Proof. The proposition holds by the exact same proof used for the analogous statement for schemes in [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 2, Corollary 1.3.5].

Proof. If $\xi $ is a piecewise trivial fibration with fibre $\mathcal {F}$ , then for every $z\in Z$ , there is a locally closed subset $Z'\subseteq Z$ containing z for which $(\mathcal {Y}\times _Z Z')_{\mathrm {red}}\cong (\mathcal {F}\times _S Z')_{\mathrm {red}}$ as $Z^{\prime }_{\mathrm {red}}$ -stacks. Then

$$ \begin{align*} \begin{split} (\mathcal{Y}\times_Z \operatorname{\mathrm{Spec}} k(z))_{\mathrm{red}} &=\left((\mathcal{Y}\times_Z Z')_{\mathrm{red}}\times_{Z^{\prime}_{\mathrm{red}}} \operatorname{\mathrm{Spec}} k(z)\right)_{\mathrm{red}}\\ &\cong\left((\mathcal{F}\times_S Z')_{\mathrm{red}}\times_{Z^{\prime}_{\mathrm{red}}} \operatorname{\mathrm{Spec}} k(z)\right)_{\mathrm{red}}=(\mathcal{F}\times_S \operatorname{\mathrm{Spec}} k(z))_{\mathrm{red}}. \end{split} \end{align*} $$

We now show that the converse holds. Since

$$ \begin{align*} (\mathcal{Y}_{\mathrm{red}}\times_Z \operatorname{\mathrm{Spec}} k(z))_{\mathrm{red}}=(\mathcal{Y}\times_Z \operatorname{\mathrm{Spec}} k(z))_{\mathrm{red}} \end{align*} $$

for every $z\in Z$ , we can assume $\mathcal {Y}$ is reduced. By Noetherian induction on Z, we need only find a nonempty open subset $U\subseteq Z$ for which $(\mathcal {Y}\times _Z U)_{\mathrm {red}}\cong (\mathcal {F}\times _S U)_{\mathrm {red}}$ . Let $z\in Z$ be the generic point of an irreducible component of Z; replacing Z by an open affine neighbourhood of z, we may further assume Z is affine. Since $\mathcal {O}_{Z,z}$ is a field, $\mathcal {Y}\times _Z \operatorname {\mathrm {Spec}} k(z)$ is reduced and we hence have a surjective closed immersion

$$ \begin{align*} \iota\colon\mathcal{Y}\times_Z \operatorname{\mathrm{Spec}} k(z)\cong(\mathcal{F}\times_S \operatorname{\mathrm{Spec}} k(z))_{\mathrm{red}}\to\mathcal{F}\times_S \operatorname{\mathrm{Spec}} k(z). \end{align*} $$

Now, $\operatorname {\mathrm {Spec}}\mathcal {O}_{Z,z}=\lim _\lambda U_\lambda $ is the inverse limit of open affine neighbourhoods $U_\lambda \subseteq Z$ of z. Since Z is affine, each map $U_\lambda \to Z$ is affine. Note also that $\mathcal {Y}$ is Noetherian, hence quasicompact and quasiseparated, and that $\mathcal {F}\times _S \operatorname {\mathrm {Spec}} k(z)\to \operatorname {\mathrm {Spec}} k(z)$ is locally of finite presentation. [Reference RydhRyd15, Proposition B.2] then shows there is some index $\lambda $ and a morphism $\iota _\lambda \colon \mathcal {Y}\times _Z U_\lambda \to \mathcal {F}\times _S U_\lambda $ whose base change to $\operatorname {\mathrm {Spec}}\mathcal {O}_{Z,z}$ is $\iota $ . Furthermore, since $\mathcal {F}\times _S \operatorname {\mathrm {Spec}} k(z)\to \operatorname {\mathrm {Spec}} k(z)$ and $\xi $ are both of finite presentation, [Reference RydhRyd15, Proposition B.3] shows that after replacing $\lambda $ by a larger index if necessary, we can assume $\iota _\lambda $ is a surjective closed immersion, and hence defines an isomorphism $(\mathcal {Y}\times _Z U_\lambda )_{\mathrm {red}}\cong (\mathcal {F}\times _S U_\lambda )_{\mathrm {red}}$ .

Proof. Let $v_i=\beta (e_i)$ for $1\leq i\leq r$ . Since $\pi $ is an isomorphism over T, each $v_i\neq 0$ . As $\sigma $ is pointed, there exists some $p \in P$ such that $\langle v_i, p \rangle> 0$ for all i. Viewing p as an element of F via the inclusion $P\subset F$ , we have $\langle e_i, p \rangle> 0$ .

Let $f_1, \dotsc , f_r$ be the basis of $\widetilde {M}$ dual to $e_1, \dotsc , e_r$ . Since the $f_i$ are generators of F, it suffices to prove the proposition for each $f_i$ . Note that

$$ \begin{align*} \langle e_1, p \rangle f_1 + \dotsb + \langle e_r, p \rangle f_r = p \in P. \end{align*} $$

Since $\langle e_i, p \rangle> 0$ , we see that

$$ \begin{align*} f^{\prime}_i:=(\langle e_i,p\rangle-1)f_i+\sum_{j\neq i}\left\langle e_j,p\right\rangle f_j\in F \end{align*} $$

and that $f_i+f^{\prime }_i\in P$ .

Proof. Let $f_1, \dotsc , f_r$ be the minimal generators of the monoid F. By Proposition 2.14, there exist $f^{\prime }_1, \dotsc , f^{\prime }_r$ such that $f_i + f^{\prime }_i \in P$ for all $i \in \{1, \dotsc , r\}$ . For any $\widetilde {w} \in \beta ^{-1}(w)$ ,

$$ \begin{align*} \left\langle \widetilde{w}, f_i \right\rangle \leq \left\langle \widetilde{w}, f_i + f^{\prime}_i \right\rangle = \left\langle w, f_i+f^{\prime}_i\right\rangle, \end{align*} $$

so there are only finitely many possible values for each $\left \langle \widetilde {w}, f_i \right \rangle $ . Thus $\beta ^{-1}(w)$ is a finite set.

