ERIC AHLQVIST
Abstract. We define stacky building data for stacky covers in the spirit of Pardini and give an equiv-alence 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.
Contents
1. Introduction 1
2. Ramified covers 4
3. Closed subgroups of groups of multiplicative type 11
4. Deligne–Faltings structures and root stacks 14
5. Special Deligne–Faltings data 19
6. Deligne–Faltings data from ramified D(A)-covers 25
7. Stacky covers as root stacks 28
8. Building data for stacky covers 34
9. Parabolic sheaves and an application 40
References 45
1. Introduction
The class of stacky covers contains flat 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 later 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 bulding data `a la Pardini [Par91].
A stacky cover π :X → S of a scheme S consists of a Deligne–Mumford stack X which has finite diagonalizable stabilizers at closed points, together with a morphism π :X → S which is
(1) flat, proper, of finite presentation, (2) a coarse moduli space, and
(3) has the property that π∗ takes line bundles to line bundles.
For instance, ifX → S is a flat stacky modification, that is, flat, proper, locally of finite presentation, and birational with finite diagonalizable stabilizers, thenX → 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 ´etale sheaf of abelian groups A over S. From A we construct two quasi-fine ´etale sheaves of monoids PA, QA,
and a flat Kummer homomorphism γA: PA → QA. Then we construct a symmetric monoidal functor
L : PA→ DivS´et where DivS´et denotes the restriction of [A
1
S/Gm,S] to the small ´etale site of S. Hence
we get a diagram
(1)
PA DivS´et
QA
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 7.14. Let π : X → S be a stacky cover. Then there exists a canonical (up to canonical isomorphism) building datum (A, L) where π∗A ∼= D(IX) is the Cartier dual of the inertia stack, 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 8.6. 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 ´etale 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 Lλ for each λ ∈ A together with global sections
sλ,λ0 ∈ Γ(S, L−1λ ⊗ L−1λ0 ⊗ Lλ+λ0) corresponding to the multiplication in OX. 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 PA and QA with a flat Kummer
homomorphism between them. These sit in an exact sequence of monoids 0 → PA→ QA→ 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 PA→ [A1S/Gm,S] such that the root stack of the diagram
PA [A1S/Gm,S]
QA
(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 Cartier dual D(IX) of the inertia stack ofX . The inertia stack will ´etale locally be a closed subgroup of a diagonalizable group and in Section 3 we show that its Cartier dual is representable, ´etale and descends to an ´etale group scheme on S.
The category of quasi-coherent sheaves on a root stackX /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 9.9), for some finite abelian group scheme G over k, generalizing the main result of [BB17]. Our theorem looks as follows:
Theorem 9.20. 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 (PA, QA, 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) → As¯is surjective, and
(ii) for every λ ∈ As¯, there exists an essentially finite, basic, parabolic vector bundle (E, ρ) on
(S, PA, QA, L) such that the morphism
M
λ0
Eeλ−eλ0|s¯
(E(eλ0)|s¯)λ0
−−−−−−−−→ Eeλ|s¯
is not surjective, where the direct sum is over all λ0∈ Γ(S, A) such that λ0 ¯ s6= 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 9.20. IfX = S(A,L)= S(PA,QA,L)is the associated root stack
then it follows thatX ' [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. We define an operation on the category of ramified Galois covers for diagonalizable groups which will be compatible with an operation defined later for root stacks.
In Section 3 we study the Cartier dual of a closed subgroup H of a multiplicative group and show that it is representable and ´etale. We use this in Proposition 7.2 to show that the Cartier dual of the inertia stack of a stacky cover descends to an ´etale group scheme on the base.
In Section 4 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 5 we define and prove statements about the universal monoids PA and QA that we use
use to model the local charts of our stacky covers. For instance, we show that the monoids PA and
QA associated to A are quasi-fine and sharp and that the action of PA on QA 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 → PA corresponding to a universal free extension
0 → PA→ QA→ A → 0.
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 (PA, QA, LX) 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. One of the key ingredients is that given a character of the inertia stack there is an essentially unique line bundle on which the inertia acts with that character and whose pushforward to the base is trivial.
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.
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 A-graded R-module, then B[λ] is the A-graded R-module with
B[λ]λ0 = Bλ+λ0.
Whenever we have a morphism s : L → L0 we denote by Lc(s) the line bundle L∨⊗ L0 and by lc(s) the
Stab(f ) for the sheaf of groups Autf. The unit object of a symmetric monoidal category will be denoted
by1.
Acknowledgements. I want to thank my supervisor David Rydh for many invaluable discussions and his great enthusiasm for the subject. I would also like to thank Angelo Vistoli and Martin Olsson for useful discussions during the MSRI program in the spring of 2019.
2. Ramified covers
Throughout this section we will always assume that A is an abelian group and G = D(A) the corre-sponding finite diagonalizable group over the base scheme S. This means that
G = DS(A) =Homgrp(AS, Gm) ∼ = Spec OS[A] . We also write OS[G] = OS[A]
for the group algebra of A over OS.
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 {Ui→ S} and an isomorphism of OUi[G]-comodules
(f∗OX)|Ui ∼= OUi[G] ,
where the comodule structure on the right hand side is the regular representation. Remark 2.2. This means that we have a splitting
f∗OX ∼=
M
λ∈A
Lλ
where each Lλ is a line bundle and L0= OS. We also have multiplication morphisms
Lλ⊗ Lλ0 → Lλ+λ0
which we think of as global sections
sλ,λ0 ∈ L−1λ ⊗ L−1λ0 ⊗ Lλ+λ0.
These global section will have the following properties: (1) s0,λ= 1 ∀λ ∈ A;
(2) sλ,λ0 = sλ0,λ ∀λ, λ0 ∈ A;
(3) sλ,λ0sλ+λ0,λ00= sλ0,λ00sλ0+λ00,λ ∀λ, λ0, λ00∈ A.
Definition 2.3. Let S be a scheme. A generalized 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−1λ ⊗ L−1λ0 ⊗ Lλ+λ0, sλ,λ0) forms a generalized Cartier divisor. Note
that the data of a generalized Cartier divisor (L, s) is equivalent to the data of a morphism of stacks S → [A1
/Gm].
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 in Tonini’s more general setup.
Remark 2.6. The ramification locus of a ramified cover which is generically a torsor has pure codimen-sion 1 [AK70, Theorem 6.8].
Remark 2.7. Any finite locally free morphism f : X → S of rank 2 is a µ2-cover if 2 is invertible in
Γ(S, OS). Indeed, there is a trace map T : f∗OX → OS sending a section x to the trace of the matrix
corresponding to multiplication by x. The composition OS→ f∗OX→ OS is multiplication by 2 and if
2 is invertible, we get that
OS → f∗OX
1 2T
−−→ OS
is the identity and hence
f∗OX∼= OS⊕ L ,
where L = ker T is a line bundle. It remains to show that the multiplication L ⊗ L → f∗OX lands in OS.
