2.1 Logarithmic schemes

We give a minimal reminder of the basic definitions regarding log schemes and root stacks. See [Reference KatoKat89, Reference Abramovich, Chen, Gillam, Huang, Olsson, Satriano, Sun, Farkas and MorrisonACGHOSS13, Reference Borne and VistoliBV12, Reference Talpo and VistoliTV18] or the [Reference Carchedi, Scherotzke, Sibilla and TalpoCSST17, Appendix] for more details.

A log scheme is a scheme $X$ together with a sheaf of monoids $M$ on the small étale site, and a homomorphism of sheaves of monoids $\unicode[STIX]{x1D6FC}:M\rightarrow {\mathcal{O}}_{X}$ that restricts to an isomorphism $\unicode[STIX]{x1D6FC}^{-1}{\mathcal{O}}_{X}^{\times }\rightarrow {\mathcal{O}}_{X}^{\times }$ , where ${\mathcal{O}}_{X}$ is equipped with the multiplication. This definition also makes sense if $X$ is only a locally ringed space, using the classical site of $X$ instead of the small étale site. With mild assumptions, a log structure in the above sense can also be seen as a symmetric monoidal functor $L:\overline{M}:=M/{\mathcal{O}}^{\times }\rightarrow \operatorname{Div}_{X}$ with trivial kernel (i.e. the only section $a$ such that $L(a)$ is invertible is zero), where $\operatorname{Div}_{X}$ is the symmetric monoidal stack on $X$ of pairs $(L,s)$ consisting of a line bundle with a global section. For details about this reformulation we refer to [Reference Borne and VistoliBV12, § 3].

A morphism of log schemes $(X,M,\unicode[STIX]{x1D6FC})\rightarrow (Y,N,\unicode[STIX]{x1D6FD})$ is a morphism of schemes $f:X\rightarrow Y$ , together with a homomorphism of sheaves of monoids $f^{-1}N\rightarrow M$ compatible with $f^{-1}{\mathcal{O}}_{Y}\rightarrow {\mathcal{O}}_{X}$ . A strict morphism is a morphism as above such that the pullback of the log structure of $Y$ is isomorphic to the log structure of $X$ (via the map $f^{-1}N\rightarrow M$ ).

A chart for a log scheme $X$ is a homomorphism of monoids $P\rightarrow {\mathcal{O}}_{X}(X)$ that induces the homomorphism $\unicode[STIX]{x1D6FC}:M\rightarrow {\mathcal{O}}_{X}$ in the following sense: from the map $P\rightarrow {\mathcal{O}}_{X}(X)$ one obtains a homomorphism of sheaves of monoids $\widetilde{\unicode[STIX]{x1D6FC}}:\text{}\underline{P}\rightarrow {\mathcal{O}}_{X}$ where $\text{}\underline{P}$ is the constant sheaf with stalks $P$ . The induced homomorphism $\text{}\underline{P}\oplus _{\widetilde{\unicode[STIX]{x1D6FC}}^{-1}{\mathcal{O}}_{X}^{\times }}{\mathcal{O}}_{X}^{\times }\rightarrow {\mathcal{O}}_{X}$ is a log structure on $X$ , that we require to be isomorphic to $\unicode[STIX]{x1D6FC}:M\rightarrow {\mathcal{O}}_{X}$ (here $\oplus$ denotes the pushout in the category of sheaves of monoids). Equivalently, a chart is a strict morphism of log schemes $X\rightarrow \operatorname{Spec}k[P]$ .

A log scheme $X$ is quasi-coherent if locally it has charts $X\rightarrow \operatorname{Spec}k[P]$ . It is coherent (respectively integral, respectively fine, respectively fine and saturated, respectively locally free) if locally it has charts as before, where the monoid $P$ is finitely generated (respectively integral, respectively finitely generated and integral, respectively finitely generated, integral and saturated, respectively free). If a log scheme is quasi-coherent, the other properties can be checked on the stalks of $\overline{M}$ on geometric points of $X$ . A log scheme $X$ is Zariski if the log structure comes from the Zariski topos (see [Reference NiziołNiz06, § 2.1]).

In the sequel we will often consider also algebraic stacks equipped with a log structure. One can define a log structure on an algebraic stack $X$ exactly as for schemes, but by using the lisse-étale site instead. By descent for fine log structures, this turns out to be the same as a log structure on a presentation, together with descent data. All the log structures on schemes and stacks in this paper will be integral and quasi-coherent (and often fine and saturated). All fibered products of fine saturated log schemes and stacks will be taken in the fine saturated category.