Notation 2.19.

• • Let $M = N^*$ .

• • Let $\widetilde {N}$ be the free abelian group with generators indexed by the rays of $\sigma $ .

• • Let $\widetilde {M} = \widetilde {N}^*$ .

• • Let $\langle \cdot , \cdot \rangle $ denote both pairings $N \otimes _{\mathbb {Z}} M \to \mathbb {Z}$ and $\widetilde {N} \otimes _{\mathbb {Z}} \widetilde {M} \to \mathbb {Z}$ .

• • Let $\widetilde {T} = \operatorname {\mathrm {Spec}}\left (k\left [\widetilde {M}\right ]\right )$ be the algebraic torus with cocharacter lattice $\widetilde {N}$ .

• • Let $\widetilde {\sigma }$ be the positive orthant of $\widetilde {N}_{\mathbb {R}}$ – that is, $\widetilde {\sigma }$ is the positive span of those generators of $\widetilde {N}$ that are indexed by the rays of $\sigma $ .

• • Let $\widetilde {X}$ be the affine $\widetilde {T}$ -toric variety associated to $\widetilde {\sigma }$ .

• • Let $\beta : \widetilde {\sigma } \cap \widetilde {N} \to \sigma \cap N$ be the monoid map taking the generator of $\widetilde {N}$ indexed by a ray of $\sigma $ to the first lattice point of that ray.

• • Let $\widetilde {\pi }: \widetilde {X} \to X$ be the toric map associated to $\beta ^{\mathrm {gp}}: \widetilde {N} \to N$ .

• • Let $P = \sigma ^\vee \cap M$ . Note that $X = \operatorname {\mathrm {Spec}}(k[P])$ .

• • Let $F = \widetilde {\sigma }^\vee \cap \widetilde {M}$ . Note that $\widetilde {X} = \operatorname {\mathrm {Spec}}(k[F])$ .

• • Identify P with its image under the injection $P \hookrightarrow F$ given by dualising $\beta $ . Note that $P \hookrightarrow F$ is injective because $\sigma $ is full-dimensional.

• • Let $A = F^{\mathrm {gp}}/P^{\mathrm {gp}} = \widetilde {M}/ M$ .

• • Let $G = \operatorname {\mathrm {Spec}}(k[A])$ be the kernel of the algebraic group homomorphism $\widetilde {T} \to T$ obtained by restricting $\widetilde {\pi }$ , and let G act on $\widetilde {X}$ by restricting the toric action of $\widetilde {T}$ on $\widetilde {X}$ .

Proof. We let $\mathcal {X}'=\left [\widetilde {X}'/G'\right ]$ , $\rho ': \widetilde {X}' \to \mathcal {X}'$ be the quotient map, $\mathcal {X}=\mathscr {R}_{S'/S}(\mathcal {X}')$ , $\widetilde {X}=\mathscr {R}_{S'/S}\left (\widetilde {X}'\right )$ and $G=\mathscr {R}_{S'/S}(G')$ . Since $\rho '\colon \widetilde {X}'\to \mathcal {X}'$ is a smooth cover, $\mathscr {R}_{S'/S}(\rho ')\colon \widetilde {X}\to \mathcal {X}$ is as well, by [Reference RydhRyd11, Proposition 3.5(v)]. Since $\rho '$ is a $G'$ -torsor, the natural map $G'\times _{S'} \widetilde {X}'\to \widetilde {X}'\times _{\mathcal {X}'} \widetilde {X}'$ induced by the $G'$ -action $G' \times _{S'} \widetilde {X}' \to \widetilde {X}'$ is an isomorphism, and applying Weil restriction, we see that the map $G\times _S \widetilde {X}\to \widetilde {X}\times _{\mathcal {X}} \widetilde {X}$ induced by the G-action $G \times _S \widetilde {X} \to \widetilde {X}$ is an isomorphism as well. Thus, $\mathscr {R}_{S'/S}(\rho ')\colon \widetilde {X}\to \mathcal {X}$ is a G-torsor, thereby inducing an isomorphism $\mathcal {X}\xrightarrow {\sim }\left [\widetilde {X}/G\right ]$ which makes the diagram in the statement of the proposition commute.

Proof. Set $n \in \mathbb {N}$ . Clearly, $\theta _n\left (\widetilde {C}\right ) \subset \mathscr {L}_n(\rho )^{-1}(\theta _n(\mathcal {C}))$ .

To prove the opposite inclusion, let $k'$ be a field extension of k, and let $\widetilde {\psi }_n \in \mathscr {L}_n\left (\widetilde {X}\right )(k')$ and $\psi \in \mathscr {L}(\mathcal {X})(k')$ be such that the class of $\psi $ in $\lvert \mathscr {L}(\mathcal {X})\rvert $ is contained in $\mathcal {C}$ and $\mathscr {L}_n(\rho )\left (\widetilde {\psi }_n\right ) \cong \theta _n(\psi )$ . We must show $\widetilde {\psi }_n\in \theta _n\left (\widetilde {C}\right )$ . Since $\rho $ is smooth, by the infinitesimal lifting criterion, we have a dotted arrow filling in the following diagram:

Then $\widetilde {\psi }\in \widetilde {C}$ , so $\widetilde {\psi }_n\in \theta _n\left (\widetilde {C}\right )$ .

