Building Data for Stacky Covers

We define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2,1)-categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne--Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme S is defined over a field, we give a criterion for when a birational building datum comes from a tamely ramified cover for a finite abelian group scheme, generalizing a result of Biswas--Borne.


Introduction
The class of stacky covers contains flat (classical) root stacks and flat stacky modifications in the sense of [Ryd].Root stacks first appeared in [MO05], [AGV08], and [Cad07].It was used by Abramovich-Graber-Vistoli in [AGV08] to define Gromov-Witten theory of Deligne-Mumford stacks and by Cadman-Chen [CC08] when counting rational plane curves tangent to a smooth cubic.Root stacks may also be used in birational geometry.For instance, Matsuki-Olsson used root stacks in the logarithmic setting to interpret the Kawamata-Viehweg vanishing theorem as an application of Kodaira vanishing for stacks [MO05].Root stacks and stacky modifications where also used by Rydh in [Ryd] to prove compactification results for tame Deligne-Mumford stacks and by Bergh in [Ber17] when constructing a functorial destackification algorithm for tame stacks with diagonalizable stabilizers.Bergh-Rydh also extended the latter result to remove the assumption that stabilizers are diagonalizable [BR19].The aim of this paper is to shed more light on these constructions in the flat case.We will do so by classifying stacky covers in terms of stacky building data à la Pardini [Par91].
A stacky cover π : X → S of a scheme S consists of a tame stack X which has finite diagonalizable stabilizers at geometric points, together with a morphism π : X → S which is (1) flat, proper, of finite presentation, (2) a coarse moduli space, and The author was supported by the Swedish Research Council 2015-05554 and the Knut and Alice Wallenberg Foundation 2021.0279.

1
(3) for any morphism of schemes T → S, the base change π| T : X T → T has the property that (π| T ) * takes invertible sheaves to invertible sheaves.For instance, if X → S is a flat stacky modification, that is, flat, proper, locally of finite presentation, and birational with finite diagonalizable stabilizers, then X → S is a stacky cover.We prove a classification of stacky covers in terms of stacky building data.To a stacky cover π : X → S we associate an étale sheaf of abelian groups A over S and construct a 2-cocycle . We show that X may be thought of as the stack parametrizing 1-cochains with boundary f X .
From A we construct two quasi-fine étale sheaves of monoids P A , Q A , and a flat Kummer homomorphism γ A : P A → Q A .To the 2-cocycle f X we associate a symmetric monoidal functor L : P A → Div S ét where Div S ét denotes the restriction of [A 1 S /G m,S ] to the small étale site of S. Hence we get a diagram (1) and we refer to this as a Deligne-Faltings datum.We denote by S (A,L) the associated root stack.The main results are the following: Theorem 7.17.Let π : X → S be a stacky cover.Then there exists a canonical (up to canonical isomorphism) building datum (A, L) where A = Pic X /S is the relative picard functor, and a canonical isomorphism of stacks X → S (A,L) where S (A,L) is the root stack associated to the building datum (A, L).
Theorem 8.8.We have an equivalence of (2,1)-categories StCov ≃ StData between the category of stacky covers and the category of stacky building data.
A stacky cover will étale locally on S look like a quotient stack of a ramified Galois cover X for a diagonalizable group D(A), where A is a finite abelian group.Such covers have been studied for example in [Par91] (Galois covers that are generically torsors) and [Ton14] (general setting) and can be described combinatorially by giving a line bundle L λ for each λ ∈ A together with global sections s λ,λ ′ ∈ Γ(S, L −1 λ ⊗ L −1 λ ′ ⊗ L λ+λ ′ ) corresponding to the multiplication in O X .These data are required to satisfy the appropriate axioms to constitute an associative and commutative algebra (see Remark 2.2).This suggests that the quotient stack [X/D(A)] can also be described in a combinatorial way using line bundles and sections, or more precisely, as a root stack.Using constructions in [Ton14] we show that the group A gives rise to two (constant) quasi-fine and sharp monoids P A and Q A with a flat Kummer homomorphism between them.These sit in an exact sequence of monoids 0 → P A → Q A → A → 0 which is the universal free extension of A. From the data of the cover X we can then construct a symmetric monoidal functor L X : P A → Div S ét such that the root stack of the diagram (compare with Diagram (1)) is isomorphic to [X/D(A)].We will use root stacks in the language of Deligne-Faltings structures as in [BV12].The monoids and the symmetric monoidal functor in the main theorem are constructed intrinsically on X using the relative Picard functor Pic X /S .When X is Deligne-Mumford then π * Pic X /S ∼ = D(I X ), where D(I X ) is the Cartier dual of the inertia stack.
The category of quasi-coherent sheaves on a root stack X /S associated to a Deligne-Faltings datum (P, Q, L) is equivalent to the category of parabolic sheaves on S with respect to (P, Q, L).When S is defined over a field and is geometrically reduced and geometrically connected, we give a criterion for when a birational building datum comes from a ramified G-cover (Definition 9.8), for some finite abelian group scheme G over k, generalizing the main result of [BB17].Our theorem looks as follows: Theorem 9.18.Let S be a scheme proper over a field k and assume that S is geometrically connected and geometrically reduced.Let (A, L) be a birational building datum and (P A , Q A , L) the associated Deligne-Faltings datum.Then the following are equivalent: (1) There exists a finite abelian group scheme G over k and a ramified G-cover X → S with birational building datum (A, L); (2) For every geometric point s in the branch locus, we have that (i) the map Γ(S, A) → A s is surjective, and (ii) for every λ ∈ A s, there exists an essentially finite, basic, parabolic vector bundle (E, ρ) on (S, P A , Q A , L) such that the morphism is not surjective, where the direct sum is over all λ ′ ∈ Γ(S, A) such that λ ′ s = 0.
Suppose that there exists a finite abelian group scheme G over k and a ramified G-cover X → S with ramification datum (A, L) as in Theorem 9.18.If X = S (A,L) = S (PA,QA,L) is the associated root stack then it follows that X ≃ [X/G].
Organization.In Section 2 we study the theory of ramified Galois covers for diagonalizable group schemes.The reader who is familiar with ramified covers may skip ahead.
In Section 3 we recall the theory developed in [BV12] involving Deligne-Faltings structures and root stacks.We also review some properties of monoids and symmetric monoidal categories.In the end we investigate what it means for a root stack to be flat in terms of the monoids defining it.
In Section 4 we define and prove statements about the universal monoids P A and Q A that we use use to model the local charts of our stacky covers.For instance, we show that the monoids P A and Q A associated to A are quasi-fine and sharp and that the action of P A on Q A is free.When A is an abelian group and P a monoid, we identify free extensions of A by P and 2-cocycles of A with values in P .We show that there is a universal 2-cocycle A × A → P A corresponding to a universal free extension 0 In Section 5 we generalize the theory developed in Section 4 to the setting where P and Q are replaced by symmetric monoidal categories.We define the stack parametrizing 1-cocycles with a fixed coboundary.
In Section 6 we look at the local structure of the stacks considered in this paper and show that to every ramified D(A)-cover X → S we may associate a Deligne-Faltings datum (P A , Q A , L X ) and that the root stack S (PA,QA,LX ) is isomorphic to [X/D(A)].
In Section 7 we show how to realize a stacky cover as a root stack using the relative Picard functor.
In Section 8 we define stacky building data and show that there is an equivalence of (2,1)-categories between the category of stacky covers and the category of stacky building data.
In Section 9 we generalize the result of Biswas-Borne and give a criterion for when building data on a scheme over a field comes from a ramified abelian cover.
In Appendix A we study the Cartier dual D(H) of a non-flat closed subgroup H of a multiplicative group.
We use this in Section 7 to show that π * D(I X ) ∼ = Pic X /S when X is Deligne-Mumford.
Notation and conventions.The letter S will always denote the base scheme which we assume to be locally Noetherian.The letter A will always denote an abelian group.If A is an abelian group, λ ∈ A, and B is an When X is a stack and f : T → X is an object, we write Stab(T ) or Stab(f ) for the sheaf of groups Aut f .The unit object of a symmetric monoidal category will be denoted by ½.
Acknowledgements.I want to thank my PhD supervisor David Rydh for many invaluable discussions and his great enthusiasm for the subject.I would like to thank Niels Borne, Magnus Carlson, Martin Olsson, Fabio Tonini, and Angelo Vistoli for useful discussions.I would also like to thank the anonymous referee for many good suggestions for improvement.