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 OS⊗ L → f∗OX 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 µ2-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]/(x2− 2) is a µ2-cover when x has weight
1.
(2) The map on spectra induced by the inclusion C[s] → C[s, x, y]/(x2− sy, y2− sx, xy − s2) is a
µ3-cover when x has weight 1 and y has weight 2.
(3) The map Proj C[x, y, z]/(z2−x2
) → P1
C= S induced by the inclusion C[x, y] → C[x, y, z]/(x 2−z2)
is a µ2-cover when z has weight 1. We have Proj C[x, y, z]/(z2− x2) ∼= Spec (OS⊕ OS(−1)) with
multiplication given by x2∈ 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 µ2-cover (flatness
follows from [Eis95, Corollary 18.17]).
(5) In particular, a K3 surface obtained as a double cover of P2branched along a sextic is a µ 2-cover.
Example 2.9 (Non-example). Here is an example suggested by Magnus Carlson, that could be confused of being a ramified cover for a diagonalizable group. Let p ≥ 3 be an odd prime number and let K be the cyclotomic field obtained by adding a primitive p:th root of unity to Q. Let OK ⊂ K be the ring of
integers. Then Spec OK → 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 2.28. However, Spec OK→ Spec Z is a ramified (Z/pZ)×-cover in the sense of [Ton14, Definition 2.1].
Despite the fact that Spec OK → Spec Z of the previous example is not a ramified D(Z/(p − 1))-cover,
we have that the stack quotient [Spec OK/Z/(p − 1)] is a stacky cover (Definition 7.1). Also see Example
8.7.
Definition 2.10. We define (D-Cov/S) to be the category with objects given by pairs (A, X) where X → S is a D(A)-cover (we denote the action by σX) and morphisms (A0, X0) → (A, X) given by pairs
(ϕ, f ) where ϕ : A → A0 is a group homomorphism and f : X0 → X is a D(ϕ)-equivariant S-morphism, i.e., the following diagram commutes:
D(A0) ×SX0 D(A) ×SX
X0 X .
D(ϕ)×f
σX0 σX
f
For a scheme T → S we define D-Cov(T ) to be the subgroupoid with the same objects as (D-Cov/T ) but where we require a morphism (ϕ, f ) : (A0, X0) → (A, X) to be such that ϕ and f are isomor-phisms. The corresponding fibered category (stack) D-Cov will have objects (T, A, X) and morphisms
(h, ϕ, f ) : (T0, A0, X0) → (T, A, X) such that the induced morphism (A0, X0) → (A, T0×T X) is an
iso-morphism in (D-Cov/T0). Note that this implies that ϕ : A → A0 is an isomorphism and the diagram
X0 X T0 T f h is cartesian. Operations on covers.
Definition 2.11 (Induced covers). Suppose we have a D(A1)-cover X1 and a D(A2)-cover X2given by
collections of line bundles L(X1)λ, L(X2)γrespectively and sections s(X1)λ,λ0 and s(X2)γ,γ0 respectively.
Given group homomorphisms ϕ1: A → A1and ϕ2: A → A2 we define a new D(A)-cover as follows: Let
X1∨ X2= X1 ϕ1∨ϕ2X2 be the D(A)-cover given by line bundles
L(X1∨ X2)λ= L(X1)ϕ1(λ)⊗ L(X2)ϕ2(λ)
and sections
s(X1∨ X2)λ,λ0 = s(X1)ϕ
1(λ),ϕ1(λ0)⊗ s(X2)ϕ2(λ),ϕ2(λ0).
In case we have morphisms α1: X1 → Y1 and α2: X2 → Y2 in (D-Cov/S), we denote the induced
morphism
α1∨ α2: X1∨ X2→ Y1∨ Y2.
The operation defined above in particular gives us induced covers, i.e., given a D(A)-cover X and a group homomorphism A0→ A, we get a D(A0)-cover X0 (take X
2= S and A2= 0).
Remark 2.12. Note that
X1∨ X2∼= X1×SX2×SD(A)/D(A1) ×SD(A2)
where the coaction is given one generators by
OX1⊗OS OX2⊗OS OS[A] → OX1⊗OS OX2⊗OS OS[A] ⊗OS OS[A1] ⊗OS OS[A2]
vλ1⊗ vλ2⊗ λ 7→ vλ1⊗ vλ2⊗ λ ⊗ ϕ1(λ)
−1λ
1⊗ ϕ2(λ)−1λ2.
The action of D(A) is given by translation on the third factor of X1×SX2×SD(A)/D(A1) ×SD(A2).
Remark 2.13. If A = A1× A2 in Definition 2.11, then X1∨ X2∼= X1×SX2.
Remark 2.14. The operation ∨ is clearly associative and commutative (up to isomorphism). However, it is only “invertible” when X1or X2is a torsor. When X1is a torsor we have that all the global sections
sλ,λ0 ∈ Γ(S, L−1
λ ⊗ L −1
λ0 ⊗ Lλ+λ0) are invertible. Hence we may construct a torsor X1−1 whose algebra is
L λ∈AL −1 λ and multiplication L−1λ ⊗ L−1λ0 → L −1 λ+λ0
given by the inverse of the morphism
L−1λ+λ0 → L
−1 λ ⊗ L
−1 λ0
corresponding to sλ,λ0. This will give X1−1∨ X1∨ X2∼= X2.
Example 2.15. The map on spectra induced by the inclusion C[s] → C[s, x, y]/(x2− sy, y2− sx, xy − s2)
is a µ3-cover where x has weight 1 and y weight 2. This cover is the composition of two covers of the form
Spec C[s, t]/(t3− s) where t has weight 1 in the first, and weight 2 in the second. Indeed, put x = t ⊗ t2
and y = t2⊗ t. Then we get x2= (t ⊗ t2)2= s(t2⊗ t) = sy and y2= (t2⊗ t)2= s(t ⊗ t2) = sx.
Example 2.16. The µ2-cover corresponding the inclusion C[x] → C[x, z]/(z2− x2) where z has weight
1 is decomposable into two copies of the µ2-cover corresponding the inclusion C[x] → C[x, z]/(z2− x)
Example 2.17. The µ2-cover Proj C[x, y, z]/(z2 − x2) → P1C induced by the inclusion C[x, y] →
C[x, y, z]/(z2 − x2) where z has weight 1 is not decomposable to pieces with reduced branch locus. Indeed,
Proj C[x, y, z]/(x2− z2) ∼
= SpecP1O ⊕ O(−1)
and the line bundle O(−1) does not have a root. However, it is decomposable over the two standard affine charts (see the previous example).