Proof. We first show that $\widetilde {C} \subset \mathscr {L}\left (\widetilde {X}\right )$ is a cylinder. Because $\mathcal {C}$ is a cylinder, there exist some $n \in \mathbb {N}$ and some constructible subset $\mathcal {C}_n \subset \lvert \mathscr {L}_n(\mathcal {X})\rvert $ such that $\mathcal {C} = (\theta _n)^{-1}(\mathcal {C}_n)$ . Then $\widetilde {C} = (\theta _n)^{-1}\left (\mathscr {L}_n(\rho )^{-1}(\mathcal {C}_n)\right )$ is a cylinder.

Now we will show that for all $n \in \mathbb {N}$ , the set $\theta _n(\mathcal {C})$ is a constructible subset of $\mathscr {L}_n(\mathcal {X})$ . Each $\theta _n\left (\widetilde {C}\right )$ is a constructible subset of $\mathscr {L}_n\left (\widetilde {X}\right )$ . Therefore each $\theta _n(\mathcal {C}) \subset \lvert \mathscr {L}_n(\mathcal {X})\rvert $ is constructible, by Chevalley’s theorem for Artin stacks [Reference Hall and RydhHR17, Theorem 5.2], Corollary 3.22 and Lemma 3.24.

Then since G is a special group, $\mathscr {L}_n(G)$ is as well, by Remark 2.4. Then Corollary 3.22 and Lemma 3.24 imply that for each $n \in \mathbb {N}$ ,

$$ \begin{align*} \mathrm{e}(\theta_n(\mathcal{C})) = \mathrm{e}\left(\theta_n\left(\widetilde{C}\right)\right)\mathrm{e}(\mathscr{L}_n(G))^{-1} = \mathrm{e}\left(\theta_n\left(\widetilde{C}\right)\right) \mathrm{e}(G)^{-1} \mathbb{L}^{-n\dim G}, \end{align*} $$

where the second equality holds because G is smooth. Therefore,

$$ \begin{align*} \mu_{\widetilde{X}}(\widetilde{C})\mathrm{e}(G)^{-1}\mathbb{L}^{\dim G} &= \lim_{n \to \infty} \mathrm{e}\left(\theta_n\left(\widetilde{C}\right)\right) \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G -(n+1)\dim \widetilde{X}}\\ &= \lim_{n \to \infty} \mathrm{e}(\theta_n(\mathcal{C})) \mathbb{L}^{-(n+1)\dim\mathcal{X}}.\\[-36pt] \end{align*} $$

Proof. The set $\mathcal {C}$ is measurable by the exact same proof used for the analogous statement for schemes in [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Proposition 3.4.2]. The remainder of the proposition follows from Corollary 3.16 and the analogous statement for schemes [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Proposition 3.4.3] applied to the scheme $\widetilde {X}$ in the statement of Corollary 3.16.

Proof. The proposition holds by the exact same proof used for the analogous statement for schemes in [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Corollary 3.5.5(a)].

Proof. By Remark 3.1, there exist $G, \widetilde {X}, \rho $ as in the statement of Theorem 3.9 such that G is a general linear group. Let $\widetilde {C} = \mathscr {L}(\rho )^{-1}(\mathcal {C})$ and $\widetilde {D} = \mathscr {L}(\rho )^{-1}(\mathcal {D})$ . Then

$$ \begin{align*} \lVert \mu_{\mathcal{X}}(\mathcal{C}) \rVert &= \left\lVert \mu_{\widetilde{X}}\left(\widetilde{C}\right)\mathrm{e}(G)^{-1}\mathbb{L}^{\dim G} \right\rVert\\ &\leq \left\lVert \mathrm{e}(G)^{-1}\mathbb{L}^{\dim G}\right\rVert \left\lVert \mu_{\widetilde{X}}\left(\widetilde{C}\right)\right\rVert\\ &\leq \left\lVert \mathrm{e}(G)^{-1}\mathbb{L}^{\dim G}\right\rVert \left\lVert \mu_{\widetilde{X}}\left(\widetilde{D}\right)\right\rVert\\ &= \left\lVert \mathrm{e}(G)^{-1}\mathbb{L}^{\dim G}\right\rVert \left\lVert \mu_{\mathcal{X}}(\mathcal{D}) \mathrm{e}(G)\mathbb{L}^{-\dim G}\right\rVert\\ &\leq \left\lVert \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G} \right\rVert \left\lVert\mathrm{e}(G) \mathbb{L}^{-\dim G}\right\rVert \lVert \mu_{\mathcal{X}}(\mathcal{D}) \rVert\\ &= \lVert \mu_{\mathcal{X}}(\mathcal{D}) \rVert, \end{align*} $$

where the first and fourth lines follow from Corollary 3.16, the third line follows from the analogous statement [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Corollary 3.3.5] for schemes applied to $\widetilde {X}$ and the last line follows from Lemma 3.17.

Proof. For each $n \in \mathbb {N}$ , let $\mathcal {C}_n \subset \lvert \mathscr {L}_n(\mathcal {X})\rvert $ be the image of $\lvert \mathscr {L}_n(\mathcal {Y})\rvert $ in $\lvert \mathscr {L}_n(\mathcal {X})\rvert $ , and let $\mathcal {C}^{(n)} = (\theta _n)^{-1}(\mathcal {C}_n)$ . By [Reference RydhRyd11, Proposition 3.5(vi)], each $\mathcal {C}_n$ is a closed subset of $\mathscr {L}_n(\mathcal {X})$ , so each $\mathcal {C}^{(n)}$ is a cylinder in $\mathscr {L}(\mathcal {X})$ .