Ramified covers
Throughout this section we will always assume that A is an abelian group and G = D(A) the corresponding finite diagonalizable group over the base scheme S.This means that Definition 2.1 ([Ton14, Definition 2.1]).Let G → S be a finite flat diagonalizable group scheme of finite presentation.A G-cover over S is a finite locally free morphism f : X → S together with an action of G such that there exists an fppf cover {U i → S} and an isomorphism of where the comodule structure on the right hand side is the regular representation.
Remark 2.2.This means that we have a splitting where each L λ is a line bundle and L 0 = O S .We also have multiplication morphisms which we think of as global sections These global sections will have the following properties: (1) Note that equality a = b here means that the element a is sent to b under the canonical isomorphism of line bundles.For instance, s λ,λ ′ = s λ ′ ,λ means that s λ,λ ′ → s λ ′ ,λ under the canonical isomorphism Let S be a scheme.A generalized effective Cartier divisor is a pair (L, s) consisting of (1) a line bundle L on S and (2) a global section s ∈ Γ(S, L).
Remark 2.4.Note that each pair (L −1 λ ⊗ L −1 λ ′ ⊗ L λ+λ ′ , s λ,λ ′ ) forms a generalized effective Cartier divisor.Note that the data of a generalized effective Cartier divisor (L, s) is equivalent to the data of a morphism of stacks Remark 2.5.By Remark 2.2 we may replace the fppf cover in Definition 2.1 by a Zariski cover.This is however not always possible if one allows non-diagonalizable group schemes as in [Ton13, Definition 2.1.2].
Remark 2.7.Any finite locally free morphism f : X → S of rank 2 is a µ 2 -cover if 2 is invertible in Γ(S, O S ).Indeed, there is a trace map T : f * O X → O S sending a section x to the trace of the matrix corresponding to multiplication by x.The composition O S → f * O X → O S is multiplication by 2 and if 2 is invertible, we get that This can be checked on stalks so we may assume that L is trivial.Take x ∈ Γ(s, L).Then multiplication by x is given by a 2 × 2-matrix 0 b x d since the multiplication O S ⊗ L → f * O X is just the module action, and hence lands in L. But L = ker T and hence d = 0. Hence we conclude that X → S is a µ 2 -cover.
Example 2.8.Here is a list of examples of ramified covers.
(1) The map on spectra induced by the inclusion Z → Z[x]/(x 2 − 2) is a µ 2 -cover when x has weight 1.
(2) The map on spectra induced by the inclusion C[s] → C[s, x, y]/(x 2 − sy, y 2 − sx, xy − s 2 ) is a µ 3 -cover when x has weight 1 and y has weight 2.
) with multiplication given by x 2 ∈ Γ(S, O S (2)).(4) In view of Remark 2.7, any degree 2 finite surjective morphism of varieties X → S over an algebraically closed field, where X is Cohen-Macaulay and S is regular, is a µ 2 -cover (flatness follows from [Eis95, Corollary 18.17]).(5) In particular, a K3 surface obtained as a double cover of P 2 branched along a sextic is a µ 2 -cover.
See also Example 2.19 for a non-example, which could be mistaken for a G-cover in the sense of Definition 2.1.

Covers and 2-cocycles.
Definition 2.9 ([Pat77b, Pat77a, Pat18]).Let A be an abelian group.A (commutative) 2-cocycle of A with values in a monoid P is a function f : A × A → P that satisfies the following properties: (1) Remark 2.10.Note that the set of 2-cocycles of A × A → P form a monoid under pointwise addition.
Remark 2.12.There is a bijection between the set of 2-cocycles A × A → N and the set Hom(P A , N) of rays as in [Ton14, Notation 3.11], where P A is the universal monoid we define in Definition 4.10.This will be explained in detail in Section 4.
Recall that if L is a line bundle on a scheme S then we have a bijection ) such that us = s ′ .Let S be a normal scheme and let X → S be a D(A)-cover with branch locus B = i∈I D i (union of irreducible components with I finite), which is generically a torsor.Then the global sections which we refer to as the 2-cocycle of the cover X.The cover X is determined by f X (see Proposition 2.13). If Structure of D(A)-covers over a normal scheme.Let p : X → S be a D(A)-cover with multiplication in p * O X given by Assume that the cover is generically a torsor.Then the global sections s λ,λ ′ are all regular since they are generically isomorphisms.If S is normal then cyc: Div(S) → WDiv(S) is injective and hence cyc( is an affine ramified G-cover.For each λ ∈ A we let v λ be a generator for the graded piece of O X,C corresponding to λ (the graded pieces are free since we are over a local ring).
Proposition 2.13.With the setup just described, we have an isomorphism sending s λ,λ ′ to the canonical global section.Let s be a uniformizer of O S,C .Then ord C (s λ,λ ′ ) can be determined by the formula ) determines an effective Cartier divisor which in turn gives the Weil divisor since s λ,λ ′ has support in B. This proves the first part.
To prove the second part we consider the cover Spec O X,C → Spec O S,C .We have This proves the second part since O X,C is free over O S,C .
Remark 2.14.Note that we could replace O S,C by its strict henselization [SP, Tag 0AP3].
Remark 2.15.The function In [Par91], Pardini gives an explicit description of Proposition 2.13 in the case when X is normal and S is smooth over over an algebraically closed field k whose characteristic does not divide |A|.In this setting, if C is an irreducible component of the branch locus B, then the stabilizer group of a component in p −1 (C) is always cyclic [Par91, Lemma 1.1] (i.e., its group of characters is cyclic).For every such C, the corresponding stabilizer group D(N ) acts via some character ψ ∈ N ∼ = D(D(N )) (which generates N ) on the cotangent space m T /m 2 T , where T is any component of p −1 (C).The character ψ is independent of the choice of T .This means that to every component C we may associate a cyclic group together with a generator.Hence we may write where we sum over cyclic quotients A ։ N and generators ψ ∈ N .Let i : A → N be the dual of the inclusion D(N ) → D(A) composed with the map N → N defined by x → min{a : ψ a = x}.For λ, λ ′ ∈ A (and N , ψ as above), Pardini defines We have that ε N,ψ (−,−) is a 2-cocycle and the following theorem is part of [Par91, Theorem 2.1].Theorem 2.16.Let C be a component with cyclic group N and generator ψ.Let For completeness, we give a proof.
Proof.Let φ : A → N be the dual of the inclusion D(N ) → D(A).The cover where the first arrow is a totally ramified D(N )-cover and the second is a trivial D(K)-torsor for K = ker φ (since char(k) Branch locus.In this subsection we consider only D(A)-covers X → S such that there is an open dense subscheme U ⊆ S such that U × S X → U is a D(A)-torsor.
We define the ramification locus R ⊂ X of a G-cover π : X → S as the set of points where X is ramified, i.e., the set of points in X where Ω X/S do not vanish.Hence there is a canonical scheme structure on the ramification locus, namely Spec O X / Ann(Ω X/S ) .
However, there are several ways to put a scheme structure on the branch locus (the set-theoretic image of the ramification locus) B ⊂ S of a G-cover.We will compare two possible choices of ideals defining the branch locus: (1) the discriminant ideal d(π) ⊆ O S , and (2) the ideal O S ∩ Ann(Ω X/S ) = Ann(π * Ω X/S ).
Remark 2.17.One could also consider the zeroth Fitting ideal Fitt 0 (Ω X/S ) ⊆ Ann(Ω X/S ) which has the same radical as Ann(Ω X/S ).But we will not do this here.
Lemma 2.18.If X → S is a D(A)-cover where S = Spec R and all line bundles defining X are trivial, then X is the spectrum of the ring and the discriminant d(π) of the cover is given by the formula Proof.The first assertion is trivial and we prove the second.The discriminant d(π) is the determinant of the map R X → (R X ) ∨ defined on the generators by where The matrix of m v λ v λ ′ in the basis {v λ ′′ } λ ′′ ∈A will have no element on the diagonal if λ + λ ′ = 0, and s λ,−λ at every entry of the diagonal otherwise.Hence the trace of the matrix will be either 0 or |A|s λ,−λ .This means that Example 2.19 (Non-example).Let p ≥ 3 be an odd prime number and let K be the cyclotomic field obtained by adding a primitive pth root of unity to Q. Let O K ⊂ K be the ring of integers.Then Spec O K → Spec Z is not a ramified D(Z/(p − 1))-cover since the discriminant of K is a power of p whereas p − 1 divides the discriminant of any D(Z/(p − 1))-cover by Lemma 2.18.
Despite the fact that Spec O K → Spec Z of the previous example is not a ramified D(Z/(p − 1))-cover, we have that the stack quotient [Spec O K /Z/(p − 1)] is a stacky cover (Definition 7.2).Also see Example 8.9.
Proof.Throughout this proof we write A multiplicatively.We may reduce to the case where X and S are affine and all line bundles of the cover are trivial.Let v λ be a generator for the graded piece of O X corresponding to λ ∈ A. The module of Kähler differentials is generated by the elements dv λ .Let n = n λ be the order of λ ∈ A. We know that v n λ lies in the zeroth piece so nv n−1 λ dv λ = 0 and hence nv n−1 λ annihilates dv λ .Hence it is enough to show that the generator of d(π) contains a factor nv n−1 λ for each λ ∈ A. We expand the part of the product (in the formula for d(π) given in Lemma 2.18) which is indexed by elements in the subgroup generated by λ.We will also group the elements two-by-two: (s λ,λ −1 s λ −1 ,λ ), (s λ 2 ,λ −2 s λ −2 ,λ 2 ), . . .except when n is even, where one element will be grouped alone.Most importantly, we use the associativity (and commutativity) of For n even, we get λ , for some elements s, s′ , and hence d(π) annihilates dv λ since n divides |A|.Note here that, in the forth equality, we simply pick the first element from each parenthesis and write these as a product, followed by that same product written out backwards (each parenthesis was raised to a power of two).All other elements in the parentheses becomes a factor of s′ .
The inclusion in Lemma 2.20 is most often strict.