2.2 Root stacks

The $n$ th root stack $\sqrt[n]{X}$ of a fine saturated log scheme (or algebraic stack) $X$ is the stack over schemes $/k$ that universally parametrizes extensions $(1/n)\overline{M}\rightarrow \operatorname{Div}_{X}$ of the symmetric monoidal functor $L:\overline{M}\rightarrow \operatorname{Div}_{X}$ . In characteristic $0$ it is a Deligne–Mumford stack, and the projection $\sqrt[n]{X}\rightarrow X$ is a coarse moduli space morphism. This construction adds to a point $x\rightarrow X$ a stabilizer $\unicode[STIX]{x1D707}_{n}^{r}$ , where $r$ is the rank of the free abelian group $\overline{M}_{x}^{\text{gp}}$ .

Quasi-coherent sheaves on the root stack $\sqrt[n]{X}$ correspond to quasi-coherent parabolic sheaves on $X$ with weights in $(1/n)\overline{M}^{\text{gp}}$  [Reference Borne and VistoliBV12, Theorem 6.1]. If $n\mid m$ we have a natural map $\sqrt[m]{X}\rightarrow \sqrt[n]{X}$ , and with these maps root stacks form an inverse system of Deligne–Mumford stacks, with index system the set $\mathbb{N}$ partially ordered by divisibility. If the log structure of $X$ is locally free, these maps are all flat.

The inverse limit $\sqrt[\infty ]{X}=:\mathop{\varprojlim }\nolimits_{n}\sqrt[n]{X}$ is called the infinite root stack of $X$ . It is a stack over schemes $/k$ with a fidèlement plat et quasi-compact (fpqc) presentation, but it is not algebraic. Nonetheless, there is a good theory of quasi-coherent sheaves on it, and they correspond to parabolic sheaves with arbitrary rational weights [Reference Talpo and VistoliTV18, Theorem 7.3]. The infinite root stack can be thought of as a purely algebro-geometric incarnation of a log structure itself: the association $X\mapsto \sqrt[\infty ]{X}$ gives a conservative faithful functor from fine saturated log schemes $/k$ to stacks $/k$  [Reference Talpo and VistoliTV18, § 5].

The formation of root stacks (finite or infinite) is compatible with strict base change.

2.3 Log blow-ups and valuativizations

The valuativization of a log scheme first appeared in the unpublished [Reference KatoKat] by Kato (see also [Reference IllusieIll94]), and it was further studied and used by Olsson [Reference OlssonOls01] and Pahnke [Reference PahnkePah05]. Nizioł [Reference NiziołNiz08a] uses a related construction of valuative sites for a log scheme, where one allows log blow-ups to be coverings (see Remark 2.20).

The valuativization $X^{\text{val}}\rightarrow X$ of a log locally ringed space $X$ (if it exists) is a valuative log locally ringed space equipped with a morphism to $X$ , such that if $Y$ is any valuative log locally ringed space, every log morphism $Y\rightarrow X$ factors uniquely as $Y\rightarrow X^{\text{val}}\rightarrow X$ (so $(-)^{\text{val}}$ would be the right adjoint of the inclusion functor from valuative log locally ringed spaces into log locally ringed spaces).

For a quasi-coherent log scheme $X$ , the valuativization exists, and can be constructed as the inverse limit $\mathop{\varprojlim }\nolimits_{{\mathcal{I}}}X_{{\mathcal{I}}}$ of all log blow-ups of $X$ , taken in the category of log locally ringed spaces. The result is rarely a scheme, but it still has some geometry that one can study, and relate to the properties of the log scheme $X$ . This construction is a logarithmic variant of Riemann–Zariski spaces.

In the next sections we will review the definition and basic properties of log blow-ups and valuativizations.

2.4 Valuativization of log stacks

In this paper it will be important to consider also the valuativization of some kinds of log stacks. To this purpose we will presently explain a variant of the process that we recalled above, that applies to stacks as well. For a log stack $X$ , in case log blow-ups make sense (for example if $X$ is algebraic, as we explain below), we will regard the inverse limit $\mathop{\varprojlim }\nolimits_{{\mathcal{I}}}X_{{\mathcal{I}}}$ , in a rather formal way, just as a fibered category with a log structure.

We will call a fibered category with a log structure a log fibered category. One can also define morphisms of log structures in the obvious way, as well as morphisms of log fibered categories (as a functor of fibered categories over schemes $/k$ , plus compatible morphisms of log structures for every scheme mapping to the source fibered category). If ${\mathcal{C}}$ is a scheme or an algebraic stack, this recovers the usual definition of log structure, and the usual properties. In those cases one can restrict the attention to the appropriate small sites (the small étale site for a scheme, the lisse-étale site for an algebraic stack), where everything becomes more manageable.

We can define the valuativization of any log fibered category ${\mathcal{C}}$ , by the universal mapping property with respect to morphisms from valuative log fibered categories.

Note that in the definition above, and in the rest of this section, we are using ‘unique’ to mean ‘unique in the $2$ -categorical sense’. Of course it is not clear why such a thing should exist in full generality. We will construct it when ${\mathcal{C}}$ is a fine log algebraic stack, or the infinite root stack of a fine and saturated log algebraic stack.

Let us check that if $X$ is a log scheme, this version of the valuativization is compatible with the locally ringed space of Proposition 2.8.