By Remark 3.1, there exist $G, \widetilde {X}, \rho $ as in the statement of Theorem 3.9. Let $\widetilde {Y} = \widetilde {X} \times _{\mathcal {X}} \mathcal {Y}$ . Then $\mathscr {L}_n(\rho )^{-1}(\mathcal {C}_n)$ is the underlying set of $\mathscr {L}_n\left (\widetilde {Y}\right )$ . Thus by Theorem 3.9,

$$ \begin{align*} \mu_{\mathcal{X}}\left(\mathcal{C}^{(n)}\right) &= \mu_{\widetilde{X}} \left(\mathscr{L}(\rho)^{-1}\left(\mathcal{C}^{(n)}\right)\right) \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G}\\ &= \mu_{\widetilde{X}}\left((\theta_n)^{-1}\left(\mathscr{L}_n\left(\widetilde{Y}\right)\right)\right)\mathrm{e}(G)^{-1} \mathbb{L}^{\dim G}. \end{align*} $$

By [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Proposition 2.3.1],

$$ \begin{align*} \lim_{n \to \infty} \mu_{\widetilde{X}}\left((\theta_n)^{-1}\left(\mathscr{L}_n\left(\widetilde{Y}\right)\right)\right) = 0, \end{align*} $$

so

$$ \begin{align*} \lim_{n \to \infty} \mu_{\mathcal{X}}\left(\mathcal{C}^{(n)}\right) = 0. \end{align*} $$

Therefore for any $\varepsilon \in \mathbb {R}_{>0}$ , we get that $\left (\emptyset , \left (\mathcal {C}^{(n)}\right )\right )$ is a cylindrical $\varepsilon $ -approximation of $\mathcal {C}$ for sufficiently large n, and we are done, by the definition of $\mu _{\mathcal {X}}$ .

Proof. The proposition holds by the exact same proof used for the analogous statement for schemes in [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Proposition 3.2.8].

Proof. As in the case of schemes, this is an easy consequence of the definitions and the fact that for all $n \in \mathbb {N}$ , the morphism $\mathscr {L}_n(\iota ): \mathscr {L}_n(\mathcal {U}) \to \mathscr {L}_n(\mathcal {X})$ is an open immersion by [Reference RydhRyd11, Proposition 3.5(vii)].

Proof. Fix $\xi _n\colon \operatorname {\mathrm {Spec}} k'\to \mathscr {L}_n(\mathcal {X})$ and let $\mathcal {Y}_{\xi _n}$ denote the fibre of the truncation map $\mathscr {L}_{n+1}(\mathcal {X})\to \mathscr {L}_n(\mathcal {X})$ over $\xi _n$ . For any $\alpha \colon \operatorname {\mathrm {Spec}} A\to \operatorname {\mathrm {Spec}} k'$ , the A-valued points $\mathcal {Y}_{\xi _n}(A)$ are the category of lifts of $\xi _n\otimes _{k'}A$ to $\mathscr {L}_{n+1}(\mathcal {X})$ . For all $m\geq 0$ , let $\mathcal {X}_m=\mathcal {X}\otimes _k k[t]/\left (t^{m+1}\right )$ and let $\alpha _m\colon \operatorname {\mathrm {Spec}} A[t]/\left (t^{m+1}\right )\to \operatorname {\mathrm {Spec}} k'[t]/\left (t^{m+1}\right )$ be the map induced by $\alpha $ ; for $m\leq n$ , let $\varphi _m\colon \operatorname {\mathrm {Spec}} k'[t]/\left (t^{m+1}\right )\to \mathcal {X}_m$ denote the map induced by $\xi _n$ . We then obtain a cartesian diagram

where the curved arrow is the structure map. Let $\mathcal {J}_n$ denote the ideal sheaf of $\operatorname {\mathrm {Spec}} k'[t]/\left (t^{n+1}\right )\to \operatorname {\mathrm {Spec}} k'[t]/\left (t^{n+2}\right )$ considered as a $k'$ -module. By [Reference OlssonOls06, Theorem 1.5] and the fact that $\mathcal {X}_{n+1}$ and $A[t]/\left (t^{n+2}\right )$ are flat over $k[t]/(t^{n+2})$ , the obstruction to the existence of a dotted arrow in this diagram lives in

$$ \begin{align*} \operatorname{\mathrm{Ext}}^1\left(L(\alpha_0\varphi_0)^*L_{\mathcal{X}/k},\alpha_0^*\mathcal{J}_n\right)=\operatorname{\mathrm{Ext}}^1\left(L\varphi_0^*L_{\mathcal{X}/k},\mathcal{O}_{k'}\right)\otimes_{k'}\mathcal{J}_n\otimes_{k'}A. \end{align*} $$

We will show that this group vanishes, and so by [Reference OlssonOls06, Theorem 1.5], the objects (resp., automorphisms) of $\mathcal {Y}_{\xi _n}(A)$ are parametrised by $\operatorname {\mathrm {Ext}}^n\left (L\varphi _0^*L_{\mathcal {X}/k},\mathcal {O}_{k'}\right )\otimes _{k'}\mathcal {J}_n\otimes _{k'}A$ where $n=0$ (resp., $n=-1$ ). In particular, if V (resp., G) denotes the affine space (resp., algebraic vector group) over $k'$ associated to the vector space $\operatorname {\mathrm {Ext}}^n\left (L\varphi _0^*L_{\mathcal {X}/k},\mathcal {O}_{k'}\right )\otimes _{k'}\mathcal {J}_n$ with $n=0$ (resp., $n=-1$ ), then we have

$$ \begin{align*} \mathcal{Y}_{\xi_n}\cong V\times_{k'} G\cong \mathbb{A}^{r\left(\xi_n\right)}\times B\mathbb{G}_a^{j\left(\xi_n\right)}, \end{align*} $$

where $r(\xi _n)=\dim \operatorname {\mathrm {Ext}}^0\left (L\varphi _0^*L_{\mathcal {X}/k},\mathcal {O}_{k'}\right )$ and $j(\xi _n)=\dim \operatorname {\mathrm {Ext}}^1\left (L\varphi _0^*L_{\mathcal {X}/k},\mathcal {O}_{k'}\right )$ . Note that this implies

$$ \begin{align*} e\left(\mathcal{Y}_{\xi_n}\right)=\mathbb{L}^{r\left(\xi_n\right)-j\left(\xi_n\right)}. \end{align*} $$

