Parabolic semi-orthogonal decompositions and Kummer flat invariants of log schemes

TL;DR

This paper develops semi-orthogonal decompositions for categories of parabolic sheaves on log schemes, generalizing Kummer flat K-theory results to broader invariants and stacks.

Contribution

It introduces a new categorical framework for decomposing parabolic sheaves on log schemes, extending existing K-theory results to more invariants and stacks.

Findings

Constructed semi-orthogonal decompositions for parabolic sheaves.

Categorified Kummer flat K-theory decomposition theorems.

Extended results to a larger class of invariants and stacks.

Abstract

We construct semi-orthogonal decompositions on triangulated categories of parabolic sheaves on certain kinds of logarithmic schemes. This provides a categorification of the decomposition theorems in Kummer flat K-theory due to Hagihara and Nizio{\l}. Our techniques allow us to generalize Hagihara and Nizio{\l}'s results to a much larger class of invariants in addition to K-theory, and also to extend them to more general logarithmic stacks.

Equations

$$\mathrm{Perf}(\sqrt[\infty ]{(X,D)})=⟨{\mathcal{A}}_{\overline{S}},\overline{S}\in S_D⟩$$

$$\mathcal{U}:{\mathrm{Cat}}_{\infty }^{\mathrm{perf}}⟶{\mathrm{Mot}}^{\mathrm{add}}.$$

$${\mathrm{Cat}}_{\infty }^{\mathrm{perf}}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathcal{U}(\sqrt[\infty]{(X,D)})\simeq\bigoplus_{\overline{S}\in S_{D}}\Big{(}\bigoplus_{(\mathbb{Q}/\mathbb{Z})^{|\overline{S}|,*}}\mathcal{U}(\overline{S})\Big{)}.$$

$$\mathrm{H}_{\mathrm{Kfl}}(X,D)\simeq\bigoplus_{\overline{S}\in S_{D}}\Big{(}\bigoplus_{(\mathbb{Q}/\mathbb{Z})^{|\overline{S}|,*}}\mathrm{H}(\overline{S})\Big{)}.$$

$$K_{\mathrm{Kfl}}(X,D)\simeq\bigoplus_{\overline{S}\in S_{D}}\Big{(}\bigoplus_{(\mathbb{Q}/\mathbb{Z})^{|\overline{S}|,*}}K(\overline{S})\Big{)}.$$

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$$D_r\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\sqrt[r]{(X,D)}\simeq [\mathrm{Spec}(A[x]/(x^r-f))/{\mu }_r]$$

$$\sqrt[\vec{r}]{(X,D)}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\sqrt[r_1]{(X,D_1)}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\sqrt[\vec{r}]{(X,D)}\simeq [\mathrm{Spec}A[x_1,\dots ,x_N]/(f_1-x_1^{r_1},\dots ,f_N-x_N^{r_n})/\prod _i{\mu }_{r_i}]$$

$$A=M_D/{\mathcal{O}}_X\to [{\mathcal{O}}_X/{\mathcal{O}}_X^{\times }]=[{\mathbb{A}}^1/{\mathbb{G}}_m{]}_X={\mathrm{Div}}_X.$$

$$\sqrt[r^{\prime }]{(X,D)}\simeq \sqrt[\frac{r^{\prime }}{r}]{(\sqrt[r]{(X,D)},D_r)}\to \sqrt[r]{(X,D)}.$$

$${\mathcal{C}}_1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Hom}}_{\mathcal{C}}(A_i,A_j)={\mathrm{Hom}}_{{\mathcal{C}}_l}({\alpha }_{l\to i}(A_i),{\alpha }_{l\to j}(A_j))$$

$$\mathrm{Qcoh}(X):=\underset{\mathrm{Spec}(A)\to X}{\underleftarrow{\lim }}\mathrm{Qcoh}(\mathrm{Spec}(A))$$

$$\mathrm{Qcoh}(X{)}^{+}\simeq \mathrm{D}(\mathrm{qcoh}(X){)}^{+}$$

$$\mathrm{Perf}(\sqrt[\infty ]{X})\simeq \underset{r}{\underrightarrow{\lim }}\mathrm{Perf}(\sqrt[r]{X}).$$

$$\mathrm{Perf}(X_{\mathrm{Kfl}})\subseteq \mathrm{Coh}(X_{\mathrm{Kfl}}),\mathrm{Perf}(\sqrt[\infty ]{X})\subseteq \mathrm{Coh}(\sqrt[\infty ]{X}).$$

$$\mathrm{Perf}(X_{\mathrm{Kfl}})\simeq \mathrm{Coh}(X_{\mathrm{Kfl}}{)}^{\mathrm{dual}},\mathrm{Perf}(\sqrt[\infty ]{X})\simeq \mathrm{Coh}(\sqrt[\infty ]{X}{)}^{\mathrm{dual}}.$$

$$\mathrm{Perf}(X_{\mathrm{Kfl}})\simeq \mathrm{Coh}(X_{\mathrm{Kfl}}{)}^{\mathrm{dual}}\simeq \mathrm{Coh}(\sqrt[\infty ]{X}{)}^{\mathrm{dual}}\simeq \mathrm{Perf}(\sqrt[\infty ]{X}).$$

$$\mathcal{A}\stackrel{F}{⟶}\mathcal{B}\stackrel{G}{⟶}\mathcal{C}$$

$$\mathrm{Ho}(\mathcal{A})\stackrel{\mathrm{Ho}(F)}{\to }\mathrm{Ho}(\mathcal{B})\stackrel{\mathrm{Ho}(G)}{\to }\mathrm{Ho}(\mathcal{C})$$

$$r_x:\mathcal{C}⟶{\mathcal{C}}_x\text{and}{l}_x:\mathcal{C}⟶{\mathcal{C}}_x.$$

$$\mathrm{Ho}(F):\mathrm{Ho}(\mathcal{A})\leftrightarrows \mathrm{Ho}(\mathcal{B})plus1.0mu\text{nonscript}:\mathrm{Ho}(G)$$

$$P\ast Q:=(P\coprod Q,{\le }_{P\ast Q})$$

$$\mathrm{ch}:K(-)\Rightarrow \mathrm{HH}(-):{\mathrm{Cat}}_{\infty ,\kappa }^{\mathrm{perf}}⟶{\mathcal{S}}_{\infty }$$

$${\mathrm{ch}}_{\mathrm{dR}}:K_0(X)\stackrel{\mathrm{ch}}{⟶}{\mathrm{HH}}_0(\mathrm{Perf}(X))\stackrel{\cong }{⟶}\underset{k\ge 0}{⨁}{\mathrm{H}}_{\mathrm{dR}}^{2k}(X),$$

Full text

1. 1 Introduction
2. 2 Preliminaries
3. 3 Perfect complexes over infinite root stacks
4. 4 Beyond simple normal crossing divisors
5. 5 Non-commutative motives of log schemes

1. Introduction

In this paper we carry forward the study of the derived category of parabolic sheaves we initiated in [46], where we established the Morita invariance of parabolic sheaves under logarithmic blow-ups. Our main result is the construction of a special kind of semi-orthogonal decompositions on derived categories of parabolic sheaves. This provides in particular a categorification of structure theorems for the Kummer flat K-theory of log schemes due to Hagihara [20] and Nizioł [36]. Additionally we generalize Hagihara and Nizioł’s result in two ways:

• •

we obtain uniform structure theorems which hold across all (Kummer flat) invariants of logarithmic schemes, including Hochschild and cyclic homology;

• •

our techniques allow us to extend these results to a much larger class of log schemes (and log stacks) than those considered by Hagihara and Nizioł.

Our main results, Theorem A, Theorem B and Corollary C, hold over an arbitrary ground ring.

1.1. Parabolic sheaves, log schemes, and infinite root stacks

Parabolic sheaves were defined by Mehta and Seshadri in the 80’s, for Riemann surfaces with marked points, as coherent sheaves equipped with flags at the marked points. They are the key ingredient to extend the Narasimhan–Seshadri correspondence to the non-compact setting. The theory over the last fifty years has undergone massive generalizations. It was extended first to pairs $(X,D)$ where $D$ is a normal crossing divisor in any dimension, and more recently to an even broader class of logarithmic schemes. The work of Borne, Vistoli and the third author shows that parabolic structures are best viewed within the framework of log geometry [6], [51], [49] and this is the perspective that we will adopt throughout the paper.

Logarithmic (log) geometry emerged in the 80’s through the collective efforts of several authors including Deligne, Faltings, Fontaine, Illusie and Kato [28]. The theory was initially designed for applications to arithmetic geometry, but over the last twenty years it has become a key organizing principle in areas as diverse as algebraic geometry, symplectic geometry, and homotopy theory. Log geometric techniques lie at the core of the Gross–Siebert program in mirror symmetry [19], and feature prominently in recent approaches to Gromov-Witten theory, see [18] and references therein.

One of the main hurdles in working with log schemes is that they encode both classical geometric and combinatorial data. For this reason transporting familiar geometric constructions to the log setting is often delicate: see for instance [40], [21] for the definition of the cotangent complex and the Chow groups of log schemes. A definition of K-theory for log schemes was first proposed by Hagihara [20] and Nizioł [36]. We will refer to it as Kummer flat (resp. étale) K-theory, and it is the algebraic K-theory of the Kummer flat (resp. étale) topos, a logarithmic analogue of the classical flat (resp. étale) topos.

Our main result is a construction of infinite semi-orthogonal decompositions on categories of parabolic sheaves. This can be viewed as a categorification of an important structure theorem due to Hagihara and Nizioł for Kummer flat K-theory: if $X$ is a regular scheme equipped with a simple normal crossings divisor $D\subset X$, the Kummer flat K-theory of $(X,D)$ splits as an an explicit direct sum indexed by the strata of $D$ [36, Theorem 1.1]. We will show that our methods yield, in particular, substantial generalizations of Hagihara and Nizioł’s results. Before stating our main result we review its two key ingredients: the infinite root stack, and semi-orthogonal decompositions.

Infinite root stacks

The infinite root stack of a log scheme was introduced in [51]: it is a limit of tame Artin stacks (Deligne–Mumford in characteristic [math]) which encodes log information as stacky data. The infinite root stack captures the geometry of the underlying log scheme, and this point of view informs several recent works by the authors and their collaborators [9, 46, 52, 50], see also [44] for recent applications to Hall algebras and quantum groups. If $X$ is a log scheme, we denote its infinite root stack by $\sqrt[\infty]{X}$. One of the key properties of the infinite root stack is that the Kummer flat topos of $X$ is equivalent, as a ringed topos, to the fppf topos of $\sqrt[\infty]{X}$ [51, Theorem 6.16]. In particular Hagihara and Nizioł’s logarithmic algebraic K-theory of $X$ coincides with the ordinary algebraic K-theory of the infinite root stack $\sqrt[\infty]{X}.$ Thus we can study logarithmic algebraic K-theory by probing the geometry and the sheaf theory of infinite root stacks.