Deligne-Faltings structures and root stacks
In this section we discuss the notions of Deligne-Faltings structures and root stacks associated to a Deligne-Faltings structure together with a homomorphism of monoids.The main reference for this section is [BV12].All monoids are assumed to be commutative.
We first recall some basic definitions about monoids: Definition 3.1.A monoid P is called (1) finitely generated if there is a number n ∈ N and a surjection N n → P ; (2) sharp if P × = {0}; (3) integral if p, q, q ′ ∈ P and p + q = p + q ′ implies that q = q ′ ; (4) u-integral if P × acts freely on P ; (5) quasi-integral if p, q ∈ P and p + q = p implies that q = 0; (6) fine if it is integral and finitely generated; (7) quasi-fine if it is quasi-integral and finitely generated.
Remark 3.2.A monoid M is integral if and only if the canonical map M → M gp is injective.
Definition 3.3.We denote by Div S ét the restriction of Div S = [A 1 S /G m,S ] to the small étale site of S. The following definition is very closely related to the notion of a log structure (see Remark 3.7): Definition 3.4.Let S be a scheme.A pre-Deligne-Faltings structure (pre-DF-structure short) (1) a presheaf P of monoids on S ét , and (2) a symmetric monoidal functor L : P → Div S ét .A pre-DF-structure is called a Deligne-Faltings structure (DF-structure short) if P is a sheaf and L has trivial kernel (Recall that the "zero" in Div S ét is (O S , 1)).
Remark 3.6.Here we view P as a symmetric monoidal category where all arrows are identities and the tensor product is given by the binary operation in P. A symmetric monoidal category is a braided monoidal category such that for each pair of objects a and b, the diagram is the braiding isomorphism.By a symmetric monoidal functor we mean a braided monoidal functor as in [ML98, IV, §2, p. 257], i.e., a monoidal functor which commutes with the braiding.
Remark 3.7.The notion of a Deligne-Faltings structure is equivalent to the notion of a u-integral log structure [BV12, Theorem 3.6], that is, a log structure ρ : on M is free.If P → Div S ét is a Deligne-Faltings structure, then the corresponding log structure is given by the projection where the map Remark 3.8.Note that L may have trivial kernel but still map different elements to isomorphic objects.For example, let P be the constant monoid N 2 and L a line bundle on S with a global section s, and assume that L is non-trivial or that s vanishes at some point of S. Then the symmetric monoidal functor which sends both (0, 1) and (1, 0) to (L, s) has trivial kernel.
A morphism of étale sheaves of monoids P → Q is called Kummer if for every geometric point x ∈ S, Definition 3.10.A chart for a sheaf of monoids P is a finitely generated monoid P together with a homomorphism of monoids P → P(S) such that the induced morphism P S → P is a cokernel in the category of sheaves of monoids.An atlas for P consists of an étale covering U i → S together with charts , and a Kummer homomorphism P → Q such that the induced diagram commutes.
Definition 3.11.A sheaf of monoids P is (1) sharp if P(U ) is sharp for every object U in the site, i.e., P(U ) has a unique invertible element, namely 0, and (2) coherent if it is sharp and has an atlas.Lemma 3.12 ([BV12, Lemma 4.7]).Let P be a coherent sheaf of monoids.Let ϕ be a chart for a Kummer homomorphism P → Q and denote by K P and K Q the kernels of P S → P and Q S → Q respectively.If U → S is étale and q ∈ K Q (U ), then there is an étale cover {U i → U }, integers n i ∈ N, and elements p i ∈ K P (U i ) for all i such that ϕ(p i ) = n i q| Ui .
Proof.Since P → Q is Kummer, there is an étale cover {U i → U }, integers n i ∈ N, and elements ) is zero and since P → Q has trivial kernel we see that p i must map to zero in P(U i ), i.e., p i ∈ K P (U i ).
A morphism of sharp monoids φ : P → Q induces a map of schemes Spec Z[Q] → Spec Z[P ] and we may ask what property A the map φ need to have for the map on spectra to have a property B. Definition 3.13.A morphism φ : P → Q of monoids is called (1) integral if it satisfies the following condition: Whenever q 1 , q 2 ∈ Q and p 1 , p 2 ∈ P satisfy φ(p 1 ) + (2) flat if it is integral and satisfy the following supplementary condition: Whenever q ∈ Q and p 1 , p 2 ∈ P satisfy φ(p 1 ) + q = φ(p 2 ) + q there exist q ′ ∈ Q and p ′ ∈ P such that Remark 3.14.One may think of the property of being integral as allowing us to complete every pair of solid arrows to a commutative square: Flatness, in addition, allows us to complete every pair of solid arrows to: such that the two compositions agree.
Remark 3.15.Let P ֒→ Q be an injective morphism of integral monoids.Note that we can complete every pair of arrows to a commutative square: in the associated group Q gp , if we put q ′′ = q 1 − p 2 = q 2 − p 1 .Hence if there is a p ∈ P such that q ′′ + p ∈ Q, p 1 − p ∈ P , and p 2 − p ∈ P , then P ֒→ Q is integral.

Root stacks.
Definition 3.17.Let L : P → Div S ét be a symmetric monoidal functor and j : P → Q a Kummer homomorphism of sheaves of monoids.This will be referred to as a Deligne-Faltings datum.
Definition 3.18 ([BV12, Definition 4.16]).Let L : P → Div S ét be a symmetric monoidal functor and j : P → Q a homomorphism of sheaves of monoids.The root stack associated to this Deligne-Faltings datum, denoted S P,Q,L or S Q/P is the fibered category over S associated with the following pseudofunctor: Let f : T → S be a morphism of schemes.This gives a symmetric monoidal functor f * L : f * P → Div T ét by pulling back L. We also get a morphism of sheaves of monoids f * P → f * Q and we define the category (f * L)(f * Q/f * P) with (1) objects: pairs (E, α), where E : f * Q → Div T ét is a symmetric monoidal functor, and α : f * L → E • f * j is an isomorphism of symmetric monoidal functors.This is pictured in the following diagram:

If we have a commutative diagram of schemes
, then we get a symmetric monoidal functor This is the pseudo-functor corresponding to the root stack S P,Q,L .
Definition 3.19.When (P, Q, L) is a Deligne-Faltings datum and L a : P a → Div S the associated Deligne-Faltings structure (see [BV12, Proposition 3.3]) then we write P a → Q a for the pushout of Proposition 3.21.Let P → Q be a homomorphism of fine sheaves of monoids.The stack S P,Q,L is flat over S if for every geometric point x ∈ S, the morphism P a x → Q a x is integral.Proof.This can be checked étale locally.Hence we may assume that there is a chart where D(P ), D(Q) denotes the Cartier duals of P and Q respectively (or equivalently, of P gr and Q gr ), and the action is given by the obvious grading of Z[P ] and .17], we may assume that P ∼ = P a x and Q ∼ = Q a x and hence we are done by Lemma 3.16.Remark 3.22.The converse of Proposition 3.21 does not hold as the flatness of S P,Q,L → S does also depend on L. Consider for example the cuspidal cubic be the submonoid generated by the elements 2 and 3. Then the inclusion P ֒→ Q is clearly not integral and hence not flat.If we take L to be the symmetric monoidal functor generated by sending 2 → (O S , t 2 ) and 3 → (O S , t 3 ), then the root stack S P,Q,L is the normalization of S which is not flat.On the other hand, if we let L be the symmetric monoidal functor generated by 2 → (O S , 0) and 3 → (O S , 0), then it is not hard to see that S P,Q,L ∼ = Spec (O S [ε]/(ε 2 )) which is clearly flat over S.