Therefore, to finish the proof of the theorem, it suffices to show that

(1) $$ \begin{align} \operatorname{\mathrm{Ext}}^1\left(L\varphi_0^*L_{\mathcal{X}/k},\mathcal{O}_{k'}\right)=0,\qquad r(\xi_n)-j(\xi_n)=\dim\mathcal{X}, \end{align} $$

and that the locus of $\varphi _0\in \lvert \mathcal {X}\rvert $ where $r(\xi _n)$ is constant is given by a constructible set. Since these remaining statements depend only on the dimension of the $\operatorname {\mathrm {Ext}}$ -groups over $k'$ , it suffices to replace $k'$ with an extension field, and hence we can assume $k'$ is algebraically closed.

Let $\rho \colon \widetilde {X}\to \mathcal {X}$ be a smooth cover. Since $k'$ is algebraically closed, we may fix a lift $\phi _0\colon \operatorname {\mathrm {Spec}} k'\to \widetilde {X}$ of $\varphi _0$ . Since $\widetilde {X}$ and $\rho $ are smooth, we have an exact triangle

$$ \begin{align*} p^*L_{\mathcal{X}/k}\to\Omega^1_{\widetilde{X}/k}\to\Omega^1_{\widetilde{X}/\mathcal{X}}, \end{align*} $$

from which we obtain an exact triangle

$$ \begin{align*} \Gamma\left(\phi_0^*T_{\widetilde{X}/\mathcal{X}}\right)\to\Gamma\left(\phi_0^*T_{\widetilde{X}/k}\right)\to \mathrm{RHom}\left(L\varphi_0^*L_{\mathcal{X}/k},\mathcal{O}_{k'}\right). \end{align*} $$

In particular, $\operatorname {\mathrm {Ext}}^n\left (L\varphi _0^*L_{\mathcal {X}/k},\mathcal {O}_{k'}\right )=0$ for $n\neq 0,-1$ , and there is an exact sequence

(2) $$ \begin{align} 0\to\operatorname{\mathrm{Ext}}^{-1}\left(L\varphi_0^*L_{\mathcal{X}/k},\mathcal{O}_{k'}\right)\to\Gamma\left(\phi_0^*T_{\widetilde{X}/\mathcal{X}}\right)\to\Gamma\left(\phi_0^*T_{\widetilde{X}/k}\right)\to\operatorname{\mathrm{Ext}}^0\left(L\varphi_0^*L_{\mathcal{X}/k},\mathcal{O}_{k'}\right)\to0. \end{align} $$

Thus,

$$ \begin{align*} r(\xi_n)-j(\xi_n)=\dim\Gamma\left(\phi_0^*T_{\widetilde{X}/k}\right)-\dim\Gamma\left(\phi_0^*T_{\widetilde{X}/\mathcal{X}}\right)= \dim\mathcal{X}, \end{align*} $$

thereby establishing equation (1). Finally, note from formula (2) that the cokernel of $\Gamma \left (\phi _0^*T_{\widetilde {X}/\mathcal {X}}\right )\to \Gamma \left (\phi _0^*T_{\widetilde {X}/k}\right )$ depends only on $\varphi _0$ and not on the choice of lift $\phi _0$ . So the locus of $\varphi _0\in \lvert \mathcal {X}\rvert $ where $r(\xi _n)$ is constant is the image under $\rho $ of the locus of $\phi _0\in \left \lvert \widetilde {X}\right \rvert $ where the dimension is constant. By Chevalley’s theorem for Artin stacks [Reference Hall and RydhHR17, Theorem 5.2], it is therefore enough to show that the locus of such $\phi _0$ is constructible. This follows by applying [Sta21, Lemma 0BDI] to the $2$ -term complex $T_{\widetilde {X}/\mathcal {X}}\to T_{\widetilde {X}/k}$ and using the fact that $\left \lvert \widetilde {X}\right \rvert $ is Noetherian, so that all locally constructible sets are constructible.

Proof. Because $\mathcal {Z}$ can be stratified by quotient stacks [Reference KreschKre99, Proposition 3.5.9], we may assume that $\mathcal {Z} = [Z / G]$ for some finite-type scheme Z over k with an action by a general linear group G over k. Let $\mathcal {Y}' = \mathcal {Y} \times _{\mathcal {Z}} Z$ . Because $Z \to \mathcal {Z}$ and $\mathcal {Y}' \to \mathcal {Y}$ are G-torsors and G is a special group, [Reference EkedahlEke09, Proposition 1.1(ii)] gives

$$ \begin{align*} \mathrm{e}(Z) &= \mathrm{e}(G)\mathrm{e}(\mathcal{Z}),\\ \mathrm{e}(\mathcal{Y}') &= \mathrm{e}(G)\mathrm{e}(\mathcal{Y}). \end{align*} $$

By the hypotheses on $\mathcal {Y} \to \mathcal {Z}$ , Proposition 2.8 implies that $\mathcal {Y}' \to Z$ is a piecewise trivial fibration with fibre $\mathcal {F}$ , so in particular,

$$ \begin{align*} \mathrm{e}(\mathcal{Y}') = \mathrm{e}(\mathcal{F})\mathrm{e}(Z). \end{align*} $$

Thus,

$$ \begin{align*} \mathrm{e}(G)\mathrm{e}(\mathcal{Y}) = \mathrm{e}(G)\mathrm{e}(\mathcal{F})\mathrm{e}(\mathcal{Z}). \end{align*} $$

Because G is a special group, $\mathrm {e}(G)^{-1} \in K_0(\mathbf {Stack}_k)$ , so we are done.