Semi-orthogonal decompositions

We view the algebraic K-theory of a stack as an invariant of its $\infty$-category of perfect complexes. We study the category of perfect complexes of the infinite root stack, and show that Nizioł’s direct sum decomposition of Kummer flat K-theory is the shadow of a factorization that holds directly at the categorical level. The appropriate concept of factorization for categories is given by semi-orthogonal decompositions (sod-s, for short). These were introduced by Bondal and Orlov in [5]. In the setting of $\infty$-categories, semi-orthogonal decompositions (of length two) were considered in [4] under the name of split-exact sequences of $\infty$-categories.

The main theorem

Recent work of Ishii and Ueda [25] and Bergh, Lunts, and Schnürer [3] shows that the categories of perfect complexes of finite root stacks of pairs $(X,D),$ where $X$ is a scheme (or stack) equipped with a simple normal crossings divisor $D\subset X,$ admit a canonical sod where the summands are labelled by the strata of $D.$ The infinite root stack is the limit of all finite root stacks, however the pull-back functors along root maps do not preserve the canonical sod-s. This issue can be obviated via a recursion that gives rise a sequence of nested sod-s as the root index grows, and that ultimately yields an infinite sod on $\mathrm{Perf}(\sqrt[\infty]{X})$.

Below we formulate our first main result for log scheme of the form $(X,D),$ where $X$ is a scheme and $D$ is a simple normal crossings divisor. We refer the reader to Theorem 3.16 in the main text for a sharper and more general statement, that applies for instance to all finite type tame algebraic stacks. Let $\{D_{i}\}_{i\in I}$ be the set of irreducible components of $D$. The divisor $D$ determines a stratification of $X$ where strata are intersections of the irreducible components of $D$. Strata are in bijection with the subsets of $I$: if $S$ is a stratum $S=\cap_{j\in J}D_{j}$ for some $J\subset I$, and $\overline{S}$ is the closure, we set $|\overline{S}|:=|J|$. If $N$ is a natural number we set $(\mathbb{Q}/\mathbb{Z})^{N,*}:=(\mathbb{Q}/\mathbb{Z}\setminus\{0\})^{N}$.

Theorem A** (Theorem 3.16).**

The $\infty$-category of perfect complexes of the infinite root stack $\sqrt[\infty]{(X,D)}$ admits a semi-orthogonal decomposition

[TABLE]

such that all objects in $\mathcal{A}_{\overline{S}}$ are supported on $\overline{S}$. Additionally, for all $\overline{S}$ the category $\mathcal{A}_{\overline{S}}$ carries a semi-orthogonal decomposition indexed by $(\mathbb{Q}/\mathbb{Z})^{|\overline{S}|,*}$ whose factors are equivalent to $\mathrm{Perf}(\overline{S}).$

1.2. Additive invariants

Theorem A recovers in particular Hagihara and Nizioł’s results, but is a much stronger statement. In order to clarify this point let us refer to the notion of additive invariants of $\infty$-categories. Let us denote by $\mathrm{Cat}^{\mathrm{perf}}_{\infty}$ the $\infty$-category of stable $\infty$-categories. A functor $\mathrm{H}\colon\mathrm{Cat}^{\mathrm{perf}}_{\infty}\to\mathcal{P}$ , where $\mathcal{P}$ is a stable presentable $\infty$-category, is an additive invariant if it preserves zero objects and filtered colimits, and it maps split exact sequences to cofiber sequences (split exact sequences are the analogue in the $\infty$-setting of a sod with two factors). Most homological invariants of algebras and categories are additive: algebraic K-theory and non-connective K-theory, (topological) Hochschild homology and negative cyclic homology are all additive invariants.

The theory of non-commutative motives was developed by Tabuada and others [48, 11, 4, 41] in analogy with the classical theory of motives. Non-commutative (additive) motives encode the universal additive invariant, exactly as classical motives are universal among Weil cohomologies. Noncommutative motives form a presentable and stable $\infty$-category $\,\mathrm{Mot}^{\mathrm{add}}$ which is the recipient of the universal additive invariant

[TABLE]

Every additive invariant $\,\mathrm{H}\colon\mathrm{Cat}^{\mathrm{perf}}_{\infty}\to\mathcal{P}$ factors uniquely as a composition

[TABLE]

If $\mathrm{H}$ is an additive invariant and $X$ is a stack we set $\,\mathcal{U}(X):=\mathcal{U}(\mathrm{Perf}(X)),\;\mathrm{H}(X):=\mathrm{H}(\mathrm{Perf}(X))$. Let $(X,D)$ be a scheme equipped with a normal crossings divisor. Then Theorem A yields a direct product decomposition of the non-commutative motive of $\sqrt[\infty]{X}$.

Theorem B** (Corollary 5.6).**

There is a canonical direct sum decomposition

[TABLE]

The K-theory of root stacks was also studied in [14], although from a different perspective.

Kummer étale K-theory

Theorem B implies uniform direct product decompositions across all additive invariants of infinite root stacks, and recovers in particular Hagihara and Nizioł’s structure theorems for Kummer flat K-theory. Let $X$ be a log scheme, and denote $X_{\mathrm{Kfl}}$ its Kummer flat topos. Let $\mathrm{Perf}(X_{\mathrm{Kfl}})$ be its $\infty$-category of perfect complexes. If $\mathrm{H}$ is an additive invariant, we set $\,\mathrm{H}_{\mathrm{Kfl}}(X):=\mathrm{H}(\mathrm{Perf}(X_{\mathrm{Kfl}}))\,$. When $\mathrm{H}(-)=K(-)$ is algebraic K-theory, this definition recovers Hagihara and Nizioł’s Kummer flat K-theory.

Work of Vistoli and the third author [51] identifies the Kummer flat topos with the “small fppf topos” of the infinite root stack. As a consequence, under suitable assumptions, there is an equivalence of stable $\infty$-categories $\mathrm{Perf}(X_{\mathrm{Kfl}})\simeq\mathrm{Perf}(\sqrt[\infty]{X}).$ This, together with Theorem B, yields the following immediate corollary.

Corollary C**.**

If $\mathrm{H}\colon\mathrm{Cat}^{\mathrm{perf}}_{\infty}\to\mathcal{P}$ is an additive invariant then there is a direct sum decomposition

[TABLE]

In particular, the Kummer flat K-theory of $(X,D)$ decomposes as a direct sum of spectra

[TABLE]

The second half of the statement recovers the first part of Nizioł’s [36, Theorem 1.1] (see Remark 5.9 for some comments about the second part). Nizioł’s result holds under the restrictive assumption that $(X,D)$ is a log smooth pair given by a regular scheme $X$ and a simple normal crossings divisor $D$. Corollary C holds with milder smoothness assumption on $X$: indeed in the main body of the paper we work with a finite type algebraic stacks $X$ equipped with a simple normal crossings divisor $D$. In particular $X$ needs not be regular outside of $D$.

In fact we can extend the the decomposition given by Corollary C to even more general log stacks. We clarify this by explaining three applications of our techniques. They require working over a field $\kappa$ of characteristic zero. Additionally for the second one we need to assume $\kappa=\mathbb{C}$.

• •

General normal crossing divisors. We extend the decomposition of Theorem A and Corollary C to general normal crossing divisors, removing the simplicity assumption required by Hagihara and Nizioł. An interesting new feature emerges in this setting. The semi-orthogonal summands appearing in the analogue of Theorem A for general normal crossing log stacks $(X,D)$ are no longer equivalent to the category of perfect complexes on the strata: instead, they are equivalent to perfect complexes on the normalization of the strata (see Theorem 4.6 in the main text). This is reflected by the Kummer flat K-theory of a general normal crossing log stack: it breaks up as a direct sum indexed by the strata, where the summands compute the K-theory of the normalization of the strata (see Theorem 5.8).

• •

Simplicial log structures. We generalize Corollary C to log smooth schemes with simplicial log structure, i.e. pairs $(X,D)$ where $D$ is a divisor with simplicial singularities. The decomposition formula holds only for the complexification of $K_{\mathrm{Kfl}}(X),$ and under suitable additional assumptions on $(X,D).$ The main differences with the normal crossings case is that the formula has additional summands keeping track of the singularities of $D$, and that it depends on the G-theory, rather than the K-theory, of the strata. As the statement is somewhat technical we do not include it in this introduction, but refer the reader directly to Proposition 5.10 in the main text.

• •

Logarithmic Chern character. Having at our disposal a general definition of additive invariants of log schemes, we introduce a construction of the logarithmic Chern character. The availability of structure theorems valid across all additive invariants of log schemes allows us to study some of its fundamental properties. This includes a Grothendieck–Riemann–Roch statement, which we establish in the restrictive setting of strict maps of log schemes, leaving generalizations to future work. These results are contained in Section 5.3 of the text.

1.3. Towards logarithmic DT invariants

Donaldson–Thomas invariants are part of the rich array of enumerative invariants inspired by string theory. One of the outstanding open questions in the area is to construct a theory of log DT invariants, analogous to the theory of log GW invariants developed in [18, 10, 1]. This would have applications to degeneration formulas for DT invariants. DT theory counts Bridgeland stable objects in triangulated categories. Thus building a theory of log DT invariants requires, first, to introduce a viable concept of derived category for log schemes; and, second, to define and study stability conditions over it. For the first requirement, one of the viable options is to try to use parabolic sheaves on $(X,D)$, which in turn are equivalent to sheaves on $\sqrt[\infty]{X}$. Thus a first step towards defining log DT invariants consists in constructing Bridgeland stability condition on $\mathrm{Perf}(\sqrt[\infty]{X})$. Results obtained in [12] give a means to glue stability conditions across semi-orthogonal decompositions. Adapting these techniques to the sod-s on $\mathrm{Perf}(\sqrt[\infty]{X})$ obtained in Theorem A, we can already obtain stability conditions in some cases, such as many toric log pairs $(X,D)$. It is too early to tell wether they will be relevant from the viewpoint of log DT theory, but this seems an interesting avenue for future investigation.

1.4. Relation to work of other authors

Several different approaches to the definition of log motives and log invariants have been considered in the literature.

• •

A definition of log motives has been proposed in [26], and in [23].

• •

Constructions of Hochschild homology and topological Hochschild homology in the log setting have been proposed by Hesselholt and Madsen [22] Rognes, Sagave, and Schlichtkrull [42], Leip [32] and Olsson [38].

It would be very interesting to compare these approaches to the one we pursue in this paper, however there is a key difference in perspective between some of these works and our own. The constructions of log Hochschild homology considered in [22], [42], and [32] are closely related to the log de Rham complex, and therefore to the cohomology of the complement of the locus where the log structure is concentrated. In this paper, on the other hand, we investigate log schemes through the lenses of their Kummer flat topos and their infinite root stack.

Acknowledgments:

We thank Daniel Bergh, David Carchedi, Olaf Schnürer, Greg Stevenson and Angelo Vistoli for their interest in this project and for useful exchanges, and the anonymous referee for useful comments. M.T. was partially supported by EPSRC grant EP/R013349/1.

Conventions

We work over an arbitrary noetherian commutative ring $\kappa$ (that could be the ring of integers $\mathbb{Z}$). In later parts of the paper (Sections 5.2 and 5.3) we will impose more restrictive assumptions on $\kappa$. All algebraic stacks (in the sense of [47, Tag 026O]) will be of finite type over $\kappa$. All monoids will be commutative and “toric”, i.e. finitely generated, sharp, integral and saturated.