Special Deligne-Faltings data
Let A be a finite abelian group.The idea of the following section is to introduce monoids P A and Q A together with a homomorphism γ A : P A → Q A such that every D(A)-cover X → S will give rise to a symmetric monoidal functor L X : P A → Div S and such that the root stack associated to the Deligne-Faltings datum ].The monoids P A , Q A and the morphism γ A will depend only on the group A but L X will depend on X and the action of D(A).
Free extensions and 2-cocycles.Whenever we write monoid we mean commutative monoid.
Remark 4.1.Recall that an action of a monoid P on a set S, written (p, s) → ps, is free if there exists a basis T ⊆ S. That is, a subset T ⊆ S such that the induced function P × T → S sending (p, t) to pt is a bijection.If Q is a monoid and P a submonoid, we get an action of P on Q by addition.
Definition 4.2.Let A be an abelian group and P a monoid.A free extension of A by P (with a chosen basis) is an exact sequence E of monoids together with a set-theoretic section ι : A → Q such that ι(0) = 0 and Q is free over P with basis ι(A).This means that the function ϕ E : Remark 4.3.Note that, given a free extension as in Definition 4.2, we get a splitting of where (p, λ) has degree λ.Hence Spec Remark 4.5.When P is sharp and Q is quasi-integral, the section ι of Definition 4.2 is uniquely determined.Indeed, if we have sections ι 1 and ι 2 , both making Q free over P with basis ι 1 (A) and ι 2 (A) respectively, then for any λ ∈ A, there are elements p 1 and p 2 in P such that p 2 + ι 1 (λ) = ι 2 (λ) and commute.We denote by Ext f (A, P) the 1-category of free extensions of A by P with 1-morphisms between them.
Recall that Z 2 (A, P) denotes the set of commutative 2-cocycles (Definition 2.11).The following proposition shows that Ext f (A, P) is naturally equivalent to a set.Proposition 4.7.There is an equivalence where A × A → Q × Q and r : Q → P are the canonical inclusion and projection respectively, obtained via ϕ E .Since Q is commutative and associative and since on objects.Two isomorphic extensions will give the same 2-cocycle and hence Ψ is in fact a functor.
Conversely, given a 2-cocycle f : A × A → P , define a monoid Q = P × f A with underlying set P × A and addition given by This gives a free A-extension 0 → P → Q → A → 0 with γ : P → Q and ι : A → Q the canonical inclusions.Hence we get a functor Θ : Z 2 (A, P) → Ext f (A, P).We leave to the reader to check that Θ and Ψ are quasi-inverse to each other.
Remark 4.8.Note that the bijection in Proposition 4.7 provides Ext f (A, P) with the structure of a monoid.
The universal extension and the universal 2-cocycle.Let A be an abelian group.We will define a universal free extension such that for any extension E : 0 → P → Q → A → 0, there exists unique morphisms P A → P and Definition 4.10.We define Remark 4.11.There is a function e (−,−) : A × A → P A sending (λ, λ ′ ) to e λ,λ ′ which by definition of R A is a 2-cocycle.This will be referred to as the universal 2-cocycle.One immediately checks that any 2-cocycle A × A → P factors uniquely through e (−,−) : Definition 4.12.We define Q A = P A × e (−,−) A , i.e., Q A is the monoid in the universal free extension of A corresponding to the universal 2-cocycle e (−,−) : Lemma 4.13.Let A be a finite abelian group.The morphism γ A : P A → Q A is a Kummer homomorphism and the induced action of P A on Q A is free.

The monoid Q +
A .There are two monoid homomorphisms Σ, Π : which are defined on the basis by Σ(e λ,λ Consider the induced monoid homomorphism We have that (N A / e 0 × N A / e 0 )/(Σ, Π)(R A ) ∼ = Z A / e 0 , where (Σ, Π)(R A ) is the induced congruence relation on N A / e 0 × N A / e 0 .Indeed, 0 ∼ e 0,λ maps to 0 ∼ (e 0 + e λ , e λ ) = (e λ , e λ ) in N A / e 0 × N A / e 0 and Z A / e 0 is obtained as the quotient of N A / e 0 × N A / e 0 by the diagonal.The induced map is Remark 4.14.The monoid P A need not be integral.For instance, it is not integral when A = Z/2Z × Z/2Z × Z/2Z (here we used Macaulay 2).This means that the morphism ϕ A need not be injective.Definition 4.17.We define Q + A = N A / e 0 .Remark 4.18.By [Ton14, Lemma 4.5] P A → ϕ A (P A ) is the associated integral monoid and hence we identify P int A with ϕ A (P A ). Similarly, the map Q A → Z A / e 0 ; (p, λ) → ϕ A (p) + e λ is a homomorphism and the image is the associated integral monoid (1, 0), (0, 1), (2, −1), (−1, 2) .This is illustrated in Figure 1.Lemma 4.21.Let M be a finitely generated monoid and v : M → Z a homomorphism.If M is generated by elements s such that v(s) ≥ 1 then M is sharp.
Lemma 4.22.The monoids P int A and Q int A are fine and sharp.
Proof.Both P int A and Q int A are finitely generated and integral (see Remark 4.18) and hence fine.It remains to show that the monoids are sharp and since A and the homomorphism q → |q| satisfies the condition of Lemma 4.21, and hence Lemma 4.23.The monoids P A and Q A are quasi-fine and sharp.
Proof.The canonical morphism Q A → Q int A sends (p, λ) to ϕ A (p)+e λ .Hence Q A and the homomorphism (p, q) → |ϕ A (p) + e λ | satisfies the condition of Lemma 4.21, and hence Q A is sharp.But then P A is also sharp since it is a submonoid.Both P A and Q A are finitely generated by definition so it remains so prove that they are quasi-integral.This follows from the fact that they are generated by elements of strictly positive value.Indeed, λ | = 0 and hence (p ′ , λ ′ ) = 0.This completes the proof.We will now define Q A as a quotient of P A ⊕ Q + A .Definition 4.24.For q ∈ Q + A , write q = n i=1 e λi where we may have λ i = λ j for i = j.Define h(q) ∈ P A by h(q) = 0 if n ≤ 1, and Remark 4.25.Note that by definition of the equivalence relation R in Definition 4.9, the element h(q) is independent of the order of the λ i 's in the representation q = n i=1 e λi .This implies that we have a set-theoretic function h : which satisfies q = ϕ A (h(q)) + e m(q) .

The function h : Q +
A → P A satisfies the relation h(q + q ′ ) = h(q) + h(q ′ ) + e m(q),m(q ′ ) .Definition 4.28.We define j : Remark 4.29.The map j is a homomorphism by Remark 4.27.Furthermore, there is a morphism Q A → Z A / e 0 sending (p, λ) to ϕ A (p) + e λ and by Remark 4.25, the composition e 0 is the canonical inclusion.Hence we conclude that j is injective and from now on we view Q + A as a submonoid of Q A .
Iterating this process we conclude that R P contains the relation (0, q) ∼ (h(q), e m(q) ) for any q ∈ Q + A .Since R P is a congruence relation, it is symmetric and closed under addition.Hence we conclude that R P contains the relation R ′ defined by (p, q) ∼ (p ′ , q ′ ) if m(q) = m(q ′ ) and p + h(q) = p ′ + h(q ′ ).It is clear that R ′ is an equivalence relation and by Remark 4.27 it follows that R ′ is a congruence relation.Note that R ′ contains the relation (e λ,λ ′ , e λ+λ ′ ) ∼ (0, e λ + e λ ′ ) and hence R P = R ′ .
Remark 4.32.We have a function τ : and by Remark 4.27 we have τ (p, q) + τ (p ′ , q ′ ) = (p + p ′ + h(q) + h(q ′ ) + e m(q),m(q ′ ) , m(q) + m(q ′ )) = (p + p ′ + h(q + q ′ ), m(q + q ′ )) = τ (p + p ′ , q + q ′ ) and hence τ is a homomorphism of monoids.Furthermore, we have τ (p, q) = τ (p ′ , q ′ ) if and only if m(q) = m(q ′ ) and p + h(q) = p ′ + h(q ′ ).This means that τ induces a morphism ( which becomes a homomorphism when we quotient by R P .Furthermore, τ • η = id QA and we conclude that (P A ⊕ Q + A )/R P → Q A is an isomorphism.Flat Kummer homomorphisms.Now consider the following situation.Suppose that we have a flat Kummer homomorphism (Definition 3.9 and 3.13) of quasi-fine (Definition 3.1) and sharp monoids γ : P → Q.Note that A = Q/P is a finite abelian group.For q, q ′ ∈ Q we write q ≤ q ′ if there exists a p ∈ P such that q ′ = q + p.The relation ≤ is a partial order since Q and P are quasi-integral and sharp.
Corollary 4.34.Let P ֒→ Q be a flat Kummer morphism of quasi-fine and sharp monoids and let λ ∈ A = Q/P with m : Q → A the quotient.There is a set-theoretic section ι : A → Q sending λ to the unique minimal element ι(λ) in m −1 (λ).Furthermore, 0 → P → Q → A → 0 is a free extension of A by P with basis A.
Proof.By Proposition 4.33 we need only show that the map P × A → Q sending (p, λ) to p + ι(λ) is injective.Suppose that p 1 + ι(λ) = p 2 + ι(λ).Then the flatness hypothesis says that there exists a q ∈ Q and p ′ ∈ P such that q + p ′ = ι(λ) and p ′ + p 1 = p ′ + p 2 .But q ∈ m −1 (λ) and q ≤ ι(λ) implies that q = ι(λ) by minimality and unicity of ι(λ).Hence ι(λ) + p ′ = ι(λ) so p ′ = 0 since Q is quasi-integral.Hence p 1 = p 2 .This proves the claim.Proposition 4.35.Let γ : P → Q be a flat Kummer homomorphism of quasi-fine and sharp monoids P and Q, and let A = Q/P .Then there exists canonical morphisms Q A → Q and P A → P such that the following diagram commutes and the upper square is a pushout: Proof.This follows from Proposition 4.7, Remark 4.11, and Corollary 4.34.We leave the proof to the reader.
Remark 4.36.Proposition 4.35 in particular implies that, whenever we have a Deligne-Faltings datum (P, Q, L) with P and Q (constant) quasi-fine and sharp, and with P → Q a flat Kummer homomorphism, we have a canonical diagram inducing an isomorphism of root stacks S QA/PA ≃ S Q/P .This also works when we have sheaves of monoids instead of constant monoids (see Definition 5.6).