Remark 2.18 (Quotients). If p : X → S is a D(A)-cover and A0 ⊆ A a subgroup, we get a D(A0)-cover
X0 → S as follows: We have a decomposition p
∗OX ∼= Lλ∈ALλ which gives an algebra structure on
the submodule L
λ0∈A0Lλ0. The cover X0 → S is the spectrum of this algebra. It is clear that X → S
factors as X → X0 → S and that X admits an action of D(A/A0) over X0. It is also clear that X → X0
is finite. However, X → X0 need not be flat.
This means that whenever we have a D(A)-cover X → S and a decomposition A ∼= Z/pe1
1 × . . . Z/p el
l
for prime numbers p1, . . . pl, there exists a sequence X = Xn → Xn−1 → · · · → X1 → X0 = S, where
each Xi→ Xi−1is a finite (possibly non-flat) cover of prime degree where n =Plj=1ej.
Covers and 2-cocycles.
Definition 2.19 (cf. [Ton14, Section 4.1]). 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) f (0, λ) = 0 ∀λ ∈ A;
(2) f (λ, λ0) = f (λ0, λ) ∀λ, λ0∈ A;
(3) f (λ, λ0) + f (λ + λ0, λ00) = f (λ0, λ00) + f (λ0+ λ00, λ) ∀λ, λ0, λ00∈ A.
A morphism of 2-cocycles f0→ f consists of a morphism of monoids h : P0→ P such that the diagram
A × A
P0 P
f0 f
h
commutes.
Remark 2.20. Note that the set of 2-cocycles of A × A → P form a monoid under pointwise addition. Definition 2.21. We denote the monoid of 2-cocycles A × A → P by Z2c(A, P).
Remark 2.22. There is a bijection between the set of 2-cocycles A × A → N and the set Hom(PA, N) of
rays as in [Ton14, Notation 3.11], where PA is the universal monoid we define in Definition 5.10. This
will be explained in detail in Section 5.
Recall that if L is a line bundle on a scheme S then we have a bijection {s ∈ Γ(S, L) : s is regular}/ ∼ →(effective Cartier divisors
D such that OS(D) ∼= L
) , where s ∼ s0 if there is a unit u ∈ Γ(S, O×S) such that us = s0.
Let S be a normal scheme and let X → S be a D(A)-cover with branch locus B =S
i∈IDi (union of
irreducible components with I finite), which is generically a torsor. Then the global sections sλ,λ0 ∈ Γ(S, L−1
λ ⊗ L −1
λ0 ⊗ Lλ+λ0)
are all regular and correspond to divisors Dλ,λ0 such that
OS(Dλ,λ0) ∼= L−1
λ ⊗ L −1
Hence we get a 2-cocycle
fX: A × A → NI
(λ, λ0) 7→ ordDi(Dλ,λ0) ,
which we refer to as the 2-cocycle of the cover X. The cover X is determined by fX (see Proposition
2.23). If X1 ϕ1∨ϕ2X2 then fX(λ, λ0) = fX1(ϕ1(λ), ϕ1(λ 0)) + f X2(ϕ2(λ), ϕ2(λ 0)) .
Structure of D(A)-covers over a locally factorial scheme. Let p : X → S be a D(A)-cover with multiplication in p∗OX given by
sλ,λ0 ∈ Γ(S, L−1
λ ⊗ L −1
λ0 ⊗ Lλ+λ0) .
Assume that the cover is generically a torsor. Then the global sections sλ,λ0 are all regular since they
are generically isomorphisms. If S is locally factorial then cyc : Div(S) → WDiv(S)
is an isomorphism and hence cyc(L−1λ ⊗ L−1λ0 ⊗ Lλ+λ0, sλ,λ0) determines (L−1λ ⊗ L−1λ0 ⊗ Lλ+λ0, sλ,λ0).
Let C be an irreducible component of the branch locus B and let OX,C= X ×SSpec OS,C. Then
Spec OX,C→ Spec OS,C
is an affine ramified G-cover. For each λ ∈ A we let vλ be a generator for the graded piece of OX,C
corresponding to λ (the graded pieces are free since we are over a local ring). Proposition 2.23. With the setup just described, we have an isomorphism
L−1λ ⊗ L−1λ0 ⊗ Lλ+λ0 ∼= OS(
X
C⊆B irred
ordC(sλ,λ0)[C])
sending sλ,λ0 to the canonical global section. Let s be a uniformizer of OS,C. Then ordC(sλ,λ0) can be
determined by the formula
ordC(sλ,λ0) = min{n ∈ N : sn∈ (vλvλ0 : vλ+λ0)}
where (vλvλ0 : vλ+λ0) is the ideal quotient.
Proof. The pair (L−1λ ⊗ L−1λ0 ⊗ Lλ+λ0, sλ,λ0) determines an effective Cartier divisor which in turn gives
the Weil divisor
X
C⊆B irred
ordC(sλ,λ0)[C]
since sλ,λ0 has support in B. This proves the first part.
To prove the second part we consider the cover Spec OX,C → Spec OS,C. We have
OX,C= OS,C[{vλ}λ∈A]/({vλvλ0− sord(sλ,λ0)vλ+λ0}(λ,λ0)∈A2) .
This proves the second part since OX,C is free over OS,C.
Remark 2.24. Note that we could replace OS,C by its strict henselization [SP, Tag 0AP3].
Remark 2.25. The function
A × A → N
(λ, λ0) 7→ ordC(sλ,λ0)
In [Par91], Pardini gives an explicit description of Proposition 2.23 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 mT/m2T, 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
B =X
N
X
ψ
DN,ψ,
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 7→ min{a : ψa= x}. For λ, λ0∈ A (and N , ψ as above), Pardini defines
εN,ψλ,λ0 =
(
0 , if i(λ) + i(λ0) < |N | , 1 , otherwise .
We have that εN,ψ(−,−)is a 2-cocycle and the following theorem is part of [Par91, Theorem 2.1]. Theorem 2.26. Let C be a component with cyclic group N and generator ψ. Let
sλ,λ0 ∈ Γ(S, L−1
λ ⊗ L −1
λ0 ⊗ Lλ+λ0)
be the global section corresponding to the multiplication Lλ⊗ Lλ0 → Lλ+λ0. Then
ordC(sλ,λ0) = εN,ψλ,λ0.
For completeness, we give a proof.