2. Preliminaries

2.1. Log structures from boundary divisors and root stacks

In this section we briefly recall how certain boundary divisors $D$ on a scheme or algebraic stack $X$ give rise to log structures and root stacks.

Definition 2.1**.**

Let $X$ be a scheme over $\kappa$, and $D\subset X$ an effective Cartier divisor. Recall that the divisor $D$ is:

• •

simple normal crossings if for every $x\in D$ the local ring $\mathcal{O}_{X,x}$ is regular, and there is a system of parameters $a_{1},\ldots,a_{n}\in\mathcal{O}_{X,x}$ such that the ideal of $D$ in $\mathcal{O}_{X,x}$ is generated by $a_{1},\ldots,a_{k}$ for some $1\leq k\leq n$, and

• •

normal crossings if every $x\in D$ has an étale neighbourhood where $D$ becomes simple normal crossings.

Note that if $D$ is a normal crossings divisor on $X$, then every point of $D$ is a regular point of $X$, but away from $D$ the scheme $X$ can well be singular.

If $X$ is an algebraic stack and $D\subset X$ is an effective Cartier divisor, we say that $D$ is (simple) normal crossings if the pull-back of $D$ to some smooth presentation $U\to X$, where $U$ is a scheme, is a (simple) normal crossings divisor on $U$.

Remark 2.2**.**

If $\kappa$ is a field, then the divisor $D\subset X$ is normal crossings if and only if étale locally around every point $x\in D$, the pair $(X,D)$ is isomorphic to the pair $(\mathbb{A}^{n},\{x_{1}\cdots x_{k}=0\})$ for some $n\in\mathbb{N}$ and $0\leq k\leq n$.

This is not necessarily true if $\kappa$ is not a field. For example, if $\kappa=\mathbb{Z}$ the divisor $V(p)\cup V(x)$ for a prime number $p$ is simple normal crossings on $X=\mathbb{A}^{1}=\mathrm{Spec}\,\mathbb{Z}[x]$, but X is not étale locally isomorphic to $\mathbb{A}^{2}$ around the point $(p,x)$.

Later on (Section 4.2) we will consider a generalization of this notion, where the divisor $D$ is allowed to have simplicial singularities. Over a field $\kappa$ one can define this by asking that étale locally around every point $x\in D$, the pair $(X,D)$ is isomorphic to the pair $(\mathrm{Spec}\,\kappa[P]\times\mathbb{A}^{n},\Delta_{P}\times\mathbb{A}^{n})$ for some simplicial monoid $P$ and $n\in\mathbb{N}$. Recall that a sharp fine saturated monoid $P$ is simplicial if the extremal rays of the rational cone $P_{\mathbb{Q}}\subset P^{\mathrm{gp}}\otimes_{\mathbb{Z}}\mathbb{Q}$ are linearly independent, and we denote by $\Delta_{P}$ the toric boundary in the affine toric variety $\mathrm{Spec}\,\kappa[P]$, i.e. the complement of the torus $\mathrm{Spec}\,\kappa[P^{\mathrm{gp}}]$, equipped with the reduced subscheme structure. This weaker notion allows for some kinds of singularities along the divisor $D$ itself.

It is not clear to us how to formulate this notion in the “absolute” case (i.e. for schemes over $\mathrm{Spec}\,\mathbb{Z}$), so we will circumvent this problem by using a canonically defined root stack of a log scheme with simplicial log structure, and reducing to Definition 2.1 on this root stack (see Definition 4.11).

Next we recall how ((simple) normal crossings) divisors induce associated log structures and root stacks. We focus on the construction of root stacks, since the full formalism of logarithmic geometry will not play an important role in the paper. We refer the reader to [37] for an extensive introduction on log geometry, and to the appendix of [9] for a quick overview of the basic concepts. For more on root stacks, the reader can consult [7, 6, 51, 49].

Any effective Cartier divisor $D$ in a scheme $X$ induces a log structure, usually called the compactifying log structure of the open embedding $X\setminus D\subset X$, as follows. The sheaf $M_{D}=\{f\in\mathcal{O}_{X}\mid f|_{X\setminus D}\in\mathcal{O}_{X\setminus D}^{\times}\}$ is a sheaf of submonoids of $\mathcal{O}_{X}$ (seen as a sheaf on the small étale site of $X$), where the monoid operation is multiplication of regular functions. The inclusion $\alpha\colon M_{D}\to\mathcal{O}_{X}$ gives rise to a log structure on $X$. If $D$ is normal crossings, this log structure admits local charts, and in fact will also be fine and saturated. Morally, the sheaf $M_{D}$ keeps track of how many branches of $D$ intersect at a point of $X$, and how their local equations fit together in the local ring $\mathcal{O}_{X,x}$ (more precisely, in its strict henselization, since we are using the étale topology).

If $X$ is a stack rather than a scheme, the above procedure gives a log structure on any given smooth presentation of $X$, and descent for fine log structures [39, Appendix] gives a fine saturated log structure on $X$ itself. We will denote the resulting log scheme or stack by $(X,D)$.

2.1.1. Root stacks along a single regular divisor

Assume that $D\subset X$ is a simple normal crossings divisor with only one component, i.e. a regular connected divisor, and $X$ is an algebraic stack. Recall that the quotient stack $[\mathbb{A}^{1}/\mathbb{G}_{m}]$ functorially parametrizes pairs $(L,t)$ where $L$ is a line bundle and $t$ is a global section of $L$.

As specified above, the effective Cartier divisor $D$ on $X$ induces a canonical log structure, that in this simple case can be described as follows: the divisor $D$ determines a line bundle $\mathcal{O}_{X}(D)$ on $X$ together with a section $\sigma\in\Gamma(\mathcal{O}_{X}(D))$ having $D$ as its zero locus, and this yields a tautological morphism $s\colon X\longrightarrow[\mathbb{A}^{1}/\mathbb{G}_{m}].$ This equips $X$ with a log structure by pulling back the canonical log structure of $[\mathbb{A}^{1}/\mathbb{G}_{m}]$ (corresponding to the regular divisor $[\{0\}/\mathbb{G}_{m}]\subset[\mathbb{A}^{1}/\mathbb{G}_{m}]$) via $s$.

In this case, for $r\in\mathbb{N}$ the $r$-th root stack of the pair $(X,D)$ can be described as the functor that associates to a scheme $T\to X$ over $X$ the groupoid $\sqrt[r]{(X,D)}(T)$ of pairs $(L,t)$ consisting of a line bundle on $T$ with a global section, and with an isomorphism $(\mathcal{O}_{X}(D),\sigma)|_{T}\cong(L^{\otimes r},t^{\otimes r})$.

The stack $\sqrt[r]{(X,D)}$ fits in a fiber square

[TABLE]

where $(-)^{r}$ is induced by the $r$-th power maps on $\mathbb{A}^{1}$ and on $\mathbb{G}_{m}$, or equivalently is the functor sending a pair $(L,t)$ of a line bundle with a global section to the pair $(L^{\otimes r},t^{\otimes r})$. The reason for the notation $g_{r,1}$ will be apparent later (see Section 2.1.4). The construction of the stack $\sqrt[r]{(X,D)}$ is compatible with respect to pull-back along smooth morphisms towards $X$ (in particular with respect to Zariski and étale localization on $X$), i.e. if $Y\to X$ is smooth, we have a canonical isomorphism $\sqrt[r]{(Y,D|_{Y})}\simeq\sqrt[r]{(X,D)}\times_{X}Y$.

We denote by $D_{r}$ the effective Cartier divisor on $\sqrt[r]{(X,D)}$ obtained by taking the reduction of the closed substack $g_{r,1}^{-1}(D)\subset\sqrt[r]{(X,D)},$ and we denote $i_{r}\colon D_{r}\to\sqrt[r]{(X,D)}$ the inclusion. We will refer to $D_{r}$ as the universal effective Cartier divisor on $\sqrt[r]{(X,D)}$, since it is the universal $r$-th root of the divisor $D$ on $X$.

Consider the commutative (non-cartesian) diagram

[TABLE]

where the left vertical arrow is induced by the $r$-th power map $(-)^{r}\colon\mathbb{G}_{m}\longrightarrow\mathbb{G}_{m}.$ As explained in [3], $D_{r}$ can be defined equivalently as the top left vertex of the base change of diagram (2) along the tautological map $s\colon X\to[\mathbb{A}^{1}/\mathbb{G}_{m}]$: in particular, there is a fiber product

[TABLE]

This implies that $D_{r}\to D$ is a $\mu_{r}$-gerbe.

Zariski locally on $X$, where the line bundle $\mathcal{O}_{X}(D)$ is trivial, the stack $\sqrt[r]{(X,D)}$ admits the following explicit description: assume also that $X=\mathrm{Spec}\,A$ is affine, and let $f\in A$ correspond to the section $\sigma$ of $\mathcal{O}_{X}(D)$ (so that $D$ has equation $f=0$). Then we have an isomorphism

[TABLE]

where $\mu_{r}$ acts by multiplication on $x$. The divisor $D_{r}\subset\sqrt[r]{(X,D)}$ is given by the global equation $x=0$, and is therefore isomorphic to $[\mathrm{Spec}\,(A/f)/\mu_{r}]\simeq D\times\mathscr{B}\mu_{r}$.

2.1.2. Root stacks along a simple normal crossings divisor

Assume now that $D$ is a simple normal crossings divisor on $X$, and denote by $D_{1},\ldots,D_{N}$ the irreducible components of $D$. In this case root stacks of $(X,D)$ are indexed by elements of $\mathbb{N}^{N}$. For $\vec{r}=(r_{1},\ldots,r_{N})\in\mathbb{N}^{N}$, the root stack $\sqrt[\vec{r}]{(X,D)}$ parametrizes tuples $((L_{1},t_{1}),\ldots,(L_{N},t_{N}))$, where $(L_{i},t_{i})$ is a $r_{i}$-th root of $(\mathcal{O}_{X}(D_{i}),\sigma_{i})$. Each pair $(\mathcal{O}_{X}(D_{i}),\sigma_{i})$ determines a morphism $s_{i}\colon X\to[\mathbb{A}^{1}/\mathbb{G}_{m}]$, and the stack $\sqrt[\vec{r}]{(X,D)}$ is the fiber product of the diagram

[TABLE]

where $s\colon X\to[\mathbb{A}^{1}/\mathbb{G}_{m}]^{N}$ is determined by the $s_{i}$ and $(-)^{\vec{r}}$ is the map induced by $(-)^{r_{i}}\colon[\mathbb{A}^{1}/\mathbb{G}_{m}]\to[\mathbb{A}^{1}/\mathbb{G}_{m}]$ on the $i$-th component.

Equivalently, $\sqrt[\vec{r}]{(X,D)}$ can be constructed by iteration from the previous case: from $X$ we first construct the stack $\sqrt[r_{1}]{(X,D_{1})}$ as in the previous section. The preimages $\widetilde{D_{2}},\ldots,\widetilde{D_{N}}$ of $D_{2},\ldots,D_{N}$ to this stack give a simple normal crossings divisor $\widetilde{D}$, and we can replace $(X,D)$ by the log stack $(\sqrt[r_{1}]{(X,D_{1})},\widetilde{D})$, and continue the process.