Proof of Theorem 3.33

By definition, there exist some $n_0 \in \mathbb {N}$ and some constructible subset $\mathcal {C}_{n_0} \subset \left \lvert \mathscr {L}_{n_0}(\mathcal {X})\right \rvert $ such that $\mathcal {C} = \left (\theta _{n_0}\right )^{-1}\left (\mathcal {C}_{n_0}\right )$ . Because $\mathcal {X}$ is smooth, infinitesimal lifting implies that the truncation maps $\theta _n: \lvert \mathscr {L}(\mathcal {X})\rvert \to \lvert \mathscr {L}_n(\mathcal {X})\rvert $ are all surjective, so

$$ \begin{align*} \theta_n(\mathcal{C}) = \begin{cases} \left(\theta^n_{n_0}\right)^{-1}\left(\mathcal{C}_{n_0}\right), & n \geq n_0, \\ \theta^{n_0}_n\left(\mathcal{C}_{n_0}\right), & n < n_0. \end{cases} \end{align*} $$

Thus all $\theta _n(\mathcal {C})$ are constructible – immediately for $n \geq n_0$ , and by Chevalley’s theorem for Artin stacks [Reference Hall and RydhHR17, Theorem 5.2] for $n < n_0$ .

The remainder of the theorem then follows from the fact that Lemma 3.34 and Lemma 3.35 imply that for any $n \geq n_0$ ,

$$ \begin{align*} \mathrm{e}(\theta_n(\mathcal{C})) = \left(\theta^n_{n_0}\right)^{-1}\left(\mathcal{C}_{n_0}\right) = \mathrm{e}\left(\mathcal{C}_{n_0}\right)\mathbb{L}^{\left(n-n_0\right)\dim\mathcal{X}}.\\[-39pt] \end{align*} $$

Proof. Let $q\colon \operatorname {\mathrm {Spec}}(\Omega [[t]])\to \operatorname {\mathrm {Spec}}(\Omega )$ denote the structure map. Since G is an étale group scheme, G-torsors on any $\Omega $ -scheme Y are classified up to isomorphism by $H^1_{et}(Y,G)=H^1_{et}(Y,\mathbb {G}_m)^{\oplus r}\oplus \bigoplus _{i=1}^N H^1_{et}\left (Y,\mu _{n_i}\right )$ . In particular, it suffices to show that the pullback map $q^*\colon H^1_{et}(\operatorname {\mathrm {Spec}}(\Omega ),G)\to H^1_{et}(\operatorname {\mathrm {Spec}}(\Omega [[t]]),G)$ is an isomorphism when G is either $\mathbb {G}_m$ or $\mu _n$ .

We first handle the case $G=\mathbb {G}_m$ . Since $H^1_{et}(Y,\mathbb {G}_m)=\mathrm {Pic}(Y)$ and since both $\mathrm {Pic}(\operatorname {\mathrm {Spec}}(\Omega ))$ and $\mathrm {Pic}(\operatorname {\mathrm {Spec}}(\Omega [[t]]))$ are trivial, we see that $q^*$ is an isomorphism.

We next handle the case $G=\mu _n$ . From the Kummer sequence

$$ \begin{align*} 1\to\mu_n\to\mathbb{G}_m\xrightarrow{\times n}\mathbb{G}_m\to1, \end{align*} $$

we see that if Y is any $\Omega $ -scheme with trivial Picard group, we have

$$ \begin{align*} H^1_{et}(Y,\mu_n)=\mathcal{O}_Y(Y)^*/(\mathcal{O}_Y(Y)^*)^n \end{align*} $$

(see, for example, [Reference MilneMil80, p. 125]). So it remains to show that $k^*/(k^*)^n\to k[[t]]^*/(k[[t]]^*)^n$ is an isomorphism. Since every element $f(t)\in k[[t]]^*$ can be written uniquely as $ag(t)$ with $a\in k^*$ and $g(t)\in k[[t]]^*$ with $g(t)-1\in tk[[t]]$ , it is enough to prove that such $g(t)$ are in $(k[[t]]^*)^n$ . This follows immediately from Hensel’s lemma: the polynomial $P(x)=x^n-g(t)\in k[[t]][x]$ has a root, since $P(1)=0$ mod t and $P'(1)\neq 0$ mod t.

Proof. For ease of notation, write $F:=\sigma ^\vee \cap \widetilde {N}^*$ and $G:=G_\nu =\operatorname {\mathrm {Spec}}(k[A])$ , where A is a finitely generated abelian group. Note that the G-action on $X_\sigma $ corresponds to a monoid map $\eta \colon F\to A$ . The arc $\psi $ corresponds to a G-torsor and G-equivariant map $Q\to X_\sigma $ . By Lemma 4.3, Q is isomorphic to the pullback of a G-torsor over $\operatorname {\mathrm {Spec}}(k')$ , which can be itself be trivialised after a finite field extension $k''$ . Thus, after base change to , we obtain a trivialisation of Q and hence a lift $\widetilde {\psi }$ .