Proof. Let φ : A → N be the dual of the inclusion D(N ) → D(A). The cover XCsh= X ×SSpec OshS,C → S sh C = Spec O sh S,C factors as XCsh→ X0 C → S sh C
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) - |A|). Hence Γ(XC0 , OX0
C) is just a product of copies of O
sh
S,C and we may replace
Ssh
C by one of the connected components of XC0 . Hence we may assume that X sh
C → S
sh
C is connected
which implies that R = Γ(XCsh, OshX
C) is a local ring. By [Par91, Lemma 1.2], we may choose a generator
xψ for the line bundle of XCsh of weight ψ such that xψ is a generator of the maximal ideal of R. Let
s = x|N |ψ . Then s is a generator for the maximal ideal in Osh
S,C since R is normal by assumption. It
follows that the line bundle Lλ,C is generated by xλ= x i(λ) ψ and since xλxλ0 = sλ,λ0xλ+λ0 we get that sλ,λ0 = ( 1 , if i(λ) + i(λ0) < |N | , s , otherwise .
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 ×SX → 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 ΩX/S do not vanish. Hence there is a canonical scheme
structure on the ramification locus, namely
Spec OX/ Ann(ΩX/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) ⊆ OS, and
(2) the ideal OS∩ Ann(ΩX/S) = Ann(π∗ΩX/S).
Remark 2.27. One could also consider the zeroth Fitting ideal Fitt0(ΩX/S) ⊆ Ann(ΩX/S) which has the
same radical as Ann(ΩX/S). But we will not do this here.
Lemma 2.28. 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
R[{vλ}λ∈A]/({vλvλ0 − sλ,λ0vλ+λ0}λ,λ0∈A)
and the discriminant d(π) of the cover is given by the formula
d(π) = |A||A| Y
λ∈A
sλ,−λ
! .
Proof. The first assertion is trivial and we prove the second. Multiplication by a generator vλ sends
vλ0 to sλ,λ0vλ+λ0. The corresponding matrix will have no element on the diagonal if λ 6= 0, and a 1 at
every entry of the diagonal otherwise. Hence the trace of the matrix will be either 0 or |A|. We want to compute the determinant of the map defined on the generators by
vλ7→ Tr(− · vλ)
and by the previous argument we may write
Tr(− · vλ) = |A|sλ,−λv∗−λ
where v∗
−λ is the dual of v−λ. Hence the matrix T of
vλ7→ Tr(− · vλ)
has determinant
det(T ) = ±|A||A| Y
λ∈A
sλ,−λ.
Lemma 2.29. If π : X → S is a D(A)-cover, then
d(π) ⊆ Ann(π∗ΩX/S) .
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 vλ be a generator for the graded piece of
OX corresponding to λ ∈ A. The module of K¨ahler differentials is generated by the elements dvλ. Let
n = nλ be the order of λ ∈ A. We know that vλn lies in the zeroth piece so nv n−1
λ dvλ = 0 and hence
for each λ ∈ A. For n even, we have d(π) = |A||A| Y λ∈A sλ,λ−1 = |A||A|˜s(sλ,λn−1sλn−1,λ)(sλ2,λn−2sλn−2,λ2) . . . = |A||A|˜s(sλ,λsλ2,λn−1)2(sλ2,λsλ3,λn−2)2. . . (sλn/2−1,λsλn/2,λn/2)2sλn/2,λn/2 = |A||A|˜s0sλ,λsλ2,λ. . . sλn/2−1,λsλ,λn/2−1. . . sλ2,λsλ,λ = |A||A|˜s0((. . . (vλvλ)vλ) . . . (vλ(vλvλ)) . . . ) = |A||A|˜s0vλn
which annihilates dvλ since n divides |A|. Similarly, when n is odd we get
d(π) = |A||A| Y λ∈A sλ,λ−1 = |A||A|˜s(sλ,λn−1sλn−1,λ)(sλ2,λn−2sλn−2,λ2) . . . = |A||A|˜s(sλ,λsλ2,λn−1)2. . . (s λn−12 −1,λsλn−12 ,λn+32 ) 2s λn−12 ,λn+12 = |A||A|˜s00sλ,λsλ2,λ. . . s λn−12 ,λsλn−12 ,λn+12 sλn−12 ,λ. . . sλ,λ
which again annihilates dvλ. Since λ was arbitrary we conclude that d(π) ⊆ Ann(π∗ΩX/S).
The inclusion in Lemma 2.29 is most often strict.
Example 2.30. Consider the µ3,S-cover π : X := Spec Z[s, x, y]/(x2 − sy, y2− sx, xy − s2) → S :=
Spec Z[s] where x has weight 1 and y weight 2. We have Ann(π∗ΩX/S) = (3s2)
d(π) = (27s4) .
3. Closed subgroups of groups of multiplicative type
In the following section we investigate the Cartier dual of a closed subgroup H of a diagonalizable group G = D(A). We will see that the Cartier dual D(H) is represented by an ´etale group scheme. We begin with a definition.
Definition 3.1. If K is a group scheme over S the Cartier dual of K is the functor DS(K) =Homgrp(K, Gm,S) .
If K is affine over S, DS(K) is isomorphic to the functor
T 7→ Γ(KT, OKT)
gr
= {g ∈ Γ(KT, OKT) : g is group-like}
(see Remark 3.3). We often ignore the subscript S and write D(K) = DS(K). We call a group morphism
λ : K → Gma character.
Remark 3.2. Note that H ,→ G is a closed subgroup scheme if and only if (1) it is a closed subscheme and
(2) I = ker(R[G] → R[H]) is a Hopf ideal, i.e., if ε, σ, ∆ denotes the counit, coinverse and comulti-plication respectively, we require that
ε(I) = 0 , σ(I) ⊆ I , and
Remark 3.3. Recall that an element g of a Hopf algebra is group-like if ∆(g) = g ⊗ g and ε(g) = 1. We denote the subset of group-like elements of a Hopf algebra E by Egr. It is immediate to check that Egr form a group under multiplication. Note that when A is an abstract abelian group then R[A]gr= A. Remark 3.4. Let G be a group scheme over S acting on a line bundle L on S (this also works when S is a stack). This corresponds to a group morphism (character) λ : G →AutOS(L) ∼= Gm,S and we say
that G acts on L via the character λ.
To understand what happens, choose an ´etale covering {Ui→ S} such that L|Ui is trivial for every i.
The coaction
ci: OUi ∼= L|Ui → LUi⊗ O[GUi] ∼= OUi⊗ O[GUi] ∼= O[GUi]
is completely determined by ci(1). Using the axioms of a coaction we get that ε(ci(1)) = 1 and ∆(ci(1)) =
ci(1) ⊗ ci(1). This means that ci(1) is group-like for every i. Since these ci’s are restrictions of a global
coaction
c : L → L ⊗OS O[G]
we get that ci and cj agree over the intersection Ui×S Uj and hence the ci(1)’s glues to a group-like
element in Γ(G, OG) corresponding to a character λ : G → Gm. It follows that the global coaction is
given by c(x) = x ⊗ λ for every local section x of L.