Proposition 2.16. Let $X$ be a quasi-coherent log scheme, and consider the valuativization $X^{\text{val}}$ as a ( $\operatorname{log}$ ) locally ringed space. We can consider the induced fibered category $F_{X^{\text{val}}}$ over schemes, by defining the category $F_{X^{\text{val}}}(S)$ to be the set of morphisms $S\rightarrow X^{\text{val}}$ as locally ringed spaces.

If we temporarily denote by $\mathop{\varprojlim }\nolimits_{{\mathcal{I}}}X_{{\mathcal{I}}}$ the valuativization as a fibered category, then there is a canonical isomorphism

$$\begin{eqnarray}\underset{{\mathcal{I}}}{\varprojlim }X_{{\mathcal{I}}}\simeq F_{X^{\text{val}}}\end{eqnarray}$$

of log fibered categories over schemes $/k$ .

2.5 Toric stacks

We recall the basic definitions of stacky fans and toric stacks from [Reference Geraschenko and SatrianoGS15]. Recall that if $N$ is a lattice and $\unicode[STIX]{x1D6F4}$ is a rational polyhedral fan in the vector space $N_{\mathbb{R}}=N\otimes _{\mathbb{Z}}\mathbb{R}$ , there is an associated normal toric variety $X_{\unicode[STIX]{x1D6F4}}$ .

To a stacky fan one can associate a toric stack as follows: the homomorphism of lattices $f$ gives a homomorphism of tori $D(f^{\vee }):D(N^{\vee })\rightarrow D((N^{\prime })^{\vee })$ , where we are denoting by $N^{\vee }$ the lattice $\operatorname{Hom}(N,\mathbb{Z})$ , and by $D(-)$ the Cartier dual $\operatorname{Hom}(-,\mathbb{G}_{m})$ . Set $G:=\ker D(f^{\vee })$ . Then, since $G$ is a subgroup of the torus $D(N^{\vee })$ of the toric variety $X_{\unicode[STIX]{x1D6F4}}$ , there is a natural action of $G$ on  $X_{\unicode[STIX]{x1D6F4}}$ .

Toric stacks and stacky fans will be used in the main text to describe root stacks and log blow-ups of some local models for our constructions. We end this subsection with a few examples that can be useful to keep in mind.

2.6 Dg-categories and categories of sheaves

Throughout the paper a dg-category will be a $k$ -linear differential graded category. We refer the reader to [Reference KellerKel07, Reference DrinfeldDri04] for additional information on dg-categories and their basic properties. All limits and colimits of diagrams of dg-categories are to be understood as homotopy limits or colimits in the category of dg-categories, equipped with its Morita model structure of [Reference TabuadaTab05, Reference ToënToë07]. Most dg-categories appearing in this paper will be triangulated, see [Reference ToënToë11, § 4.4] for a discussion of this notion. Let ${\mathcal{C}}$ be a dg-category and let $A$ and $B$ be objects of ${\mathcal{C}}$ . We denote $\operatorname{Hom}_{{\mathcal{C}}}(A,B)$ the complex of morphisms between $A$ and $B.$ We will also make use of the concept of a t-structure of a triangulated dg-category. By definition, a triangulated dg-category ${\mathcal{C}}$ has a t-structure if its homotopy category $Ho({\mathcal{C}})$ has a t-structure in the ordinary sense. A reference for the theory of t-structures of stable $(\infty ,1)$ -categories is [Reference LurieLur16, § 1.2.1]. The discussion there applies without variations to the case of triangulated dg-categories. Given an abelian category $A$ , we denote the canonical dg enhancement of its bounded derived category by $D_{dg}^{b}(A)$ .

In the richer homotopical context of dg-categories, quasi-coherent sheaves can be defined in complete generality for arbitrary (higher and derived) stacks, and they enjoy good descent properties. A discussion of these aspects can be found in the introduction of [Reference Ben-Zvi, Francis and NadlerBFN10]. If $X$ is a stack, we denote by $\operatorname{Qcoh}(X)$ the triangulated dg-category of quasi-coherent sheaves on $X$ , and by $\operatorname{Perf}(X)$ its full subcategory of perfect complexes. The tensor product of quasi-coherent sheaves equips $\operatorname{Qcoh}(X)$ with a symmetric monoidal structure, and $\operatorname{Perf}(X)$ can be defined as the full subcategory of dualizable objects. Passing to the homotopy categories we recover the ordinary unbounded derived category of quasi-coherent sheaves, and the triangulated category of perfect complexes on $X$ .