Finally, the stack $\sqrt[\vec{r}]{(X,D)}$ can also be identified with the fibered product of the diagram

[TABLE]

As in the case of one divisor, the construction of $\sqrt[\vec{r}]{(X,D)}$ is compatible with smooth base change.

On the stack $\sqrt[\vec{r}]{(X,D)}$ there are $N$ universal effective Cartier divisors $D_{i,r_{i}}$, obtained as the reduction of the preimage of $D_{i}$ to $\sqrt[\vec{r}]{(X,D)}$ via the projection to $X$, or equivalently as the preimage of the corresponding divisor $D_{i,r_{i}}$ on $\sqrt[r_{i}]{(X,D_{i})}$ constructed in the previous section (we abuse notation slightly and denote the two by the same symbol).

Zariski locally where $X=\mathrm{Spec}\,A$ is affine and each $D_{i}$ has a global equation $f_{i}=0$ we have an isomorphism

[TABLE]

where each factor $\mu_{r_{i}}$ acts on $x_{i}$ by multiplication and trivially on the other variables.

2.1.3. Root stacks for non-simple normal crossings

If $D$ is only normal crossings, then the “correct” notion of root stack of $X$ along $D$ is not the naive generalization of the simple normal crossings case, but rather the subtler one put forward in [6]. We refer to that paper and to [49, Section 2.2] for details about what follows.

We start by explaining with an example why the naive notion of root stacks along the lines of the previous two cases is not the one that we want to work with.

Example 2.3**.**

Consider an irreducible nodal cubic $D\subset\mathbb{P}^{2}$, and the pair $(\mathbb{P}^{2},D)$. We could, as in Section 2.1.1, start with the morphism $\mathbb{P}^{2}\to[\mathbb{A}^{1}/\mathbb{G}_{m}]$ that classifies the divisor $D$ (i.e. determined by the pair ($\mathcal{O}_{\mathbb{P}^{2}}(D),\sigma_{D})$, where $\sigma_{D}$ is the tautological section of $\mathcal{O}_{\mathbb{P}^{2}}(D)$ cutting out $D$), and set $\sqrt[r]{(\mathbb{P}^{2},D)}$ to be the fibered product of diagram (1). The unpleasant feature of this construction is that it adds a stabilizer $\mu_{r}$ along all of $D$, including at the node, where one would expect to have a $\mu_{r}\times\mu_{r}$ accounting for the two branches of $D$.

The point here is that the morphism $\mathbb{P}^{2}\to[\mathbb{A}^{1}/\mathbb{G}_{m}]$ does not induce to the “correct” logarithmic structure on $\mathbb{P}^{2}$ corresponding to the pair $(\mathbb{P}^{2},D)$, which is the compactifying log structure briefly mentioned in Section 2.1, and does distinguish the branches of $D$ at the node.

The trouble with the previous example is remedied by working with sheaves of weights (as explained in [6]), that allow to locally distinguish the branches of $D$ at the node, and to locally take roots separately along the branches.

We briefly recall what this entails: in general, the log structure of a fine saturated log scheme $X$ can be seen as a “Deligne–Faltings” structure, a symmetric monoidal functor $L\colon A\to\mathrm{Div}_{X}$ from a sheaf of saturated sharp monoids $A$ on the small étale site of $X$ to the symmetric monoidal stack $\mathrm{Div}_{X}$ of pairs $(L,t)$ of line bundles with global section. The monoidal operation of $\mathrm{Div}_{X}$ is given by tensor product.

In the case of simple normal crossings, the canonical (compactifying) log structure $M_{D}\to\mathcal{O}_{X}$ induced by $D$ can be completely described in the language of Deligne–Faltings structures by the symmetric monoidal functor $\mathbb{N}^{N}\to\mathrm{Div}_{X}(X)$ sending the generator $e_{i}$ to the pair $(\mathcal{O}_{X}(D_{i}),\sigma_{i})$ (this is a “chart” for the log structure, in the sense of [6, Section 3.3]). If $D$ is only normal crossings though, the global components of the divisor $D$ will not capture the geometry of the situation faithfully (as for the nodal cubic of Example 2.3), and the compactifying log structure is not described by the symmetric monoidal functor $\mathbb{N}^{N}\to\mathrm{Div}_{X}(X)$ defined above.

In both cases (simple and non-simple normal crossings), the sheaf $A$ is the quotient $M_{D}/\mathcal{O}_{X}^{\times}$ of the sheaf $M_{D}\subset\mathcal{O}_{X}$ consisting of regular functions that only possibly vanish along $D$ (mentioned in Section 2.1) by the subgroup $\mathcal{O}_{X}^{\times}$ of invertible functions, and the symmetric monoidal functor $L\colon A\to\mathrm{Div}_{X}$ is induced by the inclusion $M_{D}\subseteq\mathcal{O}_{X}$ by passing to (stacky) quotients by $\mathcal{O}_{X}^{\times}$, obtaining

[TABLE]

If $D$ has simple normal crossings, then the sheaf $A$ is isomorphic to the direct sum $\bigoplus_{i=1}^{N}{j_{i}}_{*}\mathbb{N}_{D_{i}}$, where $D_{i}$ are the irreducible components of $D$, $j_{i}\colon D_{i}\to X$ is the inclusion and $\mathbb{N}_{D_{i}}$ is the constant (étale) sheaf of monoids with stalk $\mathbb{N}$ on $D_{i}$. The map $A\to\mathrm{Div}_{X}$ in this case sends the generator $1_{D_{i}}$ of the summand ${j_{i}}_{*}\mathbb{N}_{D_{i}}$ to the pair $(\mathcal{O}_{X}(D_{i}),\sigma_{i})$ (where as usual $\sigma_{i}$ is the section that cuts out $D_{i}$). If $D$ is only normal crossings but non-simple, this description does not hold anymore (it only holds étale locally where $D$ becomes simple normal crossings). For the nodal cubic of Example 2.3, the stalk of the sheaf $A$ at the node is $\mathbb{N}^{2}$ (accounting for the two branches) rather than just $\mathbb{N}$.

For a general log scheme $X$ and $r\in\mathbb{N}$, the root stack $\sqrt[r]{X}$ (in the sense of Borne and Vistoli) parametrizes lifts of the symmetric monoidal functor $L\colon A\to\mathrm{Div}_{X}$ to a symmetric monoidal functor $\frac{1}{r}A\to\mathrm{Div}_{X}$, where we are embedding $A$ into $\frac{1}{r}A\cong A$ via the map $\cdot r\colon A\to A$. The image of the section $\frac{1}{r}a$ via this lift is an $r$-th root of the pair $L(a)=(L_{a},s_{a})$. In the case of the nodal cubic $D$ of Example 2.3, i.e. of $\mathbb{P}^{2}$ equipped with the compactifying log structure induced by $D$, this results in trivial stabilizers outside of $D$, a stabilizer $\mu_{r}$ over points of $D$ that are not the node, and a stabilizer $\mu_{r}\times\mu_{r}$ at the node.

If $D\subset X$ is a normal crossings divisor and $U\to X$ is a surjective étale morphism such that the pull-back $D|_{U}$ is simple normal crossings, then the stack $\sqrt[r]{(X,D)}$ can be obtained from the root stack $\sqrt[\vec{r}]{(U,D|_{U})}$ constructed in the previous section (where $\vec{r}$ is the vector $(r,\ldots,r)$) by descent.

Moreover, In place of $\frac{1}{r}A$ one can use an arbitrary Kummer extension $A\to B$ of sheaves of monoids, an injective morphism such that every section of $B$ locally has a multiple coming from $A$. This more general construction of the root stack $\sqrt[B]{X}$ for Kummer extensions $A\to B$, where $B$ is not of the form $\frac{1}{r}A$, will come up only in Section 4.2. In general, the stack $\sqrt[B]{X}$ is a tame Artin stack (Deligne–Mumford in characteristic [math]), and the projection $\sqrt[B]{X}\to X$ is a coarse moduli space morphism.

2.1.4. The infinite root stack and the Kummer flat topos

In all the cases considered above, the various root stacks of $(X,D)$ form an inverse system. Let us temporarily use the letter $r$ to denote either a natural number, or a vector of natural numbers $\vec{r}\in\mathbb{N}^{N}$, depending on the context that we are considering. We write $r\mid r^{\prime}$ to denote divisibility in the first case, and that $r_{1}\mid r_{1}^{\prime},\ldots,r_{N}\mid r_{N}^{\prime}$ in the second case. We also write $\frac{r^{\prime}}{r}$ for the vector $\left(\frac{r^{\prime}_{1}}{r_{1}},\ldots,\frac{r^{\prime}_{N}}{r_{N}}\right)$.

With these conventions, if $(X,D)$ is a pair where $D$ is normal crossings and $r\mid r^{\prime}$, there is a natural projection $g_{r^{\prime},r}\colon\sqrt[r^{\prime}]{(X,D)}\to\sqrt[r]{(X,D)}$, roughly defined by raising the roots parametrized by the source stack to the $r^{\prime}/r$-th power. Since $\sqrt[1]{(X,D)}\simeq X$, the map $g_{r,1}\colon\sqrt[r]{(X,D)}\to X$ is the natural projection. If $D$ is a regular divisor, then the maps $g_{r^{\prime},r}$ restrict to maps $f_{r^{\prime},r}\colon D_{r^{\prime}}\to D_{r}$ between the universal Cartier divisors on the two stacks (and analogously in the case where $D$ is simple normal crossings).

Moreover, $g_{r^{\prime},r}$ is a “relative root stack” morphism, in the sense that for a morphism $T\to\sqrt[r]{(X,D)}$ where $T$ is a scheme, the pull-back of $g_{r^{\prime},r}$ can be seen as the projection from a root stack of the scheme $T$. Consequently, the maps $g_{r^{\prime},r}$ have all the properties of a projection to a coarse moduli space of a tame algebraic stack. More precisely, $\sqrt[r^{\prime}]{(X,D)}$ is canonically isomorphic to the $\frac{r^{\prime}}{r}$-th root stack of $\sqrt[r]{(X,D)}.$ Under this isomorphism $g_{r^{\prime},r}$ is identified with the projection

[TABLE]

The maps $g_{r^{\prime},r}$ equip the stacks $\{\sqrt[r]{(X,D)}\}_{r}$ with the structure of an inverse system. The inverse limit $\sqrt[\infty]{(X,D)}:=\varprojlim_{r}\sqrt[r]{(X,D)}$ is the infinite root stack of $(X,D)$ [51]. This, contrarily to the finite root stacks, is not algebraic, but has a local description as a quotient stack, that allows some control over quasi-coherent sheaves (and in particular perfect complexes) on it.