Proposition 3.5. Let S = Spec R with R a strictly Henselian local ring with closed point x : Spec k → S and let A be a finite abelian group. Let H ,→ DS(A) be a closed subgroup scheme which is unramified
over S. Then there exists a quotient A0 of A such that H ⊆ DS(A0) ⊆ DS(A) with DS(A0) ´etale over S
and Hx= DSpec k(A0).
Proof. We have that k[Hx] ∼= k[A]/Ix for some Hopf ideal Ix. Hence k[Hx] is generated by the image of
A = k[A]gr in k[Hx]. But k[Hx]gr is a k-linearly independent set in k[Hx] (see e.g. [Abe80, Theorem
2.1.2]) and hence A = k[A]gr → k[Hx]gr = A0 is surjective. It follows that k[Hx] = k[A0] and hence
Hx∼= Dk(A0) and |A0| is invertible.
By writing A ∼= Z/pl1
1Z ⊕ · · · ⊕ Z/p ls
sZ for primes pi we see that
DS(A) ∼= DS(Au) ×SDS(Ar)
where Au has invertible order in R and no prime factor in |Ar| is invertible. The morphism H ,→
DS(Au) ×S DS(Ar) corresponds to a morphism R[A] ∼= R[Au] ⊗R R[Ar] → R[A]/I. Suppose that
λ ∈ R[Ar] is group-like and that 1 ⊗ λ 6= 0 (mod I). Then the image of 1 ⊗ λ in k[H
x] ∼= Dk(A0) is a
group-like element of order which is not invertible. This is impossible since A0 is invertible. Hence we conclude that the morphism R[Ar] → R[H] sends every group-like element to 1. Hence H ,→ DS(A)
factors through DS(Au) ,→ DS(A). By replacing DS(A) by DS(Au) we may assume that |A| is invertible
in R.
Since H is finite over S we have Γ(H, OH) ∼=Q d
i=1Ri where each Ri is a local ring [Mil80, Theorem
4.2]. Similarly, we have R[A] =Qn
j=1R where n is the order of A.
At this point we have two closed subgroup schemes H ,→ DS(A) and DS(A0) ,→ DS(A) with Hx∼=
(DS(A))x. But the open and closed subscheme of DS(A) consisting of the components hit by H is the
same as the components hit by Hx∼= Dk(A0) and since DS(A) is ´etale, this is DS(A0).
Lemma 3.6. Let S = Spec R with (R, m) a strictly Henselian local ring with k = R/m and let x : Spec k → S be the closed point. Let A be an abelian group whose order is invertible in R and let H ,→ DS(A) be a closed subgroup scheme which induces an isomorphism Hx∼= (DS(A))x. Then
R[H]gr∩ (1 + mR[H]) = {1} .
Proof. Put d = |A|. Since d is invertible it follows that R contains a primitive dth root of unity which means that there exists an isomorphism D(A) ∼= A where A is the constant group scheme defined
by A. Hence the Hopf algebra R[A] has a basis {eλ}λ∈A with eλeλ0 = δλ,λ0 (Kronecker’s delta) and
1 =P
λ∈Aeλ. The counit is given by ε(eλ) = δ0,λ and the comultiplication by
∆(eλ) =
X
λ0∈A
eλ0⊗ eλ−λ0.
Let π : R[A] → R[H] be the quotient map and let I = ker(π). Let α =P
λ∈Aaλeλ∈ R[A] be an element
such that α (mod I) is group-like and whose image in R[H]⊗Rk is 1. Then a0= ε(α) = 1 since α (mod I)
is group-like and I ⊆ ker ε. Also for every λ ∈ A we may write aλ = 1 + bλ for some bλ ∈ m since α
maps to 1 =P
λ∈Aeλ∈ R[H] ⊗Rk ∼= k[A]. We denote the multiplication by ∇ : R[A] ⊗ R[A] → R[A].
Define for every positive integer n morphisms ∆n: R[A] → R[A]⊗n+1 = R[A] ⊗R⊗ · · · ⊗RR[A] and
∆n: R[A]⊗n+1→ R[A] which are iterations of comultiplication and multiplication respectively: ∆n = (idR[A]⊗(n−1)⊗∆) ◦ (idR[A]⊗(n−2)⊗∆) ◦ · · · ◦ (idR[A]⊗∆) ◦ ∆
∇n = ∇ ◦ (∇ ⊗ id
R[A]) ◦ · · · ◦ (∇ ⊗ idR⊗(n−1) A
) .
Since α (mod I) is group-like we get that ∇n−1◦ ∆n−1(α) = αn (mod I) for all n ≥ 1. Clearly
αn=X
λ∈A
anλeλ
and we have that
∇n−1◦ ∆n−1(α) = X λ∈A anλeλ. Indeed, ∇n−1◦ ∆n−1(α) = ∇n−1◦ ∆n−1 X λ∈A aλeλ ! = ∇n−1◦ ∆n−2 X λ∈A aλ X λ1∈A eλ1⊗ eλ−λ1 ! = ∇n−1◦ ∆n−3 X λ∈A aλ X λ1∈A eλ1⊗ X λ2∈A eλ2⊗ eλ−λ1−λ2 ! .. . = ∇n−1 X λ∈A aλ X λ1∈A eλ1⊗ · · · ⊗ X λn−1∈A eλn−1⊗ eλ−λ1−···−λn−1 and ∇n−1(e λ1⊗ eλ2⊗ · · · ⊗ eλn−1⊗ eλ−λ1−···−λn−1) is eλ1 if λ1= λ2= · · · = λn−1= λ − λ1− · · · − λn−1,
i.e., λ = nλ1= nλ2= · · · = nλn−1, and zero otherwise. Hence we get that
αn = ∇n−1◦ ∆n−1(α) = X
λ∈A
anλeλ (mod I)
and in particular αd= 1 (mod I). We have
1 + α + α2+ · · · + αd−1= 1 + α + ∇ ◦ ∆(α) + · · · + ∇d−2◦ ∆d−2(α) (mod I) = X λ∈A eλ+ X λ∈A aλeλ+ X λ∈A a2λeλ+ · · · + X λ∈A a(d−2)λeλ(mod I) = X λ∈A cλeλ (mod I)
where each cλ is a sum of d units aλ0 = 1 + bλ0. This implies each cλ is of the form cλ = d + b for
some b ∈ m and we conclude that 1 + α + α2+ · · · + αd−1 is a unit. Hence α = 1 (mod I) since (α − 1)(1 + α + α2+ · · · + αd−1) = αd− 1 ∈ I. This completes the proof. Proposition 3.7. Let S = Spec R with (R, m) a strictly Henselian local ring with k = R/m and let x : Spec k → S be the closed point. Let A be an abelian group whose order is invertible in R and let H ,→ DS(A) be a closed subgroup scheme which induces an isomorphism Hx ∼= (DS(A))x. Then the
quotient π : R[A] → R[H] is surjective (and hence an isomorphism) on group-like elements. Proof. Let T =P
λ∈Aaλλ ∈ R[A] be such that π(T ) ∈ R[H] is group-like. Let λ0 be the image of π(T )
in k[H]gr= A. Then λ−10 π(T ) ∈ R[H]gr∩ (1 + mR[H]) = {1} by Lemma 3.6 and hence π(T ) = λ0.