By [Reference GaitsgoryGai11, Theorem 1.3.4], the dg-category of quasi-coherent sheaves $\operatorname{Qcoh}(-)$ satisfies faithfully flat descent. Namely, let $Y\rightarrow X$ be a faithfully flat cover, and let $Y^{\bullet }$ be the simplicial object given by the Čhech nerve of $Y\rightarrow X$ . Then we have

$$\begin{eqnarray}\varprojlim \operatorname{Qcoh}(Y^{\bullet })\simeq \operatorname{Qcoh}(X),\end{eqnarray}$$

where the (homotopy) limit on the left is computed in the category of symmetric monoidal dg-categories. It is easy to see that dualizable objects in the limit are the same as the limit of dualizable objects. Thus, passing to dualizable objects on both sides, we obtain

$$\begin{eqnarray}\varprojlim \operatorname{Perf}(Y^{\bullet })\simeq \operatorname{Perf}(X).\end{eqnarray}$$

As a consequence $\operatorname{Perf}(-)$ also satisfies faithfully flat descent. This will be an important technical ingredient in our arguments.

Quasi-coherent sheaves on pro-stacks. We will work with pro-objects in stacks. We make the following formal definition of the category of perfect complexes on a pro-stack.

Definition 2.26. Let $\{X_{i}\}_{i\in I}$ be a pro-object in stacks. We will formally define

$$\begin{eqnarray}\operatorname{Perf}\Bigl(\underset{i}{\varprojlim }X_{i}\Bigr):=\underset{i}{\varinjlim }\operatorname{Perf}(X_{i})\end{eqnarray}$$

as a dg-category.

We do not claim that Definition 2.26 is a reasonable definition of the category of perfect complexes over an arbitrary pro-object in stacks. However, in the concrete case of the kind of pro-stacks that we will be interested in, this definition can be justified. We devote the rest of this section to a discussion of this aspect.

We will work with two kinds of inverse systems, given by infinite root stacks and valuativizations, which were introduced in the previous section. Infinite root stacks are inverse limits of the inverse system $\{\!\sqrt[n]{X}\}_{n\in \mathbb{N}}$ of finite root stacks of the log scheme or stack $X.$ These stacks have groupoid presentations $R_{n}\rightrightarrows U_{n}\rightarrow \sqrt[n]{X}$ that fit very nicely together (the projections lift to affine morphisms $R_{m}\rightarrow R_{n}$ and $U_{m}\rightarrow U_{n}$ ). Because of this we have a concrete description of quasi-coherent and coherent sheaves on $\sqrt[\infty ]{X}$ (see [Reference Talpo and VistoliTV18, § 4]), and thus also of the category of perfect complexes $\operatorname{Perf}(\!\sqrt[\infty ]{X})$ . We will check below (Proposition 2.30) that, under some assumptions, this gives the same dg-category as the direct limit of the categories of perfect complexes $\operatorname{Perf}(\!\sqrt[n]{X})$ of the finite root stacks. Thus, in the case of the infinite root stack, Definition 2.26 gives a reasonable notion, since it captures perfect complexes on the actual limit of the inverse system.

The structure maps of the second inverse system, which computes the valuativization, are blow-ups. Thus they are very far from being affine as required by Remark 2.27. In this case it is maybe less clear that our definition for the category of perfect complexes is reasonable. We do not explore this question any further, but we point out the following related result. If $X$ is a log regular quasi-compact log scheme $X$ , then Nizioł [Reference NiziołNiz08a, Corollary 3.9] proves that the direct limit of the category of locally free sheaves on the Kummer-flat site of log blow-ups of $X$ is equivalent to the category of locally free sheaves on the valuative Kummer-flat site of $X$ . Thus, at least under these assumptions and in the ‘Kummer-flat’ context, vector bundles on the valuativization are modeled by the colimit of the categories of vector bundles on log blow-ups.

To conclude this section we will check that, as anticipated, under reasonable assumptions we have an equivalence

$$\begin{eqnarray}\operatorname{Perf}(\!\sqrt[\infty ]{X})\simeq \underset{n}{\varinjlim }\operatorname{Perf}(\!\sqrt[n]{X}).\end{eqnarray}$$

We start by proving a general result on colimits of pretriangulated dg-categories equipped with a t-structure.

Theorem 2.28. Let $I$ be a filtered category, and let $\{{\mathcal{C}}_{i}\}_{i\in I}$ be a filtered system of dg-categories. Assume that all the structure maps

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{i\rightarrow j}:{\mathcal{C}}_{i}\rightarrow {\mathcal{C}}_{j}\end{eqnarray}$$

are fully faithful. Then the colimit ${\mathcal{C}}:=\varinjlim {\mathcal{C}}_{i}$ is the following dg-category.

• ∙ The objects of ${\mathcal{C}}$ are given by the union $\bigcup _{i\in I}\operatorname{Ob}({\mathcal{C}}_{i})$ .

• ∙ The Hom complex between $A_{i}\in {\mathcal{C}}_{i}$ and $A_{j}\in {\mathcal{C}}_{j}$ is given by

  $$\begin{eqnarray}\operatorname{Hom}_{{\mathcal{C}}}(A_{i},A_{j})=\operatorname{Hom}_{{\mathcal{C}}_{l}}(\unicode[STIX]{x1D6FC}_{l\rightarrow i}(A_{i}),\unicode[STIX]{x1D6FC}_{l\rightarrow j}(A_{j})),\end{eqnarray}$$
  where $l$ is any object of $I$ that is the source of morphisms $j\leftarrow l\rightarrow i$ .