It will be convenient for us to work with a directed subsystem of root stacks which is cofinal in $\{\sqrt[r]{(X,D)}\}_{r}.$ Namely we will consider the subsystem $\{\sqrt[n!]{(X,D)}\}_{n\in\mathbb{N}},$ where $\vec{n}!:=(n!,\ldots,n!)$ if $D$ has more than one component. Note that the restriction of the ordering given by divisibility on $\mathbb{N}$ to the subset $\{n!\}_{n\in\mathbb{N}}\subseteq\mathbb{N}$ coincides with the usual ordering of the naturals (i.e. $r!\mid s!$ if and only if $r\leq s$), and that this subsystem is cofinal, since given an index $\vec{r}=(r_{1},\ldots,r_{N})$, we have $\vec{r}\mid\vec{M}!$ with $M=\mathrm{lcm}(r_{i})$. Therefore we have a canonical isomorphism $\sqrt[\infty]{(X,D)}\simeq\varprojlim_{n}\sqrt[n!]{(X,D)}.$

Let us also recall that every log scheme $X$ has an associated Kummer flat topos $X_{\mathrm{Kfl}}$. We omit a discussion of this construction, since for the present paper it can be safely taken as a black box. We refer the reader to [51, Section 6.2] or [36, Section 2] for the definition and basic properties. One can also define a “small fppf site” of the infinite root stack $\sqrt[\infty]{X}$, and as proven in [51, Theorem 6.16], the resulting topos $\sqrt[\infty]{X}_{\mathrm{fppf}}$ is isomorphic to the Kummer flat topos $X_{\mathrm{Kfl}}$ of the fine saturated log scheme $X$.

We will also consider the Kummer flat topos $X_{\mathrm{Kfl}}$ for $X$ a log algebraic stack. To the best of our knowledge this has not been discussed in the literature before. The conscientious reader can take the equivalence with the small fppf topos of the infinite root stack $\sqrt[\infty]{X}$ as a definition for $X_{\mathrm{Kfl}}$ in this setting. One can also write down a definition for the Kummer flat topos in analogy with the one for schemes, and use [51, Theorem 6.16] to prove that this is indeed equivalent to the small fppf topos of the infinite root stack.

2.2. $\infty$-categories and categories of sheaves

Throughout the paper we will use the formalism of $(\infty,1)$-categories, the standard reference is Lurie’s work [34, 35]. The main reason for working with $\infty$-categories is that additive invariants, and in particular algebraic K-theory, cannot be computed from the underlying triangulated categories alone: they are not invariants of triangulated categories but rather of their enhancements. We will work with stable $(\infty,1)$-categories as an enhancement of triangulated categories. We will need very little from the theory of $\infty$-categories, and the reader could replace stable $(\infty,1)$-categories with $\kappa$-linear dg categories throughout.

From now on we will refer to $(\infty,1)$-categories just as $\infty$-categories. If $\mathcal{C}$ is an $\infty$-category, and $A$ and $B$ are objects in $\mathcal{C},$ we denote by $\mathrm{Hom}_{\mathcal{C}}(A,B)$ the mapping space between $A$ and $B$. All limits and colimits of $\infty$-categories appearing in the paper are to be understood in the $\infty$-categorical sense. We say that a diagram of $\infty$-categories

[TABLE]

is commutative if there is a natural transformation $\alpha\colon HF\Rightarrow KG$ which is an equivalence when passing to homotopy categories.

We will be mostly interested in stable idempotent-complete $\infty$-categories. Stable $\infty$-categories are an enhancement of triangulated categories: if $\mathcal{C}$ is a stable $\infty$-category its homotopy category $\mathrm{Ho}(\mathcal{C})$ is triangulated. An important source of stable $\infty$-categories are the $\infty$-categorical derived categories of Grothendieck abelian categories, see [35] Definition 1.3.5.8. We refer the reader to Section 2.2 of [4] for a summary of the theory of stable $\infty$-categories which contains most of the facts that we will need in this paper. Small stable idempotent-complete $\infty$-categories form a presentable $\infty$-category which is denoted $\mathrm{Cat}^{\mathrm{perf}}_{\infty}.$ In particular $\mathrm{Cat}^{\mathrm{perf}}_{\infty}$ has all small limits and colimits. As explained in [4] there is a well-defined notion of tensor product of stable idempotent-complete $\infty$-categories. This endows $\mathrm{Cat}^{\mathrm{perf}}_{\infty}$ with a symmetric monoidal structure. If $\kappa$ is a commutative ring we denote by $\mathrm{Perf}(\kappa)$ the symmetric monoidal stable $\infty$-category of perfect $\kappa$-modules: if $\kappa-mod$ is the abelian category of $\kappa$-modules, $\mathrm{Perf}(\kappa)$ is the subcategory of compact objects in the the $\infty$-categorical derived category $\mathrm{D}(\kappa-mod)$ (see Definition 1.3.5.8 of [35]). We denote $\mathrm{Cat}_{\infty,\kappa}^{\mathrm{perf}}$ the symmetric monoidal $\infty$-category of idempotent-complete stable $\infty$-categories tensored over $\mathrm{Perf}(\kappa)$, see Section 4.1 of [24] for more details on this construction. We will refer to objects in $\mathrm{Cat}_{\infty,\kappa}^{\mathrm{perf}}$ as $\kappa$-linear $\infty$-categories.

For later reference we recall from [43] a general result on colimits of $\infty$-categories.

Proposition 2.4**.**

Let $I$ be a filtered category, $\{\mathcal{C}_{i}\}_{i\in I}$ be a filtered system of $\infty$-categories and assume that all the structure maps $\alpha_{i\to j}\colon\mathcal{C}_{i}\to\mathcal{C}_{j}$ are fully faithful. Then the colimit $\mathcal{C}:=\varinjlim\mathcal{C}_{i}$ is the $\infty$-category with

• •

objects given by the union $\bigcup_{i\in I}\mathrm{Ob}(\mathcal{C}_{i})$, and

• •

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

[TABLE]

where $l$ is any object of $I$ that is the source of morphisms $j\leftarrow l\rightarrow i$.

We will be working with stable categories of quasi-coherent sheaves on schemes and stacks. A survey of all basic facts and definitions on $\infty$-categories of quasi-coherent sheaves can be found in Section 2 and 3 of [2]. Let $X$ be a stack. We denote:

• •

by $\mathrm{qcoh}(X)$ the abelian category of quasi-coherent sheaves on $X$, and by $\mathrm{Qcoh}(X),$ the stable $\infty$-category of quasi-coherent sheaves on $X$;

• •

by $\mathrm{coh}(X)$ the abelian category of coherent sheaves on $X$, and by $\mathrm{Coh}(X)$ the stable $\infty$-category of coherent sheaves on $X$.

Remark 2.5**.**

Recall from [2] that $\mathrm{Qcoh}(X)$ is defined as the limit

[TABLE]

where $\mathrm{Spec}(A)\to X$ runs over all maps from derived affine schemes with target $X$. In general $\mathrm{Qcoh}(X)$ is different from the $\infty$-categorical derived category $\mathrm{D}(\mathrm{qcoh}(X))$. By [17, Proposition 2.4.3], however, if $X$ is a quasi-compact and quasi-separated classical Artin stack the respective subcategories of objects that are bounded below are equivalent

[TABLE]

The tensor product of quasi-coherent sheaves equips $\mathrm{Qcoh}(X)$ with a symmetric monoidal structure. We define $\mathrm{Perf}(X)$, the $\infty$-category of perfect complexes, as the full subcategory of dualizable objects in $\mathrm{Qcoh}(X)$. As proved in Proposition 3.6 of [2] this is equivalent to the ordinary definition of perfect complexes as objects that are locally equivalent to complexes of vector bundles. Let $I$ be a small cofiltered category, and let $\{X_{i}\}_{i\in I}$ be a pro-object in stacks.

Definition 2.6**.**

We set $\mathrm{Perf}(\{X_{i}\}_{i\in I}):=\varinjlim_{i}\mathrm{Perf}(X_{i})$ as an $\infty$-category.

Let $X$ be a log scheme. We will apply Definition 2.6 to the pro-object in stacks given by the root stacks of $X$ together with the root maps between them, $\{\sqrt[r]{X}\}_{r\in\mathbb{N}}.$ As we prove in Proposition 2.25 of [46], which we recall below, under appropriate assumptions there is an equivalence $\mathrm{Perf}(\sqrt[\infty]{X})\simeq\varinjlim_{r}\mathrm{Perf}(\sqrt[r]{X}).$ Although in [46] Proposition 2.25 was stated for the dg categories of perfect complexes, the proof given there works without variations for $\infty$-categories of perfect complexes over an arbitrary ground ring.

Proposition 2.7** ([46, Proposition 2.25]).**

Let $X$ be a noetherian fine saturated log algebraic stack with locally free log structure over $\kappa$. Then there is an equivalence of $\infty$-categories

[TABLE]

Proposition 2.8**.**

Let $X$ be a noetherian fine saturated log algebraic stack with locally free log structure over $\kappa$ (this holds in particular if the log structure comes from a normal crossings divisor). Then there is an equivalence of $\infty$-categories $\mathrm{Perf}(X_{\mathrm{Kfl}})\simeq\mathrm{Perf}(\sqrt[\infty]{X}).$

Proof.

The proof is very similar to the proof of Proposition 2.7 given in [46]. By Corollary 6.17 of [51] there is an equivalence of abelian categories $\mathrm{coh}(X_{\mathrm{Kfl}})\simeq\mathrm{coh}(\sqrt[\infty]{X}),$ which yields an equivalence between the stable $\infty$-categories $\mathrm{Coh}(X_{\mathrm{Kfl}})\simeq\mathrm{Coh}(\sqrt[\infty]{X}).$ Also, the structure sheaves $\mathcal{O}_{X_{\mathrm{Kfl}}}$ and $\mathcal{O}_{\sqrt[\infty]{X}}$ are coherent, see Proposition 4.9 of [51]. Thus, in the terminology of [16, Section 1.5], the ringed topoi $X_{\mathrm{Kfl}}$ and $\sqrt[\infty]{X}$ are eventually coconnective. As in the proof of Proposition 1.5.3 in [16], this implies that there are fully-faithful inclusions

[TABLE]

Further, the categories of perfect complexes are the full subcategories of dualizable objects. In formulas, we can write

[TABLE]

We obtain the following chain of equivalences, which implies the statement

[TABLE]

∎

Remark 2.9**.**

Let $X$ be a noetherian fine saturated log algebraic stack with locally free log structure over $\kappa$. In this paper we take $\mathrm{Perf}(\sqrt[\infty]{X})$ as our chosen model for the category of perfect complexes over the log scheme $X$. By Proposition 2.8 this coincides with the category of perfect complexes on the Kummer flat topos of $X$ and is therefore well suited to our applications to Kummer flat invariants of $X$. However we could have obtained an a priori different model of perfect complexes on $X$ by working with the abelian category of quasi-coherent parabolic sheaves with rational weights on $X$, $\mathrm{Par}(X)$, and then taking compact objects inside its $\infty$-categorical derived category $\mathrm{D}(\mathrm{Par}(X))$. It turns out that this actually gives an equivalent category. Let us briefly explain why.

• •

By Theorem 7.3 of [51] the abelian categories $\mathrm{Par}(X)$ and $\mathrm{qcoh}(\sqrt[\infty]{X})$ are equivalent.