2-cocycles in symmetric monoidal categories
The theory of 2-cocycles and extensions can be generalized to the setting of categories fibered in symmetric monoidal groupoids.Throughout the section we ignore the base category, which will in all applications be the category of schemes over our base scheme S. A category fibered in symmetric monoidal groupoids which is a stack will be referred to as a symmetric monoidal stack.
Definition 5.2.Let A be a sheaf of abelian groups and let M be a symmetric monoidal stack.A (commutative) 2-cocycle of A with values in M consists of (1) a morphism f : A × A → M of (ordinary) fibered categories , (2) natural isomorphisms: Furthermore, this data is required to satisfy the following commutativities (analogous to a symmetric monoidal category): (1) (5) (Hexagon axiom) , where Σ : A × A → A is the addition.This means that we have an isomorphism, natural in λ, λ ′ , for all local sections λ, λ ′ of A .Furthermore, τ is required to commute with the morphisms u, s, and a.
and τ in the obvious sense.
We denote by Z 2 (A, M) the 1-category with 2-cocycles as objects and natural isomorphisms f ′ → f , which commutes with the morphisms u, s, and a, as morphisms.
The underlying category is M × A, the tensor product and the unit is The natural isomorphisms are defined by for all x, y, z ∈ M and all λ, λ ′ , λ ′′ ∈ A.
Proposition 5.3.The data Proof.This is tedious but straight forward and left to the reader.
Definition 5.4.Let A be a sheaf of abelian groups and let M be a symmetric monoidal stack.A free extension of A by M is a short exact sequence E of symmetric monoidal stacks together with a section ι : A → C of m such that the induced morphism γ ⊗ ι : M × A → C is an equivalence of fibered categories.
A 1-morphism of free extensions E → E ′ of A by M is a triple (ϕ, α, β) consisting of a morphism of symmetric monoidal stacks ϕ : C → C ′ such that m ′ • ϕ = m, and natural isomorphisms α : ϕ • γ ≃ γ ′ , and ) is a natural isomorphism ϕ ≃ ϕ ′ which commutes with α, β, α ′ , β ′ in the obvious sense.We denote the 2-category of free extensions by Ext f (A, M).
We now state the equivalent of Proposition 4.7, which shows that Ext f (A, M) is naturally equivalent to a 1-category.
Proposition 5.5.We have an equivalence of groupoids Proof.This is of course very similar to the proof of Proposition 4.7.
Given a 2-cocycle f : A × A → M we get an associated symmetric monoidal stack M × f A as defined above.Given a natural isomorphism of 2-cocycles τ : f ′ → f , we define ϕ : M × f ′ A → M × f A to be the identity on objects and with comparison isomorphism This defines a functor ψ : Z 2 (A, M) → Ext f (A, M).To see that ψ is essentially surjective, consider an extension 0 may be given the structure of a 2-cocycle using the symmetric monoidal structure of C and we get an isomorphism of extensions C → M × f A.
We leave to the reader to check that ψ is fully faithful.
The universal extension and the universal 2-cocycle.We need a sheafified version of the monoid P A .
Definition 5.6.Let A be an étale sheaf of abelian groups of finite type on a scheme S. Define N A 2 and P A to be the sheafification of , and respectively (see Definition 4.10).Furthermore, we define We also define groupoid versions of the monoids P A and Q A (Definition 4.9, Definition 4.10, Definition 4.12).Definition 5.7.We define As in the case of monoids, there is a universal 2-cocycle e (−,−) : A × A → P A and a universal extension Lemma 5.8.The canonical maps P A → P A and Q A → Q A are equivalences of stacks.
Proof.We have that is the stackification of the fibered category {R A ⇒ N A 2 } whose fiber over a scheme U is the groupoid {R A (U ) ⇒ N A 2 (U )}.But this groupoid has at most one arrow between any two objects and hence {R A ⇒ N A 2 } is equivalent to a category fibered in sets, i.e., an ordinary presheaf.Applying the stackification functor to a presheaf gives the associated sheaf and hence we conclude that P A is equivalent to a sheaf.The canonical functor P A → P A just identifies isomorphic objects and is hence an equivalence.This also shows that Q A → Q A is an equivalence.
We will now compare 2-cocycles f : A × A → M and symmetric monoidal functors L : P A → M. We assume that our base category is (Sch/S) and that A is a sheaf of abelian groups on the small étale site of S. Passing to the espace étalé we think of A as an étale algebraic space over S.
Remark 5.9.Let Φ : L ′ → L be a morphism of symmetric monoidal functors L ′ , L : P A → M. Then for every p, p ′ ∈ P A , we have that the diagram commutes.This means that the morphism Φ is determined on generators of P A .
Proof.We first show that Θ is fully faithful.This may be checked on fibers.
To see that Θ is faithful, suppose that Θ(Φ ′ ) = Θ(Φ).Then Φ ′ and Φ agree on generators and again, by Remark 5.9 they must be equal.This shows that Θ is faithful and completes the proof.
The stack associated to a 2-cocycle.We will now associate to a 2-cocycle f : A× A → Div S ét a stack S A,f which will be equivalent to the root associated of the universal extension P A → Q A → A and the symmetric monoidal functor L f : P A → Div S ét corresponding to f .Definition 5.12.Let f : A × A → M be a 2-cocycle.We define S A,M,f to be the fibered category over S with objects (T, ζ, κ), consisting of an S-scheme h : T → S, a 1-cochain ζ : A| T → M| T , and a natural isomorphism κ : commutes.
In case M = Div S we write S A,f = S A,M,f .
Any 2-cocycle f : A × A → Div S ét factors through a symmetric monoidal functor L f : P A → Div S ét from which we get a root stack S PA,QA,L f .Proposition 5.13.We have a canonical equivalence of fibered categories Proof.Let us define a functor ϑ : S A,f ≃ S PA,QA,L f on objects as follows: given an object (T, ζ, κ) of S A,f , we define a symmetric monoidal functor where µ is the comparison isomorphism of L f .Let γ : P A → Q A be the usual inclusion.Then we get an equality We leave to the reader to check compatibility with κ ′ and κ.
The functor ϑ is essentially surjective since every object in S PA,QA,L f is isomorphic to an object whose diagram is strictly commuting.The functor ϑ is full since a natural isomorphism ϕ * E → E ′ where comes from a morphism in S A,f .We leave to the reader to check that ϑ is faithful.