Combining Proposition 3.7 and 3.5, we obtain:
Corollary 3.8. Let S = Spec R with R a strictly Henselian local ring with closed point x : Spec k → S and let A be a finite abelian group. Let H ,→ DS(A) be a closed subgroup scheme which is unramified
over S. Then
R[H]gr∼= k[Hx]gr.
The following is the main result of this section:
Theorem 3.9. Let G → S be a finite diagonalizable group scheme and let H ,→ G be a closed subgroup scheme. If H is unramified then DS(H) is locally constructible, i.e., represented by an ´etale group
scheme.
Proof. We have that DS(H) is limit preserving since Gmis locally of finite type. Let π∗be the restriction
functor from the big ´etale site to the small ´etale site and let π∗ be its left adjoint. We want to show that the counit π∗π∗DS(H) → DS(H) is an isomorphism. If t : T → S is a scheme, then DS(H)(T ) =
DT(HT)(T ) and π∗π∗DS(H)(T ) = (t∗π∗DS(H))(T ). Hence it is enough to verify that the induced
morphism t∗π∗DS(H) → π∗DT(HT) is an isomorphism for every t : T → S. This can be checked on
stalks. Let ¯x : Spec k → T be a geometric point and ¯s = t ◦ ¯x. Since DS(H) is limit preserving, we have
that
(t∗π∗DS(H))x¯∼= (π∗DS(H))¯s∼= DS(H)(Spec OS,¯s)
and
(π∗DT(HT))x¯∼= DT(HT)(Spec OT ,¯x) .
But the morphism
DS(H)(Spec OS,¯s) → (π∗D(Hx¯))(¯x) = DS(H)(Spec k)
is an isomorphism by Corollary 3.8 and factors through the isomorphism DT(HT)(Spec OT ,¯x) → DS(H)(Spec k) .
Hence we conclude that t∗π∗DS(H) → π∗DT(HT) is an isomorphism. This completes the proof.
4. 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 4.1. A monoid P is called
(1) finitely generated if there is a number n ∈ N and a surjection Nn→ P ; (2) sharp if P×= {0};
(3) integral if p, q, q0 ∈ P and p + q = p + q0 implies that q = q0;
(4) u-integral if P× acts freely on P ;
(6) fine if it is integral and finitely generated;
(7) quasi-fine if it is quasi-integral and finitely generated.
Remark 4.2. A monoid M is integral if and only if the canonical map M → Mgp is injective.
Deligne–Faltings structures.
Definition 4.3. We denote byDivS´et the restriction ofDivS = [A
1
S/Gm,S] to the small ´etale site of S.
The following definition is very closely related to the notion of a log structure (see Remark 4.7): Definition 4.4. Let S be a scheme. A pre-Deligne–Faltings structure (pre-DF-structure short)
L : P →DivS´et
consists of
(1) a presheaf P of monoids on S´et, and
(2) a symmetric monoidal functor L : P →DivS´et.
A pre-DF-structure is called a Deligne–Faltings structure (DF-structure short) if P is a sheaf and L has trivial kernel.
Remark 4.5. Given a pre-Deligne–Faltings structure there is a notion of the associated Deligne–Faltings structure [BV12, Proposition 3.3].
Remark 4.6. 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
a ⊗ b id ## γa,b // b ⊗ a γb,a a ⊗ b
commutes, where γa,b: a ⊗ b → b ⊗ a 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 4.7. 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 : M → OS such that the action of ρ−1OS×' O
× S
on M is free. If P →DivS´et is a Deligne–Faltings structure, then the corresponding log structure is given
by the projection
P ×DivS´etOS → OS,
where the map OS →DivS´et sends f ∈ Γ(U, OU) to (OU, f ).
Remark 4.8. Note that L may have trivial kernel but still map different elements to isomorphic objects. For example, let P be the constant monoid N2 and L a line bundle on S with a global section s. Then
the symmetric monoidal functor which sends both (0, 1) and (1, 0) to (L, s) has trivial kernel. Definition 4.9. A morphism of monoids ϕ : P → Q is called Kummer if
(1) it is injective and
(2) for every q ∈ Q the is an n ∈ N and p ∈ P such that ϕ(p) = nq.
A morphism of ´etale sheaves of monoids P → Q is called Kummer if for every geometric point x ∈ S, Px→ Qx is Kummer.
Definition 4.10. 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 PS → P is a cokernel in the category of sheaves of monoids. An atlas for P consists of an ´etale covering Ui→ S together with charts
Pi → P(Ui) for each i. If ϕ : P → Q is Kummer then a chart for ϕ consists of charts P → P(S),
Q → Q(S), and a Kummer homomorphism P → Q such that the induced diagram
P Q
P(S) Q(S)
commutes.
Definition 4.11. 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 4.12 ([BV12, Lemma 4.7]). Let P be a coherent sheaf of monoids. Let
P Q
P(S) Q(S)
ϕ
be a chart for a Kummer homomorphism P → Q and denote by KP and KQ the kernels of PS → P and
QS → Q respectively. If U → S is ´etale, then q ∈ KQ(U ) if and only if there is an ´etale cover {Ui→ U },
integers ni∈ N, and elements pi∈ KP(Ui) for all i such that ϕ(pi) = niq|Ui.
Proof. Assume that q ∈ KQ(U ). Since P → Q is Kummer, there is an ´etale cover {Ui → U }, integers
ni∈ N, and elements pi∈ PS(Ui) for all i such that ϕ(pi) = niq|Ui. But the image of pi in Q(Ui) is zero
and since P → Q has trivial kernel we see that pi must map to zero in P(Ui), i.e., pi ∈ KP(Ui). The
converse is easy.
A morphism of 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 4.13. A morphism φ : P → Q of monoids is called
(1) integral if it satisfies the following condition: Whenever q1, q2∈ Q and p1, p2∈ P satisfy φ(p1) +
q1= φ(p2) + q2 there exist q0 ∈ Q and p01, p02∈ P such that
q1= φ(p01) + q0,
q2= φ(p02) + q0,
p1+ p01= p2+ p02;
(2) flat if it is integral and satisfy the following supplementary condition: Whenever q ∈ Q and p1, p2∈ P satisfy φ(p1) + q = φ(p2) + q there exist q0∈ Q and p0∈ P such that
q = φ(p0) + q0, p1+ p0= p2+ p0.