• •

From Remark 2.5 and Proposition 2.7 it easily follows that the subcategories of bounded-below objects in $\mathrm{D}(\mathrm{Par}(X))\simeq\mathrm{D}(\mathrm{qcoh}(\sqrt[\infty]{X}))$ and in $\mathrm{QCoh}(\sqrt[\infty]{X})$ are equivalent. Since compact objects are bounded, this implies that the subcategory of compact objects in $\mathrm{D}(\mathrm{Par}(X))$ coincides with $\mathrm{Perf}(\sqrt[\infty]{X})$, as desired.

2.3. Exact sequences of $\infty$-categories

Let $\mathcal{C}$ be a stable $\infty$-category. We say that two objects $A$ and $A^{\prime}$ are equivalent if there is a map $A\to A^{\prime}$ that becomes an isomorphism in the homotopy category of $\mathcal{C}$. If $\iota\colon\mathcal{C}^{\prime}\to\mathcal{C}$ is a fully faithful functor, we often view $\mathcal{C}^{\prime}$ as a subcategory of $\mathcal{C}$: accordingly, we will usually denote the image under $\iota$ of an object $A$ of $\mathcal{C}^{\prime}$ simply by $A$ rather than $\iota(A).$ We will always assume that subcategories are closed under equivalence. That is, if $\mathcal{C}^{\prime}$ is a full subcategory of $\mathcal{C},$ $A$ is an object of $\mathcal{C}^{\prime},$ and $A^{\prime}$ is an object of $\mathcal{C}$ which is equivalent to $A,$ we will always assume that $A^{\prime}$ lies in $\mathcal{C}^{\prime}$ as well.

Recall that if $\mathcal{D}$ is a full subcategory of $\mathcal{C},$ $(\mathcal{D})^{\bot}$ denotes the right orthogonal of $\mathcal{D}$, i.e. the full subcategory of $\mathcal{C}$ consisting of the objects $B$ such that the Hom-space $\mathrm{Hom}_{\mathcal{C}}(A,B)$ is contractible for every object $A\in\mathcal{D}$. Let $\{\mathcal{C}_{1},\ldots,\mathcal{C}_{n}\}$ be a finite collection of stable subcategories of $\mathcal{C}$ such that, for all $1\leq i\leq n-1,$ $\,\mathcal{C}_{i}\subseteq(\mathcal{C}_{i+1})^{\bot}.$ Then we denote by $\langle\mathcal{C}_{1},\ldots,\mathcal{C}_{n}\rangle$ the smallest stable subcategory of $\mathcal{C}$ containing all the subcategories $\mathcal{C}_{i}$.

An exact sequence of stable $\infty$-categories is a sequence

[TABLE]

which is both a fiber and a cofiber sequence in $\mathrm{Cat}^{\mathrm{perf}}_{\infty}.$ This concept captures the classical notion of Verdier localization of triangulated categories in the setting of $\infty$-categories: as shown in Section 5.1 of [4], (3) is an exact sequence if and only if the sequence of homotopy categories

[TABLE]

is a classical Verdier localization of triangulated categories (up to idempotent-completion of $\mathrm{Ho}(\mathcal{C})$). This implies in particular that the fully-faithfulness of a functor between stable $\infty$-categories can be checked at the level of homotopy categories.

The functor $F$ admits a right adjoint $F^{R}$ exactly if $G$ admits a right adjoint $G^{R}$, and similarly for left adjoints. This is proved in [29, Proposition 4.9.1] for triangulated categories but the extension to $\infty$-categories is straightforward. If $F$ (or equivalently $G$) admits a right adjoint we say that (3) is a split exact sequence. In this case the functor $G^{R}$ is fully faithful and we have that $\mathcal{B}=\langle G^{R}(\mathcal{C}),\mathcal{A}\rangle.$ As we indicated earlier, since $G^{R}$ is fully faithful we will drop it from our notations whenever this is not likely to create confusion: thus we will denote $G^{R}(\mathcal{C})$ simply by $\mathcal{C},$ and write $\mathcal{B}=\langle\mathcal{C},\mathcal{A}\rangle.$

2.4. Preordered semi-orthogonal decompositions

In this section we introduce preordered semi-orthogonal decompositions of $\infty$-categories. This concept was also discussed in [3]. See [31] for a survey of semi-orthogonal decompositions (sod-s). Let $\mathcal{C}$ be a stable $\infty$-category, and let $P$ be a preordered set. Consider a collection of full stable subcategories $\iota_{x}\colon\mathcal{C}_{x}\longrightarrow\mathcal{C}$ indexed by $x\in P$.

Definition 2.10**.**

We say that the subcategories $\mathcal{C}_{x}$ form a preordered semi-orthogonal decomposition (psod) of type $P$ if they satisfy the following three properties.

• •

For all $x\in P,$ $\mathcal{C}_{x}$ is a non-zero admissible subcategory: that is, the embedding $\iota_{x}$ admits a right adjoint and a left adjoint, which we denote by

[TABLE]

• •

If $y<_{P}x$, i.e. $y\leq_{P}x,$ and $x\not=y,$ then $\mathcal{C}_{y}\subseteq\mathcal{C}_{x}^{\bot}.$

• •

$\mathcal{C}$ is the smallest stable subcategory of $\mathcal{C}$ containing all the subcategories $\mathcal{C}_{x},$ $x\in P.$

Definition 2.11**.**

If $\mathcal{C}$ is equipped with a psod of type $(P,\leq)$, we write $\mathcal{C}=\langle\mathcal{C}_{x},x\in(P,\leq)\rangle.$

Definition 2.12**.**

Let $\mathcal{C}=\langle\mathcal{C}_{x},x\in(P,\leq)\rangle$ and $\mathcal{D}=\langle\mathcal{D}_{y},y\in(Q,\leq)\rangle$ be categories equipped with psod-s. Let $F:\mathcal{C}\to\mathcal{D}$ be a fully-faithful functor. We say that $F$ is compatible with the psod-s if for all $x\in P$ there exists $y\in Q$ such that $F(\mathcal{C}_{x})=\mathcal{D}_{y}$.

Remark 2.13**.**

The definition of psod makes sense also for classical triangulated categories, and not just for $\infty$-categories. We mentioned in Section 2.3 that a sequence of stable $\infty$-categories is exact if and only if the corresponding sequence of homotopy categories is a Verdier localization. Also, $F\colon\mathcal{A}\leftrightarrows\mathcal{B}\nobreak\mskip 6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0muG$ is an adjoint pair of functors between stable $\infty$-categories if and only if

[TABLE]

is an adjoint pair between the homotopy categories. As a consequence psod-s can be checked at the level of homotopy categories: that is, a collection of admissible subcategories $\mathcal{C}_{x}$ gives a psod of type $P$ of the $\infty$-category $\mathcal{C}$ if and only if $\mathrm{Ho}(\mathcal{C}_{x})$ gives a psod of type $P$ of the triangulated category $\mathrm{Ho}(\mathcal{C}).$

Definition 2.14**.**

Let $P$ and $Q$ be preordered sets. The join of $P$ and $Q$ is the preordered set

[TABLE]

defined as follows. Let $x$ and $y$ be in $P\coprod Q$:

• •

if $x,y\in P,$ then $x\leq_{P*Q}y$ if and only if $x\leq_{P}y$;

• •

if $x,y\in Q,$ then $x\leq_{P*Q}y$ if and only if $x\leq_{Q}y$;

• •

if $x\in P$ and $y\in Q,$ then $x\leq_{P*Q}y$.

Lemma 2.15**.**

Let $\,\,\mathcal{A}\stackrel{{\scriptstyle F}}{{\longrightarrow}}\mathcal{B}\stackrel{{\scriptstyle G}}{{\longrightarrow}}\mathcal{C}\,\,$ be an exact sequence of stable idempotent-complete $\infty$-categories. Assume that $\mathcal{A}$ is admissible, and that $\mathcal{A}$ and $\mathcal{C}$ carry psod-s of type $P_{\mathcal{A}}$ and $P_{\mathcal{C}},$ respectively. Then $\mathcal{B}$ carries a canonical psod of type $\,\,P_{\mathcal{B}}:=P_{\mathcal{A}}*P_{\mathcal{C}}$ such that for all $x\in P_{\mathcal{B}}$ the component $\mathcal{B}_{x}$ is equivalent to $\mathcal{A}_{x},$ if $x$ is in $P_{\mathcal{A}},$ and to $\mathcal{C}_{x},$ if $x$ is in $P_{\mathcal{C}}.$

Proof.

It follows from the assumptions that the functor $G$ has a right adjoint $G^{R}\colon\mathcal{C}\to\mathcal{B}$. The admissible subcategories $\mathcal{C}_{x},$ $x\in P_{\mathcal{C}},$ are admissible subcategories of $\mathcal{B}$ under the image of $G^{R},$ and they are all right orthogonal to the image of $\mathcal{A}$ under $F$. In particular, they are right orthogonal to $F(\mathcal{A}_{x}),$ $x\in P_{\mathcal{A}},$ and this concludes the proof. ∎

2.5. The Chern character

In section 5 we will study the Chern character in the logarithmic setting. For later reference, we give below an account of the Chern character that follows closely [4].

Let $\mathcal{S}_{\infty}$ be the $\infty$-category of spectra. It follows from [4] that the Chern character can be defined in the abstract setting of $\infty$-categories as a natural transformation between additive invariants

[TABLE]

where:

• •

$K(-)$ is the algebraic K-theory,

• •

$\mathrm{HH}(-)$ is the Hochschild complex viewed as an object in spectra.

As explained in Section 10 of [4] the Chern character (4) is uniquely determined by the choice of the element $1\in\mathrm{HH}(\mathrm{Perf}(\kappa))\simeq\kappa$. Assume now that $\kappa$ is a field of characteristic $0.$ Then the Chern character (4) captures the ordinary de Rham Chern character, we refer to [8] for a thorough discussion of these aspects. More precisely, let $X$ be a smooth and proper scheme over $\kappa$. Denote by $\mathrm{H}_{\mathrm{dR}}(X)$ be the de Rham cohomology of $X,$ which is the hypercohomology of the de Rham complex. The HKR theorem gives an isomorphism $\,\mathrm{HH}_{0}(\mathrm{Perf}(X))\cong\bigoplus_{k\geq 0}\mathrm{H}_{\mathrm{dR}}^{2k}(X)\,$. Then the composition

[TABLE]

recovers the ordinary Chern character taking values in the even de Rham cohomology of $X$.

3. Perfect complexes over infinite root stacks

In this section we construct sod-s for infinite root stacks. In 3.1 and 3.2 we treat separately the case of root stacks of a single Cartier divisor, and of a simple normal crossings divisor with an arbitrary number of components. We start by reviewing the results obtained in [25] and [3] for finite root stacks of simple normal crossings divisors. We construct recursively compatible sod-s for the $\infty$-categories of perfect complexes of these root stacks, for a cofinal subset of indices. This will be key to constructing sod-s on the infinite root stack.

In Section 4 we explain how to extend these results beyond the normal crossing case: in Section 4.1 we extend our investigation to root stacks of general (not necessarily simple) normal crossings divisors, and in Section 4.2 we discuss the case of log stacks with simplicial log structure.