Next, it is clear that if $\widetilde {\psi }\in \mathscr {L}(X_\sigma )(k'')$ is a lift of $\psi $ and $k'''$ is a finite field extension of $k''$ , then $\operatorname {\mathrm {trop}}\left (\widetilde {\psi }\right )=\operatorname {\mathrm {trop}}\left (\widetilde {\psi }\otimes _{k''}k'''\right )$ . So it suffices to show that if $\widetilde {\psi }_1,\widetilde {\psi }_2\in \mathscr {L}(X_\sigma )(k'')$ are both lifts of $\psi $ , then $\operatorname {\mathrm {trop}}\left (\widetilde {\psi }_1\right )=\operatorname {\mathrm {trop}}\left (\widetilde {\psi }_2\right )$ . In this case, there exists such that $g\cdot \widetilde {\psi }_1=\widetilde {\psi }_2$ . Letting $g^*(a)$ denote the pullback of $a\in A$ under the map , we therefore have

$$ \begin{align*} g^*(\eta(f))\psi_1^*(f)=\psi_2^*(f). \end{align*} $$

Since $g^*(\eta (f))$ is a unit, the power series $\psi _1^*(f)$ and $\psi _2^*(f)$ have the same t-order of vanishing – that is, $\operatorname {\mathrm {trop}}\left (\widetilde {\psi }_1\right )=\operatorname {\mathrm {trop}}\left (\widetilde {\psi }_2\right )$ .

Proof. Let $\rho ^*\colon \sigma ^\vee \cap N^*\to \widetilde \sigma ^\vee \cap \widetilde {N}^*$ denote the pullback map on monoids. First note that if $k'$ is a field extension of k, $\psi \in \mathscr {L}\left (X_{\widetilde {\sigma }}\right )(k')$ and $p\in \sigma ^\vee \cap N^*$ , then

$$ \begin{align*} \operatorname{\mathrm{trop}}(\mathscr{L}(\rho)(\psi))(p)=\operatorname{\mathrm{ord}}_t(\psi^*\rho^*(p))=(\operatorname{\mathrm{trop}}(\psi))(\rho^*(p)). \end{align*} $$

To prove part (1), let $\operatorname {\mathrm {trop}}(\psi )=\widetilde {w}\in \widetilde \sigma ^\vee \cap \widetilde {N}^*$ . Then by these equalities, we see

$$ \begin{align*} \operatorname{\mathrm{trop}}(\mathscr{L}(\rho)(\psi))(p)=\langle\widetilde{w},\rho^*(p)\rangle=\langle\beta(\widetilde{w}),p\rangle, \end{align*} $$

so $\operatorname {\mathrm {trop}}(\mathscr {L}(\rho )(\psi ))=\beta (\widetilde {w})$ .

Part (2) follows immediately from part (1), provided we can show that $\operatorname {\mathrm {trop}}(\mathscr {L}(\rho )(\psi ))\in \sigma \cap N$ implies $\operatorname {\mathrm {trop}}(\psi )\in \widetilde \sigma \cap \widetilde {N}$ . Set $f\in \widetilde {\sigma }^\vee \cap \widetilde {N}^*$ . By hypothesis, there exists $f'\in \widetilde {\sigma }^\vee \cap \widetilde {N}^*$ such that $f+f'=\rho ^*(p)$ for some $p\sigma ^\vee \cap N^*$ . Then

$$ \begin{align*} (\operatorname{\mathrm{trop}}(\psi))(f)+(\operatorname{\mathrm{trop}}(\psi))(f')=(\operatorname{\mathrm{trop}}(\psi))(\rho^*(p))=\operatorname{\mathrm{trop}}(\mathscr{L}(\rho)(\psi))(p)\neq\infty. \end{align*} $$

So $(\operatorname {\mathrm {trop}}(\psi ))(f)\neq \infty $ for all f, and hence $\operatorname {\mathrm {trop}}(\psi )\in \widetilde \sigma \cap \widetilde {N}$ .

Proof. We keep the notation listed in Notation 2.19 and let $\mathcal {X}=\left [\widetilde {X}/G_\nu \right ]$ . We know there exist a finite field extension $k''$ of $k'$ and a lift $\widetilde {\psi }\in \mathscr {L}\left (\widetilde {X}\right )(k'')$ of $\psi $ . By construction, $\operatorname {\mathrm {trop}}(\psi )=\operatorname {\mathrm {trop}}\left (\widetilde {\psi }\right )$ . From Proposition 2.14, we know the hypotheses of Lemma 4.7(2) are satisfied, so $\operatorname {\mathrm {trop}}\left (\widetilde {\psi }\right )\in \beta ^{-1}(w)$ .

Proof. Let $\psi _1, \psi _2 \in \mathscr {L}(\mathcal {X})(k')$ correspond to G-torsors and G-equivariant maps $\gamma _1: Q_1 \to \widetilde {X}$ and $\gamma _2: Q_2 \to \widetilde {X}$ , respectively, and assume that $\mathscr {L}(\pi )(\psi _1) = \mathscr {L}(\pi )(\psi _2) = \varphi $ and $\operatorname {\mathrm {trop}}(\psi _1) = \operatorname {\mathrm {trop}}(\psi _2)$ . We need to prove that $\psi _1$ and $\psi _2$ are isomorphic – that is, that there exists an isomorphism of G-torsors $\alpha : Q_1 \to Q_2$ such that $\gamma _1 = \gamma _2 \circ \alpha $ . In fact, we prove the stronger statement that there exists a unique such $\alpha $ .

To prove this stronger statement, by descent, it is enough to show the existence of a unique such $\alpha $ étale locally on . By Lemma 4.3, the $Q_i$ are isomorphic to pullbacks of torsors over $k'$ , which can themselves be trivialised after base change to a finite field extension $k''$ of $k'$ . Since , after replacing $k'$ by $k''$ , we may therefore assume that the $Q_i$ are trivial G-torsors. Since $\operatorname {\mathrm {trop}}(\psi _i)$ depends only on the isomorphism class of $\psi _i$ , we may further assume .

