrow \mathscr {Z}$ is a closed immersion and $\mathscr {Z} \subseteq \mathscr {W}$ is an open immersion. In this generality, however, $\mathscr {Z}$ is not necessarily fundamental (it can be arranged to be if $\mathscr {W}_0 \hookrightarrow \mathscr {W}$ is a closed immersion, however). But we can now apply Theorem 5.1 to the closed immersion $\mathscr {W}_\beta \hookrightarrow \mathscr {Z}$ . We thus obtain an étale neighborhood $p \colon \mathscr {W}' \to \mathscr {Z}$ of $\mathscr {W}_\beta $ such that $\mathscr {W}'$ is fundamental and the induced morphism $\mathscr {W}_\beta ' = p^{-1}(\mathscr {W}_\beta ) \simeq \mathscr {W}_\beta \to \mathscr {Z} \subseteq \mathscr {W} \to \mathscr {X}$ is representable. The result now follows from [Reference Alper, Hall and RydhAHR19, Prop. 5.7] applied to the pair $(\mathscr {W}',\mathscr {W}_\beta ')$ and the morphism $\mathscr {W}' \to \mathscr {X}$ .

Proof of Theorem 1.6

As in the proof of Theorem 1.5, we may assume that there is a factorization of $\mathscr {W}_0 \hookrightarrow \mathscr {W}$ through an immersion $\mathscr {W}_\beta \hookrightarrow \mathscr {W}$ such that $\mathscr {W}_{\beta }$ is linearly fundamental [Reference Alper, Hall and RydhAHR19, Thm. 7.3]. Likewise, if $\mathscr {W}_0=[\operatorname {\mathrm {Spec}} A_0/G_0]$ as in (1), then we can arrange so that $\mathscr {W}_\beta =[\operatorname {\mathrm {Spec}} A_\beta /G_\beta ]$ with $G_\beta $ embeddable and linearly reductive or nice [Reference Alper, Hall and RydhAHR19, Lem. 2.12 and Thm. 7.3]. Finally, if $\mathscr {W}_0=[\operatorname {\mathrm {Spec}} A_0/G]$ as in (2), then $\mathscr {W}_\beta =[\operatorname {\mathrm {Spec}} A_\beta /G]$ by [Reference RydhRyd15, Thm. C].

We have a closed then open factorization $\mathscr {W}_\beta \hookrightarrow \mathscr {Z} \subseteq \mathscr {W}$ . Apply Theorem 5.1 to $\mathscr {W}_\beta \hookrightarrow \mathscr {Z}$ to replace $\mathscr {W}$ with an étale neighborhood of $\mathscr {W}_\beta $ that is fundamental. We can now apply [Reference Alper, Hall and RydhAHR19, Props. 7.16, 7.18(3) and 7.20] and the result follows.

Proof of Theorem 1.9

We apply Theorem 1.4 to every point of $|\mathscr {X}|$ : For each $x\in |\mathscr {X}|$ we obtain an étale morphism $f_x \colon \mathscr {W}_x \to \mathscr {X}$ such that $f_x|_{\mathscr {G}_x}$ is an isomorphism and $\mathscr {W}_x$ is fundamental. If $\mathscr {X}$ has affine (resp. separated) diagonal, then Theorem 1.5 says that we can arrange that $f_x$ is affine (resp. representable). By Theorem 1.6, we may further assume that $\mathscr {W}_x$ is nicely fundamental. Set $\mathscr {W} = \amalg _{x\in |\mathscr {X}|} \mathscr {W}_x$ , and take $f=\amalg _{x} f_x \colon \mathscr {W} \to \mathscr {X}$ , then f is a quasi-separated Nisnevich covering. By [Reference Hall and RydhHR18, Prop. 3.3], we may shrink $\mathscr {W}$ so that it is quasi-compact (a monomorphic splitting sequence must factor through finitely many of the $\mathscr {W}_x$ ), remains nicely fundamental and f is a Nisnevich covering.

Proof of Theorem 1.11

We apply Theorem 5.1 to every closed point of $|\mathscr {X}|$ : For each closed point x of $\mathscr {X}$ we obtain an étale morphism $g \colon \mathscr {W} \to \mathscr {X}$ such that $\mathscr {W}=[U/\mathrm {GL}_n]$ is fundamental and $g|_{\mathscr {G}_x}$ is an isomorphism. If $\mathscr {X}$ has affine (resp. separated diagonal), then Theorem 1.5 says that we can arrange that g is affine (resp. representable).

For an integer $d\geq 1$ , let $\mathscr {W}^d$ be the dth fiber product of g; then the symmetric group $S_d$ acts on $\mathscr {W}^d$ by permuting the factors. Let e be the maximum rank of a fiber of g. Then there is an induced Nisnevich covering $f \colon \coprod _{1 \leq d \leq e} [\mathscr {W}^d/S_d] \to \mathscr {X}$ since g is representable.