3.1. Root stacks of a regular divisor

Let $X$ be an algebraic stack, and $D\subset X$ a regular Cartier divisor. We use the notations of Section 2.1.1 in the preliminaries. Recall in particular that we denote by by $g_{r^{\prime},r}$ the projection $\sqrt[r^{\prime}]{(X,D)}\to\sqrt[r]{(X,D)}$ for $r\mid r^{\prime}$, by $D_{r}$ the universal Cartier divisor on $\sqrt[r]{(X,D)}$, i.e. the reduction of the preimage $g_{r,1}^{-1}(D)\subset\sqrt[r]{(X,D)}$. Further, we denote by $i_{r}\colon D_{r}\to\sqrt[r]{(X,D)}$ the closed embedding.

Lemma 3.1**.**

The category $\mathrm{Perf}(D_{r})$ splits as the direct sum of $r$ copies of $\mathrm{Perf}(D).$ More precisely, if $\mathbb{Z}_{r}$ is the Cartier dual of $\mu_{r},$ there are natural equivalences

[TABLE]

Proof.

This is a well-known fact, that applies more generally to gerbes banded by a diagonalizable group scheme, so we limit ourselves to a brief sketch. Equivalence $(1)$ comes from the character decomposition of $\mathrm{Perf}(D_{r}).$ If $\chi$ is in $\mathbb{Z}_{r},$ let $\kappa(\chi)$ be the corresponding $\mu_{r}$-representation. Then equivalence $(2)$ maps the twisted structure sheaf $\mathcal{O}_{D}^{\chi}$ to $\mathcal{O}_{D}\otimes\kappa(\chi).$ ∎

Lemma 3.2**.**

Let $r,r^{\prime}\in\mathbb{N},$ and $r\mid r^{\prime}.$ Then the pull-back functors

[TABLE]

are fully faithful.

Proof.

Since $g_{r^{\prime},r}$ is a relative coarse moduli space map, the natural map $\mathcal{O}_{\sqrt[r]{(X,D)}}\longrightarrow g_{r^{\prime},r,*}\mathcal{O}_{\sqrt[r^{\prime}]{(X,D)}}$ is an equivalence in $\mathrm{Perf}(\sqrt[r]{(X,D)}).$ This easily implies that $g_{r^{\prime},r}^{*}$ is fully faithful by the adjunction with ${g_{r^{\prime},r}}_{*}$ and the projection formula (see for instance [3, Lemma 4.4]). ∎

Let $\mathbb{Z}_{r}^{*}=\mathbb{Z}_{r}\setminus\{0\}$ be the set of non-trivial characters of $\mu_{r}.$ In order to keep track of indices it will be convenient to identify $\mathbb{Z}_{r}$ and $\mathbb{Z}_{r}^{*}$ with subsets of $\mathbb{Q}/\mathbb{Z}=\mathbb{Q}\cap(-1,0]$ as follows:

[TABLE]

We equip $\mathbb{Z}_{r}$ and $\mathbb{Z}_{r}^{*}$ with the total order $\leq$ given by

[TABLE]

Here $-\frac{k}{r}$ should be thought of as the element $r-k$ in $\{1,\ldots,r\}$, equipped with the standard ordering. This (perhaps unusual) identification will be convenient when we pass to the limit for $r\to\infty$.

Following Theorem 4.7 of [3], for every $\chi\in\mathbb{Z}_{r}^{*}$ we consider the fully faithful embedding

[TABLE]

given by the composite

[TABLE]

Note that if we see $\chi$ as an element of $\{1,\ldots,r\}$, in the notation of [3] the objects of the image of $\Phi_{\chi}$ are actually equipped with the action corresponding to the character $-\chi\equiv r-\chi\in\mathbb{Z}_{r}$.

Remark 3.3**.**

Each individual summand $\mathrm{Perf}(D_{r})_{\chi}$ of $\mathrm{Perf}(D_{r})$ embeds fully faithfully in the category $\mathrm{Perf}(\sqrt[r]{(X,D)})$ via $\Phi_{\chi},$ which is a restriction of $i_{r,*}$ to $\mathrm{Perf}(D_{r})_{\chi}.$ However the functor $i_{r,*}$ itself is not fully faithful.

We introduce the following notations:

• •

Let $\chi$ be in $\mathbb{Z}_{r}.$ If $\chi\neq 0$ we denote by $\mathcal{A}_{\chi}\subset\mathrm{Perf}(\sqrt[r]{(X,D)})$ the image of $\mathrm{Perf}(D)$ under $\Phi_{\chi},$ and we denote by $\mathcal{A}_{0}\subset\mathrm{Perf}(\sqrt[r]{(X,D)})$ the image of $\mathrm{Perf}(X)$ under $g_{r,1}^{*}.$

• •

We denote by $Q_{r}$ the subcategory of $\mathrm{Perf}(\sqrt[r]{(X,D)})$ generated by the subcategories $\mathcal{A}_{\chi}$ for $\chi\in\mathbb{Z}_{r}^{*}$.

The next theorem is proved in [3] using the language of classical triangulated categories, but the proof applies without variations to the $\infty$-setting.

Proposition 3.4** ([3, Theorem 4.7]).**

---

1. (1)

The category $\mathcal{A}_{0}$ is an admissible subcategory of $\mathrm{Perf}(\sqrt[r]{(X,D)})$. 2. (2)

The right orthogonal of $\mathcal{A}_{0}$ inside $\mathrm{Perf}(\sqrt[r]{(X,D)})$ is the subcategory $Q_{r}$ introduced above. There is a psod of type $(\mathbb{Z}_{r}^{*},\leq)$

[TABLE] 3. (3)

By items (1) and (2), the category $\mathrm{Perf}(\sqrt[r]{(X,D)})$ has a psod of type $(\mathbb{Z}_{r},\leq)$

[TABLE]

Consider the directed system of root stacks $\{\sqrt[n!]{(X,D)}\}_{n\in\mathbb{N}}$ whose indices are the factorials, with the natural projections

[TABLE]

We point out once again that this subsystem of indices is identified with $\mathbb{N}$ with its standard ordering, i.e. if $n\leq m$ we have a map $\sqrt[m!]{(X,D)}\xrightarrow{g_{m!,n!}}\sqrt[n!]{(X,D)}$, and these are compatible in the obvious sense. We will inductively construct a tower of compatible sod-s on the categories of perfect complexes on these root stacks. This is done in Proposition 3.5 below by applying iteratively Proposition 3.4. For $n\geq 2$, the sod-s we will construct on $\mathrm{Perf}(\sqrt[n!]{(X,D)})$ will be different from the sod-s on the category of perfect complexes on the root stacks of $(X,D)$ given directly by Proposition 3.4 (see Example 3.6).

We will equip the set $\mathbb{Q}/\mathbb{Z}=\mathbb{Q}\cap(-1,0]$ with a total order $\leq^{!}$ which is not the restriction of the usual ordering of the real numbers. First of all we define the order $\leq^{!}$ on $\mathbb{Z}_{n!}$ recursively, as follows.

• •

On $\mathbb{Z}_{2!}=\{-\frac{1}{2},0\}$ we set $-\frac{1}{2}<^{!}0$.

• •

Having defined $\leq^{!}$ on $\mathbb{Z}_{(n-1)!}$, let us consider the natural short exact sequence

[TABLE]

where $\mathbb{Z}_{n}=\{-\frac{n-1}{n},\ldots,-\frac{1}{n},0\}$ is equipped with the standard order $\leq$ described above. Given two elements $a,b\in\mathbb{Z}_{n!}$, we set $a\leq^{!}b$ if either $\pi_{n}(a)<\pi_{n}(b)$, or $\pi_{n}(a)=\pi_{n}(b)$ and $a\leq^{!}b$ in $\mathbb{Z}_{(n-1)!}$, where we are identifying the fiber $\pi_{n}^{-1}(\pi_{n}(a))\subseteq\mathbb{Z}_{n!}$ with $\mathbb{Z}_{(n-1)!}$ in the canonical manner.

For example, on $\mathbb{Z}_{3!}=\{-\frac{5}{6},-\frac{4}{6},-\frac{3}{6},-\frac{2}{6},-\frac{1}{6},0\}$, the resulting ordering is described by

[TABLE]

Now every element in $\mathbb{Q}\cap(-1,0]$ can be written as $-\frac{p}{n!}$ for some $p\in\mathbb{N}$ and $n\in\mathbb{N}\setminus\{0\}.$ This expression is unique if we require $n$ to be as small as possible, and we call this the normal factorial form. Let $\chi=-\frac{p}{n!},\chi^{\prime}=-\frac{q}{m!}\in\mathbb{Q}\cap(-1,0]$ be in normal factorial form. We write $\chi<^{!}\chi^{\prime}$ if:

• •

$n>m$, or

• •

$n=m$ and $-\frac{p}{n!}<^{!}-\frac{q}{n!}$ in $\mathbb{Z}_{n!}$.

For example, with this ordering we have $-\frac{1}{24}<^{!}-\frac{2}{6}<^{!}-\frac{4}{6}<^{!}-\frac{1}{2}$.

Proposition 3.5**.**

---

For every $n\in\mathbb{N}$ the category $\mathrm{Perf}(\sqrt[n!]{(X,D)})$ has a psod of type $(\mathbb{Z}_{n!},\leq^{!})$*

[TABLE]

where

• •

$\mathcal{A}_{0}^{!}\simeq\mathrm{Perf}(X)$, and

• •

$\mathcal{A}_{\chi}^{!}\simeq\mathrm{Perf}(D)$* for all $\chi\in\mathbb{Z}_{n!}^{*}$.*

Further, for all $n\in\mathbb{N}$ the functor

[TABLE]

is compatible with the psod-s.

We remark again that for a given $\chi\in\mathbb{Z}_{n!}^{*}$, it is not necessarily the case that $\mathcal{A}_{\chi}=\mathcal{A}_{\chi}^{!}$ as subcategories of $\mathrm{Perf}(\sqrt[n!]{(X,D)})$ (see Example 3.6 below).

Proof.

We construct the sod with the desired properties on $\mathrm{Perf}(\sqrt[n!]{(X,D)})$ inductively.

Basis: $n=2.$ We take the sod on $\mathrm{Perf}(\sqrt[2!]{(X,D)})$ given by Proposition 3.4. We take $\mathcal{A}_{0}$ and $\mathcal{A}_{\frac{1}{2}}$ to be the same as in Proposition 3.4. This clearly gives a psod of $\mathrm{Perf}(\sqrt[2]{(X,D)})$ of type $(\mathbb{Z}_{2},\leq^{!})$ satisfying the properties of the claim.

Inductive step: $n-1\to n.$ Recall that there is a natural identification

[TABLE]

Further, the projection

[TABLE]

coincides under this identification with the map $g_{n!,(n-1)!}.$ We then apply Proposition 3.4 to (5). This yields a psod of type $(\mathbb{Z}_{n},\leq)$

[TABLE]

where:

• •

the subcategory $\mathcal{B}_{0}\simeq\mathrm{Perf}(\sqrt[(n-1)!]{(X,D)})$ is given by the image of $g_{n!,(n-1)!}^{*}$, and

• •

the subcategory $\mathcal{B}_{\zeta}\simeq\mathrm{Perf}(D_{n!})_{\zeta}\simeq\mathrm{Perf}(D_{(n-1)!})$ is given by the image of $\Phi_{\zeta}$.