Remark 4.14. An injective integral morphism of integral monoids is flat.
Remark 4.15. One may think of the property of being integral as allowing us to complete every pair of solid arrows to a commutative square:
q0 q1 q2 q0. p01 p0 2 p1 p2
Flatness, in addition, allows us to complete every pair of solid arrows to:
q0 q q0,
p0 p1
p2
such that the two compositions agree.
Remark 4.16. Let P ,→ Q be an injective morphism of integral monoids. Note that we can complete every pair of arrows to a commutative square:
q00 q1
q2 q
p2
p1 p1
p2
in the associated group Qgp, if we put q00 = q1− p2 = q2− p1. Hence if there is a p ∈ P such that
q00+ p ∈ Q, p1− p ∈ P , and p2− p ∈ P , then P ,→ Q is integral.
Lemma 4.17. 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 integral and injective.
Proof. See [Ogu18, Remark 4.6.6].
Root stacks.
Definition 4.18. Let L : P → DivS´et 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. If L : P → DivS´et is regular, i.e. sp∈ Γ(S, Lp) is regular for all p ∈ P(U ), ∀ U → S in the site, then we say that the
Deligne–Faltings datum (P, Q, L) is regular.
Definition 4.19 ([BV12, Definition 4.16]). Let L : P →DivS´et 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 SP,Q,L or SQ/P is the fibered category over S associated with the following
pseudo-functor: Let f : T → S be a morphism of schemes. This gives a symmetric monoidal functor f∗L : f∗P →
DivT´et 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 →DivT´et 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: f∗P DivT´et f∗Q , f∗L f∗j α E
(2) morphisms (E0, α0) → (E , α) given by an isomorphism E0→ E such that the diagram f∗L
E0◦ f∗j E ◦ f∗j
If we have a commutative diagram of schemes T0 T S h f0 f
and an object (E , α) in (f∗L)(f∗Q/f∗P), then we get a symmetric monoidal functor
h∗E : h∗f∗P = f0∗P →DivT0 ´ et .
We may also pull back α : f∗L → E ◦ f∗j to an isomorphism h∗α : h∗f∗L → h∗E ◦ h∗f∗j and if we
pre-compose with the natural isomorphism f0∗L → h∗f∗L we get a natural isomorphism
f0∗α : f0∗L → h∗E ◦ f0∗j
and the pair (h∗E, h∗α) defines an object in (f0∗L)(f0∗Q/f0∗P). This defines a functor
h∗: (f∗L)(f∗Q/f∗P) → (f0∗L)(f0∗Q/f0∗P)
on the level of objects. If (E0, α0) → (E , α) is a morphism in (f∗L)(f∗Q/f∗P), then via pullback we get
an isomorphism h∗E0→ h∗E such that the corresponding diagram over (f0)∗L commutes.
This is the pseudo-functor corresponding to the root stack SP,Q,L.
Definition 4.20. When (P, Q, L) is a Deligne–Faltings datum and La: Pa → Div
S the associated
Deligne–Faltings structure (see [BV12, Proposition 3.3]) then we write Pa → Qa for the pushout of
P Pa
Q .
Lemma 4.21. Let P → Q be an integral homomorphism of monoids. Then Pa → Qa is integral.
Proof. See [Ogu18, Proposition 4.6.3(1)].
Proposition 4.22. Let P → Q be a homomorphism of fine sheaves of monoids. The stack SP,Q,L is
flat over S if for every geometric point ¯x ∈ S, the morphism Pa ¯
x → Qax¯ is integral.
Proof. This can be checked ´etale locally. Hence we may assume that there is a chart
P //
Q
P // Q .
By [BV12, Proposition 4.18] we have that SP,Q,Lis isomorphic to the stack S×[Spec Z[P ]/D(P )][Spec Z[Q]/D(Q)]
where D(P ), D(Q) denotes the Cartier duals of P and Q respectively, and the action is given by the obvious grading of Z[P ] and Z[Q] respectively. We have a commutative diagram
[Spec Z[Q]/G] // [Spec Z[Q]/D(Q)] Spec Z[P ] // [SpecZ[P ]/D(P )] where G = D(Qgp/Pgp
). Since D(P ) is fppf over S, we get that Spec Z[Q] → Spec Z[P ] is flat ⇔ [Spec Z[Q]/D(Q)] → [Spec Z[P ]/D(P )] ⇒ SP,Q,L → S is flat. By [BV12, Proposition 3.17], we may
Operations on Deligne–Faltings data. Suppose we are given two diagrams P1 L1 // DivS´et P2 L2 // DivS´et Q1 , Q2
giving rise to root stacks X1 and X2. Suppose also that we have morphisms of sheaves of monoids
ϕi: P → Pi, ψi: Q → Qi for i ∈ {1, 2}, and a homomorphism γ : P → Q making the following diagram
commute P1 P P2 Q1 Q Q2 .
Then there is a Deligne–Faltings structure which on objects looks like L1⊗ L2: P →DivS´et
a 7→ L1,ϕ1(a)⊗ L2,ϕ2(a).
Definition 4.23. We define the stack (X1∨X2, Q/P, {ϕi, ψi}i∈{1,2}) as the root stack associated to
the DF-structure L1⊗ L2defined above and the homomorphism γ : P → Q.
Example 4.24. LetX be the 2nd root stack of (O(1), x) on P1
C= Proj C[x, y], let P = Q = N, put γ to
be multiplication by 2, and let ϕiand ψibe the identity. ThenX ∨X is just the root stack of (O(2), x2),
which is isomorphic to [X/µ2,P1] where X is the cover given by the inclusion C[x, y] → C[x, y, z]/(z2−x2).
5. Special Deligne–Faltings data
Let A be a finite abelian group. The idea of the following section is to introduce monoids PA and
QA together with a homomorphism γA: PA → QA such that every D(A)-cover X → S will give rise
to a symmetric monoidal functor LX: PA → Div S and such that the root stack associated to the
Deligne–Faltings datum
PA Div S
QA γA
is isomorphic to [X/D(A)]. The monoids PA, QA and the morphism γA will depend only on the group
A but LX will depend on X and the action of D(A).
Free extensions and 2-cocycles. Whenever we write monoid we mean commutative monoid. Remark 5.1. Recall that an action of a monoid P on a set S, written (p, s) 7→ 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 5.2. Let A be an abelian group and P a monoid. A free extension of A by P is an exact sequence E of monoids
0 → P −→ Qγ −→ A → 0m
together with a set-theoretic section ι : A → Q such that Q is free over P with basis ι. This means that the function ϕE: P × A → Q sending (p, λ) to γ(p) + ι(λ) is a bijection.