Deligne-Faltings data from ramified D(A)-covers
Let A be a finite abelian group.Recall that every D(A)-cover f : X → S comes with a canonical splitting and multiplication morphisms L λ ⊗ L λ ′ → L λ+λ ′ for every λ, λ ′ ∈ A, which we think of as global sections The quotient stack X = [X/D(A)] has a canonical D(A)-torsor p : X → X and we have a canonical splitting of p * O X indexed by the elements of A. Definition 6.2.With the notation above we write O X [λ] for the line bundle which is the direct summand of p * O X of weight λ, so that We explain the notation in the following remark: We have a counit , the universal divisor associated to the character λ ∈ A and we call E λ the universal line bundle associated to the character λ ∈ A.
Similarly, for every pair of universal line bundles E λ and E λ ′ with characters λ, λ ′ ∈ A, we have a morphism ) be the corresponding generalized effective Cartier divisor.Remark 6.5.Note that we have canonical isomorphisms with u, s, and a chosen to be the canonical natural isomorphisms.We define the 1-cochain associated to X as where (E λ , ε λ ) is the universal line bundle defined in Remark 6.4.Finally, we define to be the canonical isomorphism of Remark 6.5.
Proposition 6.7.We have an isomorphism of stacks where S A,fX is the stack defined in Definition 5.12.
Proof.The stack X = [X/D(A)] → S has a universal object (X, ϕ) where X → X is the canonical D(A)-torsor and ϕ : X → X × S X the canonical D(A)-equivariant morphism over X .In a diagram: ϕ Using Definition 6.6 we obtain an object (ζ X , κ X ) defining a morphism X → S A,fX .Conversely, from the universal object (f X , ζ, κ) on the stack π : Y → S, we construct an O Y -algebra whose underlying module is and with multiplication m, defines via κ and the commutative diagram This defines a D(A)-torsor Y (ζ,κ) on Y and the global sections defining ζ yields a D(A)-equivariant morphism Y (ζ,κ) → X × S X.This defines a morphism Y → X and we leave to the reader to check that the two functors are quasi-inverse to each other.
Definition 7.2.Let X be a tame stack with finite diagonalizable stabilizers at geometric points and let S be a scheme.We say that π : X → S is a stacky cover if it is (1) flat, proper, of finite presentation, (2) a coarse moduli space, and (3) for any morphism of schemes T → S, the base change π| T : X T → T has the property that (π| T ) * takes invertible sheaves to invertible sheaves.
In this section we describe how to reconstruct a stacky cover X → S from logarithmic data on S.
Remark 7.3.A flat good moduli space π : X → S takes vector bundles to vector bundles.This follows from [Alp08, Theorem 4.16].Indeed, π * preserves coherence and flatness relative to S. If E is a vector bundle on X , then it is flat over S since π is flat.Hence π * E is coherent and flat, or equivalently, locally free of finite rank.
Remark 7.4.A stacky cover π : X → S will étale locally on S look like a quotient of a ramified cover.That is, for every geometric point s ∈ S, there is an étale neighborhood U → S of s, an abelian group A, and a ramified D(A)-cover X → U such that X × S U ≃ [X/D(A)].Indeed, by the local structure theorem for tame stacks [AOV08, Theorem 3.2] there is a finite f : since X has diagonalizable stabilizers at geometric points.Since X → X × S U is finite flat of finite presentation and X → S is finite flat of finite presentation, we get that f : X → U is finite, flat, and of finite presentation.It remains to show that f * O X looks fppf locally like the regular representation.Since p : X → X × S U is a D(A)-torsor, we get a splitting and since (π| U ) * is exact and takes line bundles to line bundles, we get that The relative Picard functor.Let Pic X /S : (Sch/S) → (Ab) denote the relative Picard functor, i.e., the fppf sheafification of the functor By [Bro09, Section 2] Pic X /S sits in a exact sequence of Picard stacks 0 → BG m,S → π * BG m,X → Pic X /S → 0 .
Lemma 7.5.The stack π * BG m,X is algebraic and locally of finite presentation.
Proposition 7.6.Let X → S be a stacky cover.Then Pic X /S is representable by an étale algebraic space.
Proof.The representability is [Bro09, Proposition 2.3.3]so we need to show that it is étale.We will show that Pic X /S is formally étale.Let i : T 0 ֒→ T be a closed immersion of affine schemes over S, defined by a quasi-coherent ideal J such that J 2 = 0. Let J denote the sheaf of ideals corresponding to X T0 ֒→ X T .We have an exact sequence where the morphism J → G m,T sends a local section r to 1 + r, which is a homomorphism since (1 + r)(1 + r ′ ) = 1 + (r + r ′ ).Hence we get a long exact sequence in cohomology since the Leray spectral sequence and ).Since X → S is cohomologically of dimension zero and T is affine, it follows that H 1 (X T , J ) = H 2 (X T , J ) = 0 and hence Pic(X T ) → Pic(X T0 ) is an isomorphism.Hence Pic X /S is formally étale.By Lemma 7.5 we then conclude that Pic X /S is étale.
Deligne-Faltings data from stacky covers.Given a stacky cover π : X → S we want to realize X as a root stack over S. We will realize X as a root stack by defining a universal diagram where A = Pic X /S .
Definition 7.7.Let Div X /S be the stack of relative divisors.That is, the fibered category Div X /S = π * Div X with objects (T, D) with T → S a scheme and D an object in Div (T × S X ).
Note that we have forgetful morphisms and hence a diagram where the right column is exact by definition and the left column is exact because Γ(S, O S ) ∼ = Γ(X , O X ) since X → S is a coarse moduli space.
Definition 7.8.Let X be a stacky cover.We define Pic triv X /S to be the fibered category (which is equivalent to a sheaf in sets) with objects that are triples (T, E, t) where • T → S is a scheme, • E is a line bundle on T × S X , and Example 7.9.Let S = Spec Z[ √ −5] and let L ⊆ O S be the line bundle given by the non-principal ideal , where x has weight 1 and y has weight 2. If we let X = [X/µ µ µ 3 ] and Definition 7.10.Let Λ : Pic triv X /S → Div X /S be the functor given on objects by (T, where ε : π * π * E → E is the counit. Uniqueness.Let E be a line bundle on X and let f : T → X be an object in the lisse-étale site.A lift T → I X is equivalent to an automorphism (T, f ) → (T, f ) and hence we get a linear automorphism E(T ) → E(T ).This means that we have a linear action of I X on E and this action is via some character λ : I X → G m,X .
Lemma 7.11.Let S be the spectrum of a strictly henselian ring and let X = [X/D S (A)] be the quotient stack of a ramified cover X → S whose stabilizer over the closed point Spec k → S is D k (A).If E is a line bundle on X , then I X acts on E via a character I X → G m,X which factors through I X → D X (A), i.e., the character of E corresponds to an element of A.
Proof.Over the closed point we know that E has character corresponding to an element λ ∈ A. We have a tautological line bundle O X [λ] and E ⊗ O X [−λ] has trivial character over the closed point.Hence by [Alp08, Theorem 10.3], we get that E ⊗ O X [−λ] is the pullback of a line bundle on S, which says that it has trivial character globally.Thus E has character λ globally.
Lemma 7.12.Let λ : be a morphism of X -groups and let E and E ′ be line bundles on X on which I X acts via the character λ.Then the counit Proof.This follows from [Alp08, Theorem 10.3].
Lemma 7.13.Let λ : be a morphism of X -groups and let (E, s) and (E ′ , s ′ ) be objects in Pic triv X /S (S) on which I X acts via the character λ.Then there exists a unique isomorphism (E, s) → (E ′ , s ′ ) in Pic triv X /S (S).Proof.We have a morphism and we want to show that this is an isomorphism.Indeed, then there will be a unique morphism E → E ′ mapping to the morphism which is what we want to prove.It is enough to show that ψ is an isomorphism after passing to an étale cover and hence we may assume that X ≃ [X/D(A)] for a ramified cover X → S and that λ comes from a character of D(A), which we think of as an element of A. We have a tautological sheaf O X [λ] and Hence E is generated as an O X -module in degree −λ and any morphism E → E ′ is completely determined by what it does in degree −λ.By the projection formula we get . Hence we conclude that E → E ′ is completely determined by π * E → π * E ′ .On the other hand, since E and E ′ are both generated in degree −λ, any morphism π * E → π * E ′ induces a morphism E → E ′ .Hence we conclude that Hom(E, E ′ ) ≃ Hom(π * E, π * E ′ ) .
Corollary 7.14.Let X → S be a stacky cover.Then the morphism Pic triv X /S → Pic X /S , sending a pair (E, t) to the class [E], is an equivalence of sheaves of sets.
Proof.Let U → S be étale.Injectivity follows from Lemma 7.13 since two line bundles on X U which differ by a line bundle coming from U must have the same character.Surjectivity follows from the fact that if L is any line bundle on X U , then ((π * π * L) ∨ ⊗ L, ε) represents an element in Pic triv X /S (U ) and The canonical DF-datum.Now we are in a position to define a Deligne-Faltings datum associated to X .The goal is to obtain a pointed section Pic X /S → Div X /S , which by Proposition 5.5 would give us a 2-cocycle Pic X /S × Pic X /S → Div S as desired.The functor Λ : Pic triv X /S → Div X /S of Definition 7.10 is fully faithful.
Choose a quasi-inverse ψ : Pic X /S ∼ − → Pic triv X /S of the morphism of Corollary 7.14.We denote by ζ X the morphism Definition 7.15.The exact sequence 0 → Div S π * −→ Div X /S → Pic X /S → 0 together with the section ζ X defines a free extension of Pic X /S by Div S (Definition 5.4) which we call the free extension associated to X .The corresponding 2-cocycle is referred to as the 2-cocycle associated to X .Lemma 7.16.There exists a canonical natural isomorphism Proof.This follows from the fact that ζ X (λ) ⊗ ζ X (λ ′ ) and f X (λ, λ ′ ) ⊗ ζ(λ + λ ′ ) have the same character and since there is a canonical isomorphism between them, via the using the projection formula, after pushing down to S.
The pair (ζ X , κ X ) defines a morphism X → S A,f X which is in fact an equivalence.
Theorem 7.17.The morphism given by the pair (ζ X , κ X ) is an isomorphism of stacks, where A = Pic X /S .Hence there exists a canonical (up to canonical isomorphism) symmetric monoidal functor L : P A → Div S ét , and a canonical isomorphism of stacks X → S (A,L) where S (A,L) is the root stack associated to the building datum (A, L).
Proof.We may work locally on S and hence we may assume that X is a quotient by a ramified cover under an action of a finite diagonalizable group (Remark 7.4).By Remark 6.4 and Proposition 6.7 we conclude that η is an isomorphism.
The Cartier dual of the inertia stack.We saw in Lemma 7.13 that any two line bundles with the same character must differ by a line bundle from S. Now we will see that when X → S is a stacky cover which is Deligne-Mumford, then Pic X /S ∼ = π * D(I X ).This does not hold in general if X is not Deligne-Mumford and a counter-example is obtained by taking X to be the quotient stack of the ramified cover given in Remark A.8.
To show that there exists a line bundle with character λ for every character λ of I X when X is Deligne-Mumford, we show that it exists étale locally (on the base) and that these local bundles may be glued to a global one.
Lemma 7.18.Let λ be a character of I X .Then there exists a line bundle on X on which I X acts with character λ.
Proof.Let {U i → S} be an étale cover such that every X i := X × S U i may be written as a quotient X i ≃ [X i /D(A i )] of a ramified cover (Remark 7.4) with the property that the canonical morphism ] is an isomorphism on group-like elements (see Proposition A.7) since X is Deligne-Mumford.The character λ pulls back to a character for each Stab(X i ) which we may think of as an element Now consider the two objects (E i , t i ) and (E j , t j ), where ] and (here we write π for both maps X i → U i and X j → U j ) and t i is the trivialization where the second isomorphism is the projection morphism.Both E i and E j pulls back to a line bundle on X × S (U i × S U j ) with character λ and the trivializations t i and t j pulls back to trivializations.Hence Lemma 7.13 gives a unique isomorphism whose pushforward sends (the pullback of) t i to t j .This implies that the line bundles E i will glue to a global line bundle on X with character λ and trivial pushforward.
Example 7.20 (Stack ramified over a nodal cubic).The following is an example of a stacky cover which is not globally a quotient of a ramified cover (but of course étale locally).Consider the nodal cubic V (y 2 − x 2 (x + 1)) ⊂ Spec C[x, y] = A 2 C =: S. Let π : X → S be the 2nd root stack associated to the log structure given by f = y 2 − x 2 (x + 1) ∈ O S and let A = π * D(I X ).More precisely, let j : U ֒→ S be the complement of V (f ) ֒→ S and consider the canonical log structure O S ∩ j * O × U → O S .Let L : P → Div S ét be the associated Deligne-Faltings structure (see Remark 3.7), put Q = P, and let γ : P → Q be given by multiplication by 2. We put X = S (P,Q,L) .
We have an étale cover U 1 , U 2 → S where U 1 and U 2 are given as the spectrum of respectively.We have that X U1 ≃ [X 1 /D(Z/2Z)] and X U2 ≃ [X 2 /D(Z/2Z × Z/2Z)] respectively, where and f 1 = y − xw and f 2 = y + xw.Over U 1 × S U 2 the diagonal ∆ : Z/2Z → Z/2Z × Z/2Z gives a ∆-equivariant morphism We have and a commutative diagram (1,0,1) where α D(fi) (1) is the non-trivial element for i = 1, 2 whereas (This corresponds to projection onto the second and first Z/2Z factor respectively).Note that Γ(U is an epimorphism of étale sheaves which is not surjective on global sections.From here we see that both P Z/2Z → Div U 1 × S U 2 and P Z/2Z×Z/2Z → Div U 1 × S U 2 factor through N A 2 /R × S (U 1 × S U 2 ) and hence we can glue to a global symmetric monoidal functor Note that Γ(S, P A ) ∼ = N and hence there cannot be a global chart for this log/DF structure.