Next, the identity section of the G-torsor $Q_i$ then yields a lift $\widetilde {\psi }_i\in \mathscr {L}\left (\widetilde {X}\right )(k')$ of $\psi _i$ . Then the map

satisfies

where $u^{\overline {f}} \in k[A]$ is the monomial indexed by the image $\overline {f}$ of f in A. Since $\operatorname {\mathrm {trop}}\left (\widetilde {\psi }_i\right )=\operatorname {\mathrm {trop}}(\psi _i)$ , we see $\operatorname {\mathrm {trop}}\left (\widetilde {\psi }_1\right )=\operatorname {\mathrm {trop}}\left (\widetilde {\psi }_2\right )\in \widetilde {\sigma } \cap \widetilde {N}$ . Thus, for all $f \in F$ , the power series $\widetilde {\psi }_1^*(f)$ and $\widetilde {\psi }_2^*(f)$ are nonzero and have the same t-order of vanishing. It follows that there is a unique unit

such that

$$ \begin{align*} \psi_1^*(f) = g^{(f)} \psi_2^*(f). \end{align*} $$

Since $\mathscr {L}(\pi )\left (\widetilde \psi _1\right ) = \mathscr {L}(\pi )\left (\widetilde \psi _2\right )$ , we see that $\widetilde \psi _1^*(p) = \widetilde \psi _2^*(p)$ , and so $g^{(p)} = 1$ for all $p \in P$ .

Thus the semigroup homomorphism induces a group homomorphism , which corresponds to an element and hence an automorphism $\alpha $ of the G-torsor . By construction, $\widetilde {\psi _1} = \widetilde {\psi _2} \circ \alpha $ , and so $\gamma _1 = \gamma _2 \circ \alpha $ . Moreover, the uniqueness of $\alpha $ follows from the uniqueness of each $g^{(f)}$ .

Proof. Recall that in Corollary 4.8, we proved that the image of $\overline {\left (\mathscr {L}(\pi )^{-1}(\varphi )\right )}(k')$ under $\operatorname {\mathrm {trop}}$ is contained in $\beta ^{-1}(w)$ .

Set $\widetilde {w} \in \beta ^{-1}(w)$ . We will construct an arc $\psi \in \mathscr {L}(\mathcal {X})(k')$ satisfying $\operatorname {\mathrm {trop}}(\psi ) = \widetilde {w}$ and $\mathscr {L}(\pi )(\psi ) = \varphi $ . Let be the generic point of $\varphi $ – that is, $\eta $ is the composition . Since $w \in \sigma \cap N$ , we see that $\eta $ factors through $T \hookrightarrow X$ . Thus $\eta $ is given by a group homomorphism .

Given our inclusion $M\hookrightarrow \widetilde {M}$ , we can choose a $\mathbb {Z}$ -basis $f_1, \dotsc , f_r$ for $\widetilde {M} = F^{\mathrm {gp}}$ and $m_1, \dotsc , m_d \in \mathbb {Z}_{>0}$ such that $m_1 f_1, \dotsc , m_d f_d$ is a $\mathbb {Z}$ -basis for M. Since $\operatorname {\mathrm {trop}}(\varphi ) = w$ , the image of $m_i f_i$ under has t-order of vanishing $\langle w, m_i f_i \rangle $ , so it can be written as $t^{\left \langle w, m_i f_i \right \rangle } g_i$ with .

Now set R to be the

-algebra

and give $\operatorname {\mathrm {Spec}}(R)$ the G-action obtained by letting $x_i$ have grading $\overline {f}_i$ , where $\overline {f}_i$ is the image of $f_i$ in A. Since each $g_i$ is a unit in

, we see that

is a G-torsor.

Let $\psi \in \mathscr {L}(\mathcal {X})(k')$ be the arc corresponding to the G-torsor $\operatorname {\mathrm {Spec}}(R)$ and the G-equivariant map $\operatorname {\mathrm {Spec}}(R) \to \widetilde {X} = \operatorname {\mathrm {Spec}}(k[F])$ defined by

$$ \begin{align*} F \to R: f \mapsto t^{\left\langle \widetilde{w}, f \right\rangle} \prod_{i=1}^r x_i^{c_i}, \end{align*} $$

for $f = \sum _{i=1}^r c_i f_i$ and $c_1, \dotsc , c_r \in \mathbb {Z}$ . Note that this map is well defined, since $\langle \widetilde {w}, f \rangle \geq 0$ when $f \in F$ , and each $x_i$ is a unit in R as a consequence of the fact that each $g_i$ is a unit in . Note further that since the $x_i$ are units, the map sends f to an element whose t-order of vanishing is $\langle \widetilde {w},f\rangle $ . As a result, any lift $\widetilde {\psi }\in \mathscr {L}\left (\widetilde {X}\right )(k'')$ obtained from a trivialisation of the G-torsor after base change to $k''$ , satisfies $\operatorname {\mathrm {trop}}(\psi )=\operatorname {\mathrm {trop}}\left (\widetilde {\psi }\right )=\widetilde {w}$ .

To finish the proof, we must show $\mathscr {L}(\pi )(\psi ) = \varphi $ – that is, $\pi \circ \psi =\varphi $ . Since X is separated, it is enough to prove equality after precomposing by the generic point . But this follows from the facts that $\eta =\varphi \circ \xi $ and that for any $i \in \{1, \dotsc , d\}$ , the image of $m_i f_i$ under the pullback of $\operatorname {\mathrm {Spec}}(R) \to \widetilde {X} \xrightarrow {\widetilde {\pi }} X$ is equal to

$$ \begin{align*} t^{\left\langle \widetilde{w}, m_i f_i \right\rangle} x_i^{m_i} = t^{\left\langle w, m_i f_i \right\rangle} g_i.\\[-39pt] \end{align*} $$