We need a refinement of this result to the case where the categories ${\mathcal{C}}_{i}$ are the dg enhancements of bounded derived categories, and the structure maps are exact. Notations and assumptions in the next lemma are carried over from Theorem 2.28.

Lemma 2.29. Assume that ${\mathcal{C}}_{i}=D_{dg}^{b}(A_{i})$ , where $\{A_{i}\}_{i\in I}$ is a filtered system of abelian categories. Assume also that the structure maps preserve the abelian categories, that is, for all $i\rightarrow j$ in $I$ , $\unicode[STIX]{x1D6FC}_{i\rightarrow j}(A_{i})\subset A_{j}$ . Then there is an equivalence

$$\begin{eqnarray}\underset{i}{\varinjlim }{\mathcal{C}}_{i}=\underset{i}{\varinjlim }D_{dg}^{b}(A_{i})\simeq D_{dg}^{b}\Bigl(\underset{i}{\varinjlim }A_{i}\Bigr).\end{eqnarray}$$

Proposition 2.30. Let $X$ be a noetherian fine saturated log algebraic stack over $k$ , with locally free log structure. Then there is an equivalence of dg-categories

$$\begin{eqnarray}\underset{n}{\varinjlim }\operatorname{Perf}(\!\sqrt[n]{X})\simeq \operatorname{Perf}(\!\sqrt[\infty ]{X}).\end{eqnarray}$$

Proof. Applying Lemma 2.29, we obtain that the colimit $\mathop{\varinjlim }\nolimits_{n}D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[n]{X}))$ is equivalent to $D_{dg}^{b}(\mathop{\varinjlim }\nolimits_{n}\operatorname{Coh}(\!\sqrt[n]{X}))$ . Since $X$ is noetherian and the log structure is locally free, we can apply [Reference Talpo and VistoliTV18, Proposition 6.1] (together with [Reference Talpo and VistoliTV18, Proposition 4.19]), and conclude that the direct limit $\mathop{\varinjlim }\nolimits_{n}\operatorname{Coh}(\!\sqrt[n]{X})$ is equivalent to the abelian category $\operatorname{Coh}(\!\sqrt[\infty ]{X})$ . This implies that

$$\begin{eqnarray}\underset{n}{\varinjlim }D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[n]{X}))\simeq D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[\infty ]{X})).\end{eqnarray}$$

Note that there is an inclusion $\operatorname{Perf}(\!\sqrt[\infty ]{X})\subseteq D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[\infty ]{X}))$ . Indeed, by [Reference Talpo and VistoliTV18, Proposition 4.19], the structure sheaf ${\mathcal{O}}_{\sqrt[\infty ]{X}}$ is coherent. Thus, using [Reference GaitsgoryGai12, § 1.5], $\sqrt[\infty ]{X}$ is eventually coconnective and, by the proof of [Reference GaitsgoryGai12, Proposition 1.5.3], this implies that $\operatorname{Perf}(\sqrt[\infty ]{X})\subseteq D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[\infty ]{X}))$ . Further $\operatorname{Perf}(\!\sqrt[\infty ]{X})$ can be characterized as the full subcategory of dualizable objects inside $D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[\infty ]{X}))$ . In symbols, we write

$$\begin{eqnarray}\operatorname{Perf}(\!\sqrt[\infty ]{X})=(D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[\infty ]{X})))^{\text{dual}}.\end{eqnarray}$$

From the explicit description of $\mathop{\varinjlim }\nolimits_{n}D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[n]{X}))$ given by Theorem 2.28, and the fact that the structure maps are monoidal, we conclude that the dualizable objects in $D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[\infty ]{X}))$ are given by the image of the dualizable objects in the categories $D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[n]{X}))$ . In formulas

$$\begin{eqnarray}\underset{n}{\varinjlim }\operatorname{Perf}(\!\sqrt[n]{X})=\underset{n}{\varinjlim }(D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[n]{X})))^{\text{dual}}\simeq (D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[\infty ]{X})))^{\text{dual}}\simeq \operatorname{Perf}(\!\sqrt[\infty ]{X}),\end{eqnarray}$$

which is what we wanted to prove. ◻

Note that by [Reference Talpo and VistoliTV18, Theorem 7.3] the category of quasi-coherent sheaves on $\sqrt[\infty ]{X}$ is equivalent to the category $\operatorname{Par}(X)$ of parabolic sheaves on $X$ with rational weights. Therefore we can also identify $D_{dg}^{b}(\operatorname{Coh}(\!\sqrt[\infty ]{X}))$ with the (dg enhancement of the) bounded derived category of the category of coherent parabolic sheaves with rational weights on $X$ .