Building data for stacky covers
The goal of this subsection is to describe the 2-category of stacky covers via stacky building data.Definition 8.1.Let A be an étale sheaf of abelian groups of finite type on a scheme S and L : P A → Div S ét a symmetric monoidal functor.Then we define A ⊥ ⊆ A to be the subsheaf (of sets) defined by Lemma 8.2.The subsheaf A ⊥ is a subgroup.
Proof.Let us write A = A(U ) for an étale U → S and similarly for A ⊥ .Clearly 0 ∈ A ⊥ .Let λ ∈ A ⊥ and write s λ,λ ′ for the global section of L U (e λ,λ ′ ).Then for all λ ′ ∈ A we have This means that X is fppf locally the quotient of a ramified cover and hence a stacky cover.Let E : Q π * A → Div X ét be the universal DF-object.We have a canonical set-theoretic section ι : A → Q A sending a local section λ to (0, λ), and we define a morphisms of sheaves of sets β ′ : A → Pic triv X /S by sending λ to the class represented by (E λ , π * ε λ ) and let β : A → Pic X /S be the composition of β ′ with the canonical morphism (E λ , π * ε λ ) → [E λ ] in Corollary 7.14.To see that β is an equivalence we may work étale locally on the base and hence assume that X = [X/D(A)] ≃ S (A,L) .Then β is an epimorphism since every line bundle on X is of the form L 0 ⊗ E λ , where L 0 is the pullback of a line bundle on S and E λ is the universal line bundle associated to λ ∈ A. We also see that β is a homomorphism since E λ and O X [λ] differ by a line bundle coming from S and hence they define the same class in Pic X /S .Furthermore, since ε λ : O X → E λ is given by s λ ′ ,λ in degree λ ′ , we get that ε λ is an isomorphism if and only if λ ∈ A ⊥ = 0.This means that zero is the unique element mapped to the trivial element (O X , π * 1) by β ′ .Hence zero is the unique element mapping to zero by β and we conclude that β is an isomorphism.
Remark 8.7.There is a monoidal structure on StData S which on objects is defined by Theorem 8.8.There exists an equivalence of (2,1)-categories StCov S ≃ StData S between the 2-category of stacky covers over S and the 2-category of stacky building data on S.
Proof.We define Φ : StData S → StCov S on objects by sending (A, L) to the corresponding root stack X = S (A,L) .Hence it is clear from Theorem 7.17 that Φ will be essentially surjective.For building data D = (A, L) and is defined as follows.Put π : X = S (A,L) → S and π ′ : Y = S (A ′ ,L ′ ) → S and let (E, α) be the universal object on X and (E ′ , α ′ ) the universal object on Y .
Let (ϕ, H, τ ) : This defines a morphism of stacks f (ϕ,H,τ ) : X → Y and we define Φ on 1-morphisms by Now assume that we have a 2-morphism θ: Then we get an isomorphism x x r r r r r r r r r r π * L ′ commutes.This means that we get a well-defined natural transformation To show that Φ D,D ′ is an equivalence of categories we define a quasi-inverse Ψ.Let f : X → Y be a morphism of stacks, where X and Y are root stacks constructed from building data (A, L) and (A ′ , L ′ ) with universal Deligne-Faltings objects By pullback we get a morphism Pic Y /S → Pic X /S and by Proposition 8.6 a morphism ϕ : A ′ → A. We may pullback the universal diagram on Y to get a diagram There is a unit morphism This implies that we have a canonical morphism Since the inertia acts trivially on each π * H λ , the symmetric monoidal functor π This defines the quasi-inverse on objects by putting Ψ(f ) = (ϕ, H, τ ).
To define Ψ on morphisms, suppose that we have a natural transformation η : where X and Y are root stacks constructed from building data (A, L) and (A ′ , L ′ ) with universal Deligne-Faltings objects and from the construction of π * H and π * H ′ as in Equation (2), we see that we get a natural isomorphism θ : We leave this to the reader.
Example 8.9.The following is an example where the building datum from a stacky cover is not determined on the global sections of A. Let ξ be a primitive 5th root of unity and consider the ring of integers Z and the corresponding coaction is where e n is the standard basis of idempotents.If we invert 2 and add a square root i of −1 we get an étale morphism Spec Z[2 −1 , i] → Spec Z and we have an isomorphism of Hopf algebras where ϕ is the inverse of Z/4Z → (Z/5Z) × ; a → 2 a .Via this isomorphism we get a coaction given on the generator ξ by After some calculations, one concludes that the induced splitting is and the multiplication is given by the global sections Hence we have a building datum over the étale chart Spec Z Over the open chart Spec Z[5 −1 ] → Spec Z we have a trivial building datum.Note that all the sections s i,j above become invertible when inverting 5.One may check that the two building data glue to a global building datum (A, L) such that [Spec Z[ξ]/(Z/5Z) × ] ≃ S (A,L) .