Now note that by Lemma 3.1 for all $\zeta\in\mathbb{Z}_{n}^{*}$ the category $\mathcal{B}_{\zeta}$ splits as a direct sum of categories labelled by characters in $\mathbb{Z}_{(n-1)!},$ that is:

[TABLE]

We identify the subcategories $\mathcal{B}_{\zeta,\xi}$ with the factors $\mathcal{A}_{\chi}^{!}$ for $\chi\in\mathbb{Z}_{n!}^{*}$ appearing in the statement of the proposition by setting

[TABLE]

Note that we can make sense of the expression $\frac{\zeta}{(n-1)!}+\xi$ as an element of $\mathbb{Z}_{n!}^{*}$ because we have identified the sets of characters $\mathbb{Z}_{n!}$, $\mathbb{Z}_{(n-1)!}$ and $\mathbb{Z}_{n}$ with subsets of $\mathbb{Q}\cap(-1,0].$ It is easy to verify that we can write as sums of the form $\frac{\zeta}{(n-1)!}+\xi$ exactly the elements of $\mathbb{Z}_{n!}$ which are in the complement of $\mathbb{Z}_{(n-1)!},$ that is

[TABLE]

Thus we conclude that the subcategory $Q_{n}$ has a psod of type $(\mathbb{Z}_{n!}\setminus\mathbb{Z}_{(n-1)!},\leq^{!})$

[TABLE]

such that $\mathcal{A}_{\chi}^{!}\simeq\mathrm{Perf}(D)$ for all $\chi\in\mathbb{Z}_{n!}\setminus\mathbb{Z}_{(n-1)!}.$

By the inductive hypothesis $\mathcal{B}_{0}\simeq\mathrm{Perf}(\sqrt[(n-1)!]{(X,D)})$ carries a psod of type $(\mathbb{Z}_{(n-1)!},\leq^{!})$

[TABLE]

such that:

• •

$\mathcal{A}_{0}^{!}\simeq\mathrm{Perf}(X)$, and

• •

$\mathcal{A}_{\chi}^{!}\simeq\mathrm{Perf}(D)$ for all $\chi\in\mathbb{Z}_{(n-1)!}^{*}$.

We can thus write

[TABLE]

Note that the we have a canonical isomorphism of ordered sets

[TABLE]

Thus, by Lemma 2.15 the category $\mathrm{Perf}(\sqrt[n!]{(X,D)})$ carries a psod of type $(\mathbb{Z}_{n!},\leq^{!}),$

[TABLE]

The compatibility with the psod-s of the pull-back along root maps follows by construction. This concludes the proof. ∎

Example 3.6**.**

For $n=2$, by construction our psod coincides with the one given by Proposition 3.4. For $n=3$ though, the psod given by that proposition for $\sqrt[3!]{(X,D)}$ looks like

[TABLE]

(where $\mathcal{A}_{-\frac{k}{n}}$ is the factor $\Phi_{k}$ in Theorem 4.7 of [3]). The psod that was constructed in the previous proposition, on the other hand, has the form

[TABLE]

where $\mathcal{B}_{-\frac{k}{3}}\simeq\mathrm{Perf}(D_{2})\simeq\mathrm{Perf}(D)_{-\frac{1}{2}}\oplus\mathrm{Perf}(D)_{0}$, and $\mathrm{Perf}(\sqrt[2]{(X,D)})=\langle\mathcal{A}^{!}_{-\frac{1}{2}},\mathrm{Perf}(X)\rangle$, embedded in $\mathrm{Perf}(\sqrt[6]{(X,D)})$ via pullback along the projection $\sqrt[6]{(X,D)}\to\sqrt[2]{(X,D)}$.

Following the proof of the previous proposition, the first term $\mathcal{B}_{-\frac{2}{3}}$ gives us $\mathcal{A}^{!}_{-\frac{5}{6}}\oplus\mathcal{A}^{!}_{-\frac{2}{6}}$, the second term $\mathcal{B}_{-\frac{1}{3}}$ gives $\mathcal{A}^{!}_{-\frac{4}{6}}\oplus\mathcal{A}^{!}_{-\frac{1}{6}}$, and $\mathcal{A}^{!}_{-\frac{3}{6}}$ is defined as the image of $\mathcal{A}^{!}_{-\frac{1}{2}}\subseteq\mathrm{Perf}(\sqrt[2]{(X,D)})$. Overall, the psod looks like

[TABLE]

(note that this reflects exactly the ordering $\leq^{!}$ on $\mathbb{Z}_{3!}$).

In fact we could swap $\mathcal{A}^{!}_{-\frac{2}{6}}$ and $\mathcal{A}^{!}_{-\frac{4}{6}}$, and we have $\mathcal{A}_{-\frac{k}{6}}=\mathcal{A}^{!}_{-\frac{k}{6}}$ for all values of $k$ except $3$: the subcategory $\mathcal{A}^{!}_{-\frac{3}{6}}$ does not contain the structure sheaf of $D$ equipped with character $-\frac{3}{6}$, but only thickened versions of it.

Set $\mathbb{Q}/\mathbb{Z}^{*}:=\mathbb{Q}/\mathbb{Z}\setminus\{0\}.$

Proposition 3.7**.**

---

The category $\mathrm{Perf}(\sqrt[\infty]{(X,D)})$ has a psod of type $(\mathbb{Q}/\mathbb{Z},\leq^{!})$*

[TABLE]

where:

• •

$\mathcal{A}_{0}^{!}\simeq\mathrm{Perf}(X)$, and

• •

$\mathcal{A}_{\chi}^{!}\simeq\mathrm{Perf}(D)$* for all $\chi\in\mathbb{Q}/\mathbb{Z}^{*}$.*

Proof.

Factorials are cofinal in the filtered set of natural numbers ordered by divisibility. This together with Proposition 2.7 implies that $\mathrm{Perf}(\sqrt[\infty]{(X,D)})$ is the colimit of the directed system of fully-faithful embeddings

[TABLE]

By Proposition 3.5 the $n$-th category in the directed system (7) carries a psod of type $(\mathbb{Z}_{n!},\leq^{!}).$ Further the structure functors are compatible with the sod-s: the compatibility is witnessed by the inclusion of preordered sets

[TABLE]

which embeds the indexing set of the sod of $\mathrm{Perf}(\sqrt[(n-1)!]{(X,D)})$ into the indexing set of the sod of $\mathrm{Perf}(\sqrt[n!]{(X,D)})$.

By Proposition 2.4, $\mathrm{Perf}(\sqrt[\infty]{(X,D)})$ is given by the union of the categories making up the directed system (7). This implies that $\mathrm{Perf}(\sqrt[\infty]{(X,D)})$ carries a psod of type

[TABLE]

It is immediate to see that this sod satisfies the properties $(1)$ and $(2)$ from the statement. This concludes the proof. ∎

3.2. Root stacks of simple normal crossings divisors

As shown in [3], the categories of perfect complexes over the root stacks of simple normal crossing divisors carry a canonical sod. We review that result in a slightly different formulation, which is better adapted to our purposes. Further, we extend it to the infinite root stack. We start by fixing notations.

Let $X$ be an algebraic stack and $D$ be a simple normal crossings divisor on $X$. We denote by $D_{1},\ldots,D_{N}$ its irreducible components. As recalled in the preliminaries, this gives rise to a log stack $(X,D)$. Denote by $I$ the set $\{1,\ldots,N\}$. The stack $X$ carries a canonical stratification, given by the closed substacks of $D$ obtained as intersections of the $D_{i}$. If $J$ is a subset of $I$, we denote $D_{J}:=\cap_{j\in J}D_{j}$ if $J\neq\varnothing,$ and we set $D_{\varnothing}:=X$ otherwise. Let $\,\,S_{I}:=\{J\subseteq I\}$ be the power set of $I$, and $S_{I}^{*}$ denote the subset $S_{I}\setminus\{\varnothing\}$. We often equivalently regard $I$ as the set of irreducible components of $D$, $S_{I}$ as the set of strata of $(X,D)$, and $S_{I}^{*}$ as the set of the strata of positive codimension. We equip $S_{I}$ with a preorder $\leq$ that keeps track of the inclusions of strata, but it is finer than that. Namely we let $\leq$ be the coarsest preorder on $S_{I}$ with the following two properties:

• •

if $J\subseteq J^{\prime},$ then $J^{\prime}\leq J$, and

• •

if $d$ is the dimension of $X$, then the assignment $(S_{I},\leq)\to(\mathbb{N},\leq)$ mapping a subset $J$ to $d\,-|J|$ is an order-reflecting map.

By “order-reflecting map” we mean a map between preordered sets $f:P\to Q$ such that $f(p)\leq f(p^{\prime})$ implies $p\leq p^{\prime}$. Note in particular that these conditions impose both $J\leq J^{\prime}$ and $J^{\prime}\leq J$ if $|J|=|J^{\prime}|$.

Recall that the root stacks of $(X,D)$ are indexed by multi-indices $\,\vec{r}=(r_{1},\ldots,r_{N})\in\mathbb{N}^{N}.\,$ For elements $\vec{r},\vec{r^{\prime}}$ of $\mathbb{N}^{N}$ we write $\vec{r}=(r_{1},\ldots,r_{N})\;\mid\;(r_{1}^{\prime},\ldots,r_{N}^{\prime})=\vec{r^{\prime}}$ if $r_{1}\mid r_{1}^{\prime},\ldots,r_{N}\mid r_{N}^{\prime}.$ Recall also that the root stack $\sqrt[\vec{r}]{(X,D)}$ in this case can be realized as the limit of the diagram of stacks

[TABLE]

If $\vec{r},\vec{r^{\prime}}\in\mathbb{N}^{N},$ $\vec{r}\mid\vec{r^{\prime}},$ we denote the natural maps between root stacks by

[TABLE]

3.2.1. Root stacks and the strata of $(X,D)$

Let $\vec{r}\in\mathbb{N}^{N}$. We will use the following notation.

• •

Let $J=\{j_{1},\ldots,j_{k}\}\subseteq I$ be non-empty. We denote by $\vec{r}_{J}\in\mathbb{N}^{N}$ the index vector obtained from $\vec{r}$ by setting to $1$ all the entries whose index is not in $J$. In formulas, letting $\vec{e}_{j}$ be the size $N$ vector with $j$-th entry $1$ and all other entries equal to [math], we can write

[TABLE]

Note that $\vec{r}_{J}\mid\vec{r}$.

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With slight abuse of notation, for all $i\in I$, we denote by $D_{i,r_{i}}$ both the universal effective Cartier divisor on the stack $\sqrt[r_{i}]{(X,D_{i})}$ (obtained as reduction of the preimage of $D_{i}$), and its pull-back to $\sqrt[\vec{r}]{(X,D)}$. Whenever we use this notation we will specify which of the two meanings is the intended one.

We denote by $D_{J,\vec{r}}$ the limit of the diagram of stacks

[TABLE]

where the arrows are the embeddings $D_{j_{l},r_{j_{l}}}\subset\sqrt[\vec{r}]{(X,D)}$. In general $D_{J,\vec{r}}$ is not a gerbe over $D_{J}$, as $D_{J,\vec{r}}$ will have larger isotropy groups along the higher codimensional strata $S\in S_{I}$