4.1 Root stacks

Let us start by describing explicitly the main objects appearing in our constructions in the local case, on the root stack side.

• ∙ The root stacks of the base $\mathbb{A}^{1}$ are given by quotient stacks

  $$\begin{eqnarray}\sqrt[n]{\mathbb{A}^{1}}=[\operatorname{Spec}k[t_{n}]/\unicode[STIX]{x1D707}_{n}],\end{eqnarray}$$
  where $\operatorname{Spec}k[t_{n}]\rightarrow \sqrt[n]{\mathbb{A}^{1}}\rightarrow \mathbb{A}^{1}=\operatorname{Spec}k[t]$ corresponds to $t\mapsto t_{n}^{n}$ (so that morally $t_{n}\sim t^{1/n}$ ), and the action of $\unicode[STIX]{x1D707}_{n}$ is the obvious one.

• ∙ The root stack $\mathbb{A}_{n}^{r+1}=\sqrt[n]{(\mathbb{A}^{r+1})^{\prime }}$ of $\mathbb{A}^{r+1}$ with respect to the log structure pulled back from $\mathbb{A}^{1}$ is isomorphic to the fibered product $\mathbb{A}^{r+1}\times _{\mathbb{A}^{1}}\sqrt[n]{\mathbb{A}^{1}}$ , for which we have

  $$\begin{eqnarray}\mathbb{A}_{n}^{r+1}=\sqrt[n]{\mathbb{A}^{1}}\times _{\mathbb{ A}^{1}}\mathbb{A}^{r+1}\simeq [\operatorname{Spec}k[x_{i},t_{n}]/(x_{1}\cdots x_{r+1}-t_{n}^{n})/\unicode[STIX]{x1D707}_{n}]\end{eqnarray}$$
  where $\unicode[STIX]{x1D707}_{n}$ acts on $t_{n}$ as for $\sqrt[n]{\mathbb{A}^{1}}$ .

• ∙ The root stack $\sqrt[n]{\mathbb{A}^{r+1}}$ can be described as the quotient

  $$\begin{eqnarray}\sqrt[n]{\mathbb{A}^{r+1}}=[\operatorname{Spec}k[x_{i,n}]/\unicode[STIX]{x1D707}_{n}^{r+1}]\end{eqnarray}$$
  for new variables $x_{1,n},\ldots ,x_{r+1,n}$ , where $\unicode[STIX]{x1D707}_{n}^{r+1}$ acts diagonally, in the obvious way, and the composite $\operatorname{Spec}k[x_{i,n}]\rightarrow \sqrt[n]{\mathbb{A}^{r+1}}\rightarrow \mathbb{A}^{r+1}=\operatorname{Spec}k[x_{i}]$ is given algebraically by $x_{i}\mapsto x_{i,n}^{n}$ (so that morally $x_{i,n}\sim x_{i}^{1/n}$ ).

Remark 4.1. Note that everything in sight is either a toric variety or a toric stack.

The toric stack $\mathbb{A}_{n}^{r+1}$ can be described via another global quotient. The action of $\unicode[STIX]{x1D707}_{n}^{r+1}$ on $\mathbb{A}^{r+1}=\operatorname{Spec}k[x_{i,n}]$ induces an action of the kernel of the morphism

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{n}^{r+1}\rightarrow \unicode[STIX]{x1D707}_{n},\quad (\unicode[STIX]{x1D709}_{1},\ldots ,\unicode[STIX]{x1D709}_{r+1})\mapsto \unicode[STIX]{x1D709}_{1}\cdot \ldots \cdot \unicode[STIX]{x1D709}_{r+1},\end{eqnarray}$$

which we identify with $\unicode[STIX]{x1D707}_{n}^{r}$ via the map $(\unicode[STIX]{x1D702}_{1},\ldots ,\unicode[STIX]{x1D702}_{r})\mapsto (\unicode[STIX]{x1D702}_{1},\ldots ,\unicode[STIX]{x1D702}_{r},(\unicode[STIX]{x1D702}_{1}\cdots \unicode[STIX]{x1D702}_{r})^{-1})$ .

The quotient $\mathbb{A}^{r+1}/\unicode[STIX]{x1D707}_{n}^{r}$ has an action of the target $\unicode[STIX]{x1D707}_{n}$ , and we have a canonical isomorphism $\mathbb{A}_{n}^{r+1}\simeq [(\mathbb{A}^{r+1}/\unicode[STIX]{x1D707}_{n}^{r})\,/\,\unicode[STIX]{x1D707}_{n}]$ . In fact the quotient $\mathbb{A}^{r+1}/\unicode[STIX]{x1D707}_{n}^{r}$ is exactly $\operatorname{Spec}k[x_{i},t_{n}]/(x_{1}\cdots x_{r+1}-t_{n}^{n})$ .