Parabolic sheaves and an application
In Section 6 we saw that a D(A)-ramified cover gives rise to a Deligne-Faltings datum which we may think of as ramification data.In [BB17] Biswas-Borne consider ramification data (D, r) where D = (D i ) i∈I is a simple normal crossings divisor on a scheme S over a field k, and r = (r i ) i∈I is a family of positive integers.Then they give a criterion [BB17, Theorem 2.8] for when this birational building datum comes from a tamely ramified G-torsor [BB17, Definition 2.2] where G is a finite abelian group scheme over k.Following [BB17, Remark 2.9.(2)], we extend this result to the more general setting where (1) the ramification datum (D, r) is replaced with a birational building datum (A, L), i.e., all global sections of L are regular, and (2) tamely ramified torsor is replaced with a tamely ramified cover (Definition 9.8).
Parabolic sheaves.Before stating the theorem, we recall the notion of parabolic sheaf in [BV12].First we need to define the category of weights of a monoid.Definition 9.1.Let P be an integral monoid.We denote by P wt the partially ordered set which is the strict symmetric monoidal category with objects that are elements of P gp and whose arrows p : p ′ → p ′′ are elements p ∈ P such that p ′ + p = p ′′ .Similarly, when P is a fine sheaf of monoids, we denote by P wt the corresponding symmetric monoidal stack.
Remark 9.2.Every symmetric monoidal functor L : P → Div S ét gives rise to a symmetric monoidal functor L wt : P wt → Pic S ét and vice versa (see [BV12, Proposition 5.4 and Remark 5.5]).Definition 9.3.Let (P, Q, L) be a Deligne-Faltings datum on a scheme S. Consider the actions Σ : P wt × Q wt → Q wt , T : Pic S ét × QCoh S ét → QCoh S ét where the first one is given by addition and the second by taking tensor products (this means that Q wt and QCoh S ét are module categories over P wt and Pic S ét respectively).A parabolic sheaf (E, ρ) on (S, P, Q, L) consists of (1) a cartesian functor E : Q wt → QCoh S ét , which we write on objects as q → E q and q ′ → E(q ′ ) on arrows, and (2) an isomorphism ρ : E • Σ ≃ T • (L wt × E) realizing E as an L wt -equivariant functor.We denote by Par (S, P, Q, L) the category of parabolic sheaves on S with respect to the Deligne-Faltings datum (P, Q, L).
Let π : X → S be the root stack of a Deligne-Faltings datum (P, Q, L), with P and Q constant, and let E : Q wt → Pic X be the universal Deligne-Faltings structure on X .Let F be a quasi-coherent sheaf on X .The corresponding parabolic sheaf (E, ρ) is given as follows: (1) For q ∈ Q wt , put (3) and for q ′ ∈ Q, we let E(q ′ ) : E q → E q+q ′ be the pushforward of the morphism (2) For p ∈ P wt , q ∈ Q wt , we let ρ p,q : E q+p ≃ L p ⊗ E q be the isomorphism obtained via the projection formula and the isomorphism E q ⊗ E p ≃ E q+p .This construction may be globalized and one has the following theorem: Theorem 9.5 ([BV12, Theorem 6.1]).Let (S, P, Q, L) be a Deligne-Faltings datum where L : P → Div S ét .There is an equivalence of symmetric monoidal categories Par(S, P, Q, L) ≃ QCoh(S P,Q,L ) .
Ramified G-covers.Proposition 9.6.Let k be a field and S a k-scheme.Let π : X → S be a stacky cover and X → X a G X -torsor where X is a scheme and G is a finite abelian group scheme over k.Then for every point s ∈ S there is (1) a field extension k ′ ⊇ k, (2) an abelian group A, a monomorphism D k ′ (A) → G k ′ , (3) an fppf neighborhood V → S of s, (4) a D V (A)-cover Y → V , and (5) a G-equivariant isomorphism Y × D(A) G ∼ = V × S X.
The proof will be very much along the same lines as the proof of [BB17, Proposition 3.5].
Proof.If x ∈ X is a closed point then there is an étale neighborhood U of π(x) such that X U := X × S U ≃ [Y /H] where H = D(A) (where A is an abstract abelian group) and D(A) x is the stabilizer at x.We replace X by X U .Then we have G x ≃ BH κ(π(x)) and BH → X is a section of the morphism X → BH corresponding to the H-torsor Y → X .The torsor X → X corresponds to a representable morphism X → BG.After base change to a field extension of κ(π(x)) we may assume that the (2) For every geometric point s in the branch locus, we have that (i) the map Γ(S, A) → A s is surjective, and (ii) for every λ ∈ A s, there exists an essentially finite, basic, parabolic vector bundle (E, ρ) on (S, P A , Q A , L) such that the morphism is not surjective, where the direct sum is over all λ ′ ∈ Γ(S, A) such that λ ′ s = 0.This proof is analogous to that in [BB17].
Proof.Let π : = S (A,L) → S be the associated root stack and E : π * Q A → Div X ét be the universal Deligne-Faltings object.Any vector bundle on G s is a direct sum of E λ | Gs for λ ∈ A s.By Proposition [BB17, Proposition 3.18], (1) holds if and only if every vector bundle on G s ∼ = BA s is a quotient of a subbundle of F | Gs , for some essentially finite vector bundle F on X .This implies that (1) holds if and only if, for every point s and every λ ∈ A s, there exists an essentially finite vector bundle F on X , such that Hom Gs (F | Gs , E λ | Gs ) = 0. Let F be an essentially finite vector bundle on X .We have a commutative diagram respectively, where the first direct sum is over all λ, λ ′ ∈ Γ(S, A) such that λ ′ s , λ s = 0 and the second direct sum is over all λ ∈ Γ(S, A) such that λ s = 0. We have an exact sequence and a commutative diagram where p is a good moduli space.Note that the left diagram is Cartesian.This implies that we have a natural isomorphism i * p * ≃ π * j * , since p is a good moduli space.
Example A.5.Let (R, m) be a local ring and A a finite abelian group.We also let O X be an Ralgebra with a coaction of R[A] such that X → S is a ramified D(A)-cover, where S = Spec R and X = Spec O X .Hence we may write O X ∼ = R[{x λ } λ∈A ]/({x λ x λ ′ − s λ,λ ′ x λ+λ ′ }) where each x λ is a generator for the line bundle with character λ (see Remark 2.2).Let I be the ideal cutting out the stabilizer group H = X × X×SX D X (A) ֒→ D X (A) with respect to the action on X. ).The goal is to show that g represents the element 1 in R[H], i.e., g = 1 modulo I. Since I ⊆ ker ε and the ideal ker ε is generated by elements of the form λ − 1, we get that g must be of the form g = 1 + λ∈A\{0} a λ (λ − 1) ∈ R[A] where a λ ∈ m for all λ ∈ A \ {0}.Using that ∆(g) = g ⊗ g modulo I iteratively, we get that g d = 1 modulo I and hence (g − 1)(1 + g + g 2 + • • • + g d−1 ) ∈ I.But 1 + g + g 2 + • • • + g d−1 is a unit if d is a unit and hence g = 1 modulo I.The element g = 1 + t(T 10 − 1) is not equal to 1 (we used Macaulay 2 to check this) but group-like modulo I. Indeed, since

Lemma 3. 16 .
Let f : Spec Z[Q] → Spec Z[P ] be the morphism induced by a morphism φ : P → Q of integral monoids.Then f is flat if and only if φ is flat if and only if φ is integral and injective.Proof.See [Ogu18, Remark 4.6.6].
is a ramified D(A)-cover (Definition 2.1).Remark 4.4.Definition 4.2 also makes sense when A is just a monoid.A free extension of A with values in P without the choice of a section ι is called a Schreier extension (see [R 52, Str72, Ina65] or [Pat18, Definition 4.1]).
Definition 4.15.Define m : Z A / e 0 → A to be the group homomorphism defined on generators by e λ → λ.Remark 4.16.As in [Ton14, Definition 4.4] we may consider the short exact sequence of abelian groups 0 → K → Z A / e 0 m − → A → 0 e λ → λ .and ϕ A : P A → Z A / e 0 factors through P A → K, which is the groupification of P A [Ton14, Lemma 4.5].

Figure 1 .
Figure 1.The red dots represents the elements of P int A ⊂ Q int A and the blue dots represents the elements of Q int A \ P int A with A = Z/3Z.Definition 4.20.The group homomorphism Z A / e 0 → Z defined on generators by sending e λ to 1 is written q → |q| and we call |q| the value of q.

Remark 6. 3 .
Recall that O X -modules corresponds to equivariant O X -modules and since f : X → S is affine, these in turn corresponds to equivariant f * O X -modules on S. The module O X may hence be thought of as the f * O X -module f * O X on S together with the A-grading given by the action.The canonical D(A)-torsor p : X → X corresponds to a morphism of stacks X → BD(A) and the character λ :D(A) → G m gives a morphism λ : BD(A) → BG m .The stack BG m has a canonical G m -torsor Spec Z → BG m whichis the relative spectrum of a Z-graded BG m -algebra.The graded piece of weight 1 is just O BGm [1], that is, O BGm shifted by 1.This means that O BGm [1] is the Z-graded Z-module which is 0 in every degree except in degree −1 where it is Z.Pulling back O BGm [1] along λ : BD(A) → BG m we get the line bundle O BD(A) [λ], which is O BD(A) shifted by λ.If we now pull back O BD(A) [λ] along X → BD(A) we get the line bundle O X [λ] which is nothing but O X shifted by λ.Remark 6.4.Let π : X = [X/D(A)] → S be the structure morphism.First note that π * O X [λ] = L λ .
and if we pull back along the canonical D(P gp A )-torsor Spec Z[P A ] → [Spec Z[P A ]/D(P gp A )], we get a Cartesian diagram Remark 9.4.For a more explicit definition of a parabolic sheaf, see [BV12, Definition 5.6].

E
e λ −e λ ′ | s (E(e λ ′ )|s) λ ′ − −−−−−−− → E e λ | s OG s (F | Gs , E λ | Gs ) ≃ π * Hom OG s (F | Gs , E λ | Gs ) ≃ π * g * (F ∨ ⊗ E λ ) .Let E F ∨ be the parabolic vector bundle corresponding to F ∨ under the equivalence of Theorem 9.5.Consider the ideals J s ⊆ O S and J s ⊆ O X generated by the image of λ,λ ′ Then I is generated by {x λ (λ − 1)} λ∈A .Now let (R, m) be a local ring, let S = Spec R, let A be a finite abelian group, and let H ֒→ G := D S (A) be a closed subgroup cut out by a Hopf ideal I. Lemma A.6.If the order d of A is invertible in S, then R[H] gr ∩ (1 + mR[H]) = {1} .Proof.Let g be an element of R[A] representing an element in R[H] gr ∩ (1 + mR[H]