Let $V^d$ be the dth fiber product of $U\to \mathscr {W}\to \mathscr {X}$ . Then $\mathscr {W}^d=[V^d/(\mathrm {GL}_n)^d]$ . Let P be one of the properties: separated, quasi-affine, affine. If the diagonal of $\mathscr {X}$ has property P, then the algebraic space $V^d$ has property P. Since the Stiefel manifold $\mathrm {GL}_{dn}/(\mathrm {GL}_n)^d$ is affine, it follows that $\mathscr {W}^d=[V'/\mathrm {GL}_{nd}]$ for an algebraic space $V'$ with property P.

Let $p\colon \mathscr {W}^d\to [\mathscr {W}^d/S_d]$ . Let $\mathcal {E}$ be the vector bundle on $\mathscr {W}^d$ with frame bundle $V'$ . Then we claim that the frame bundle V of $p_*\mathcal {E}$ is an algebraic space with property P. Indeed, V is an algebraic space since the stabilizers of $[\mathscr {W}^d/S_d]$ act faithfully on $p_*\mathcal {E}$ , cf. [Reference Edidin, Hassett, Kresch and VistoliEHKV01, Lem. 2.13]. Since $p^*V\to V$ is finite, étale and surjective, it is enough to prove that $p^*V$ has property P. But since p is finite étale, we have that $p^*p_*\mathcal {E}\to \mathcal {E}$ is split surjective and it follows that $p^*V$ has property P by considering Stiefel manifolds again. We have thus shown that $[\mathscr {W}^d/S_d]=[V/\mathrm {GL}_N]$ for an algebraic space V with property P.

When $\mathscr {X}$ has affine diagonal, then $\mathscr {W}^d\to \mathscr {X}$ is affine but $[\mathscr {W}^d/S_d]\to \mathscr {X}$ is merely separated. Let $\mathrm {SEC}^d(\mathscr {W}/\mathscr {X})\subseteq \mathscr {W}^d$ be the open and closed substack that is the complement of all diagonals. Then $S_d$ acts freely on $\mathrm {SEC}^d(\mathscr {W}/\mathscr {X})$ relative to $\mathscr {X}$ and $\mathrm {\acute {E}T}^d(\mathscr {W}/\mathscr {X}):=[\mathrm {SEC}^d(\mathscr {W}/\mathscr {X})/S_d]\to \mathscr {X}$ is affine and an étale neighborhood of any point of $\mathscr {X}$ at which g has rank d. Thus, $f\colon \coprod _{1 \leq d \leq e} \mathrm {\acute {E}T}^d(\mathscr {W}/\mathscr {X}) \to \mathscr {X}$ is a fundamental Nisnevich covering with f affine.

Proof of Theorem 1.7

By Theorem 5.8, there exists an étale neighborhood $\mathscr {W} \to \mathscr {X}$ of $\mathscr {W}_0:=\mathscr {X}_0$ such that $\mathscr {W}$ is fundamental. Let $\mathscr {W}_0 \to W_0$ and $\mathscr {W} \to W$ be the good and adequate moduli spaces. We claim that the henselization $W^h$ of W along $W_0$ exists and is affine. If $\mathscr {W}_0$ is a closed substack, then this follows from [Reference RaynaudRay70, Ch. XI, Thm. 2] as W is affine. If $\mathscr {W}_0 = \mathscr {G}_x$ is the residual gerbe of a point $x \in |\mathscr {X}|$ , then $W_0 = \operatorname {\mathrm {Spec}} \kappa (x) \hookrightarrow W$ is the inclusion of a point w and $W^h = \operatorname {\mathrm {Spec}} \mathcal {O}_{W,w}^h$ . In this case, we also note that $\mathscr {W}_0$ satisfies (FC). Let $\mathscr {W}^h = \mathscr {W} \times _W W^h$ . Since $W^h \to W$ is flat, $\mathscr {W}^h \to W^h$ is an adequate moduli space. By [Reference Alper, Hall and RydhAHR19, Thm. 3.6], $(\mathscr {W}^h, \mathscr {W}_0)$ is a henselian pair and by [Reference Alper, Hall and RydhAHR19, Cor. 6.10], $\mathscr {W}^h$ is linearly fundamental since the closed points of $\mathscr {W}^h$ have linearly reductive stabilizer. To show that $\mathscr {W}^h \to \mathscr {X}$ is the henselization of $\mathscr {X}$ along $\nu \colon \mathscr {X}_0 \hookrightarrow \mathscr {X}$ , it is enough to prove that any quasi-separated étale neighborhood $g \colon \mathscr {W}'\to \mathscr {W}^h$ of $\mathscr {W}_0$ has a section. This is precisely the conclusion of [Reference Alper, Hall and RydhAHR19, Prop. 7.9].