The theory of stacky fans (recalled in § 2.5) allows to encode the difference between the toric variety $\mathbb{A}^{r+1}/\unicode[STIX]{x1D707}_{n}^{r}$ and the toric stack $\mathbb{A}_{n}^{r+1}$ in a combinatorial way. Let ${\mathcal{F}}_{n}$ be the fan of $\mathbb{A}^{r+1}/\unicode[STIX]{x1D707}_{n}^{r}$ in a lattice $L_{n}$ (see below for more details about these). Then the stacky quotient $\mathbb{A}_{n}^{r+1}$ is described by the same fan in the same lattice, with the difference that we are also keeping track of an inclusion of lattices $L_{n}\subseteq \mathbb{Z}^{r+1}$ (denoted by $N\rightarrow N^{\prime }$ in § 2.5). The kernel of the morphism between the Cartier duals $\mathbb{G}_{m}^{r+1}=D(L_{n}^{\vee })\rightarrow D((\mathbb{Z}^{r+1})^{\vee })=\mathbb{G}_{m}^{r+1}$ is exactly the copy of $\unicode[STIX]{x1D707}_{n}$ appearing in the quotient stack description of $\mathbb{A}_{n}^{r+1}$ .

4.2 Log blow-ups

Next we turn to the other side of our equivalence, which is given by the valuativization. Let us first spend some words on log blow-ups and the valuativization of $X=\mathbb{A}^{r+1}$ itself. As for any toric variety, the valuativization $(\mathbb{A}^{r+1})^{\text{val}}$ is the inverse limit of all toric blow-ups (see Remark 2.5). These are conveniently described by subdivisions of the fan ${\mathcal{F}}$ associated to the toric variety, that in this case is the cone given by the first orthant ${\mathcal{F}}=\mathbb{R}_{{\geqslant}0}^{r+1}$ of the vector space $\mathbb{R}^{r+1}$ , with lattice $\mathbb{Z}^{r+1}$ . So the valuativization is described as the inverse limit of the schemes that we get by subdividing the fan ${\mathcal{F}}$ further and further.

Rather than in $\mathbb{A}^{r+1}$ itself, we will be mostly interested in the toric stacks obtained from $\mathbb{A}^{r+1}$ by extracting roots on the base $\mathbb{A}^{1}$ . Pulling back the degeneration $\mathbb{A}^{r+1}\rightarrow \mathbb{A}^{1}$ along the $n$ th root stack morphism $\sqrt[n]{\mathbb{A}^{1}}\rightarrow \mathbb{A}^{1}$ , has the effect of replacing the lattice $\mathbb{Z}^{r+1}$ by the inclusion $L_{n}\subseteq \mathbb{Z}^{r+1}$ (the resulting object will be a toric stack), where $L_{n}$ is the lattice of vectors $(a_{1},\ldots ,a_{r+1})$ such that $a_{1}+\cdots +a_{r+1}\equiv 0\;(\text{mod}\;n)$ . This operation does not change the fan. Let us denote by ${\mathcal{F}}_{n}$ the fan given by the first orthant in $\mathbb{R}^{r+1}$ , but with lattice the subgroup $L_{n}$ described above. Because of this description, subdivisions of the original fan ${\mathcal{F}}$ will induce subdivisions of the fan  ${\mathcal{F}}_{n}$ .

For our purposes it will be important to consider certain specific kinds of subdivisions of the fan ${\mathcal{F}}_{n},$ which we describe next. Consider the hyperplane given by the equation $x_{1}+\cdots +x_{r+1}=n$ in $\mathbb{R}^{r+1}$ , and the standard simplex that the first orthant determines on it. With respect to the lattice $L_{n}$ , this simplex is isomorphic to the standard simplex $n\unicode[STIX]{x1D6E5}^{r}$ of dimension $r$ , but whose sides have $n+1$ lattice points each. We can subdivide the fan ${\mathcal{F}}_{n}$ by choosing a unimodular (i.e. such that every lattice point is contained in a face, or equivalently, every lattice point in the interior of $n\unicode[STIX]{x1D6E5}^{r}$ is a vertex) triangulation ${\mathcal{T}}$ of the simplex $n\unicode[STIX]{x1D6E5}^{r}$ , and construct the induced fan in the space $\mathbb{R}^{r+1}$ with lattice $L_{n}$ , by taking the cones over the simplices in the triangulation ${\mathcal{T}}$ .

Denote by $(\mathbb{A}_{n}^{r+1})_{{\mathcal{T}}}$ the toric blow-up of $\mathbb{A}_{n}^{r+1}$ corresponding to this fan (we trust that there will be no confusion with the notation for a log blow-up along a sheaf of ideals of the log structure). Note that the stack $\mathbb{A}_{n}^{r+1}\simeq [(\mathbb{A}^{r+1}/\unicode[STIX]{x1D707}_{n}^{r})\,/\,\unicode[STIX]{x1D707}_{n}]$ is a global quotient of a toric variety $(\mathbb{A}^{r+1}/\unicode[STIX]{x1D707}_{n}^{r})$ (as mentioned in Remark 4.1), whose fan coincides with ${\mathcal{F}}_{n}$ , and this gives a canonical way to define the log blow-up on an atlas, as described in (2.3).

By extracting further roots and subdividing further, we obtain a direct system of triangulations. More precisely, if $m\in \mathbb{N}$ and ${\mathcal{T}}$ is a unimodular triangulation of $n\unicode[STIX]{x1D6E5}^{r}$ , then ${\mathcal{T}}$ will induce a (non-unimodular) triangulation of the simplex $(mn)\unicode[STIX]{x1D6E5}^{r}$ , that we can refine further to a unimodular one in several possible ways. If ${\mathcal{T}}^{\prime }$ is such a refinement, we write $(mn,{\mathcal{T}}^{\prime })\geqslant (n,{\mathcal{T}})$ , and we will simply say that ${\mathcal{T}}^{\prime }$ is a refinement of ${\mathcal{T}}$ (see also Remark 4.5 below). In this way the set of pairs $\{(n,{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}})\}$ where $n\in \mathbb{N}$ and ${\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}}$ is a unimodular triangulation of $n\unicode[STIX]{x1D6E5}^{r}$ becomes a filtered partially ordered set.

To each $(n,{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}})$ we can associate the log algebraic stack $(\mathbb{A}_{n}^{r+1})_{{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}}}$ , and if $(mn,{\mathcal{T}}_{mn}^{\unicode[STIX]{x1D6FD}})\geqslant (n,{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}})$ we have an induced morphism $(\mathbb{A}_{mn}^{r+1})_{{\mathcal{T}}_{mn}^{\unicode[STIX]{x1D6FD}}}\rightarrow (\mathbb{A}_{n}^{r+1})_{{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}}}$ . Thus we have an inverse system of log algebraic stacks indexed by the pairs $(n,{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}})$ .

Let us consider the inverse limit of the system $\{(\mathbb{A}_{n}^{r+1})_{{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}}}\}_{(n,{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}})}$ .

Proposition 4.6. There is a canonical isomorphism of log fibered categories

$$\begin{eqnarray}(\mathbb{A}_{\infty }^{r+1})^{\text{val}}\simeq \underset{(n,{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}})}{\varprojlim }(\mathbb{A}_{n}^{r+1})_{{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}}}.\end{eqnarray}$$

Corollary 4.7. The natural maps $(\mathbb{A}_{\infty }^{r+1})^{\text{val}}\rightarrow (\mathbb{A}_{n}^{r+1})_{{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}}}$ induce an equivalence of dg-categories

$$\begin{eqnarray}\underset{(n,{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}})}{\varinjlim }\operatorname{Perf}((\mathbb{A}_{n}^{r+1})_{{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}}})\simeq \operatorname{Perf}((\mathbb{A}_{\infty }^{r+1})^{\text{val}})\end{eqnarray}$$

and the same holds for any cofinal subsystem of the family $\{(\mathbb{A}_{n}^{r+1})_{{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}}}\}_{(n,{\mathcal{T}}_{n}^{\unicode[STIX]{x1D6FC}})}$ .

4.3 Compatible cofinal systems of triangulations

The statement of Theorem 3.8 in the local case is that there is an equivalence of dg-categories $\operatorname{Perf}((\mathbb{A}_{\infty }^{r+1})^{\text{val}})\rightarrow \operatorname{Perf}(\!\sqrt[\infty ]{\mathbb{A}^{r+1}}),$ which is obtained as a limit of Fourier–Mukai functors induced by a push-pull formalism, see § 3. We will deduce this by proving that we can choose a sequence $\{k_{n}\}_{n\in \mathbb{N}}$ of positive integers and, for every $n,$ a unimodular triangulation ${\mathcal{T}}_{k_{n}}$ of the standard simplex $k_{n}\unicode[STIX]{x1D6E5}^{r}$ such that we have both the following.

• ∙ The sequence $\{k_{n}\}_{n\in \mathbb{N}}$ is cofinal in $\mathbb{N}$ ordered by divisibility, so that .

• ∙ The triangulations ${\mathcal{T}}_{k_{n}}$ are cofinal in the system of all rational triangulations of $\unicode[STIX]{x1D6E5}^{r}$ (in the sense of Remark 4.5), so that $\mathop{\varprojlim }\nolimits_{n}(\mathbb{A}_{k_{n}}^{r+1})_{{\mathcal{T}}_{k_{n}}}\simeq (\mathbb{A}_{\infty }^{r+1})^{\text{val}}$ .

We will also check (Proposition 4.12) that for every $n$ and every unimodular triangulation ${\mathcal{T}}$ of the simplex $n\unicode[STIX]{x1D6E5}^{r}$ , the Fourier–Mukai functor $\operatorname{Perf}((\mathbb{A}_{n}^{r+1})_{{\mathcal{T}}})\rightarrow \operatorname{Perf}(\!\sqrt[n]{\mathbb{A}^{r+1}})$ given by push-pull along the diagram