Remark 5.3. When P is sharp, the section ι of Definition 5.2 is uniquely determined. Definition 5.4. We denote by Extf(A, P) the set of free extensions of A by P .
Proposition 5.5. There is a bijection
Extf(A, P) ' Z2c(A, P) .
Proof. Let E be a free A-extension 0 → P −→ Qγ −→ A → 0 with basis ι : A → Q and let ϕm E: P × A → Q
be the induced bijection. Let ∇ : Q × Q → Q be the addition. Then we get a canonical function fE: A × A → Q × Q
∇
−→ Q → P
where A × A → Q × Q and r : Q → P are the canonical inclusion and projection respectively, obtained via ϕE. Since Q is commutative and associative and since
A−→ Qι −→ Pr
is zero we conclude that fE: A × A → P is a 2-cocycle. We define
Ψ : Extf(A, P) → Z2c(A, P)
e 7→ fE.
Conversely, given a 2-cocycle f : A × A → P , define a monoid Q = P ×fA with underlying set P × A
and addition given by
(p, λ) + (p0, λ0) = (p + p0+ f (λ, λ0), λ + λ0) .
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 map Θ : Z2c(A, P) → Extf(A, P). We leave to the reader to check that Θ and
Ψ are inverse to each other.
Remark 5.6. Note that the bijection in Proposition 5.5 provides Extf(A, P) with the structure of a
monoid.
Definition 5.7. A morphism β : E0→ E of extensions of A consists of a morphism Q0→ Q sending P0
into P and such that the diagrams
Q0 Q A A m0 m = and Q0 Q A = A ι0 ι commutes.
Remark 5.8. Note that a morphism β : E0 → E of extensions of A is completely determined by its restriction to P0⊆ Q0. Hence we may think of β as a morphism β : P0 → P .
The universal extension and the universal 2-cocycle. Let A be an abelian group. We will define a universal free extension EA of A
0 → PA→ QA→ A → 0
such that for any extension 0 → P → Q → A → 0, there exists a unique morphism PA→ P of extensions
of A (cf. Remark 5.8). The corresponding 2-cocycle A × A → PA is called the universal 2-cocycle of A.
Definition 5.9 ([Ton14, Definition 4.1]). Let R ⊂ NA×A× NA×Abe the congruence relation generated
by the relations
eλ,λ0 ∼ eλ0,λ
e0,λ∼ 0
eλ,λ0+ eλ+λ0,λ00 ∼ eλ0,λ00+ eλ0+λ00,λ.
Definition 5.10. We define PA= NA×A/R.
Remark 5.11. There is a function e(−,−): A × A → PA sending (λ, λ0) to eλ,λ0 which by definition of
R 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(−,−): A × A → PA.
Remark 5.12. The monoid PA need not be integral. For instance, it is not integral when A = Z/2Z ×
Z/2Z × Z/2Z (here we used Macaulay 2).
Definition 5.13. We define QA= PA×e(−,−)A , i.e., QAis the monoid in the universal free extension of
A corresponding to the universal 2-cocycle e(−,−): A × A → PA. That is, QAhas underlying set P × A
with addition (p, λ) + (p0, λ0) = (p + p0 + eλ,λ0, λ + λ0) Let γA: PA → QA be the canonical inclusion
p 7→ (p, 0).
Remark 5.14. Note that we have a splitting Z[QA] ∼=
M
λ∈A
Z[PA] ,
where (p, λ) has degree λ. Hence Spec Z[QA] → Spec Z[PA] is a ramified D(A)-cover (Definition 2.1).
There are two monoid homomorphisms
Σ, Π : NA×A→ NA
which are defined on the basis by
Σ(eλ,λ0) = eλ+ eλ0
Π(eλ,λ0) = eλ+λ0.
Consider the induced monoid homomorphism
NA×A (Σ,Π)−−−→ NA/he0i × NA/he0i .
We have that
(NA/he0i × NA/he0i)/(Σ, Π)(R) ∼= ZA/he0i ,
where (Σ, Π)(R) is the induced congruence relation on NA/he
0i × NA/he0i. Indeed, 0 ∼ e0,λ maps
to 0 ∼ (e0 + eλ, eλ) = (eλ, eλ) in NA/he0i × NA/he0i and ZA/he0i is obtained as the quotient of
NA/he0i × NA/he0i by the diagonal. The induced map is
ϕA: PA→ ZA/he0i
eλ,λ0 7→ eλ+ eλ0− eλ+λ0.
Remark 5.15. The monoid PA need not be integral. For instance, it is not integral when A = Z/2Z ×
Definition 5.16. By abuse of notation, define m : ZA/he0i → A to be the group homomorphism defined
on generators by eλ7→ λ.
Remark 5.17. As in [Ton14, Definition 4.4] we may consider the short exact sequence of abelian groups 0 → K → ZA/he0i
m
−→ A → 0 eλ7→ λ .
and ϕA: PA→ ZA/he0i factors through PA→ K, which is the groupification of PA [Ton14, Lemma 4.5].
Definition 5.18. We define Q+A= NA/he 0i .
Definition 5.19. For q ∈ Q+A, write q = Pn
i=1eλi where we may have λi = λj for i 6= j. Define
f (q) ∈ PAby f (q) = 0 if n ≤ 1, and
f (q) = eλ1,λ2+ eλ1+λ2,λ3+ · · · + eλ1+···+λn−1,λn
otherwise.
Remark 5.20. Note that by definition of the equivalence relation R in Definition 5.9, the element f (q) is independent of the order of the λi’s in the representation q =Pni=1eλi. This implies that we have a
set-theoretic function
f : Q+A → PA
q 7→ f (q) , which satisfies q = ϕA(f (q)) + em(q).
Remark 5.21. Note thatPn
i=1(pi, λi) = ( P ipi+ f ( P ieλi), P iλi) in QA.
Remark 5.22. The function f : Q+A→ PA satisfies the relation f (q + q0) = f (q) + f (q0) + em(q),m(q0).
Definition 5.23. We define
j : Q+A→ QA
q 7→ (f (q), m(q)) .
Remark 5.24. The map j is a homomorphism by Remark 5.22. Furthermore, there is a morphism QA→ ZA/he0i sending (p, λ) to ϕA(p) + eλand by Remark 5.20, the composition Q+A → QA→ ZA/he0i
is the canonical inclusion. Hence we conclude that j is injective and from now on we view Q+A as a submonoid of QA.
Lemma 5.25. Let A be a finite abelian group. The morphism γA: PA → QA is a Kummer
homomor-phism and the induced action of PA on QA is free.
Proof. For any (p, λ) ∈ QA we have ord(λ)(p, λ) = (p + f (ord(λ)eλ), 0). The rest follows readily from
the definitions.
Definition 5.26. Let RPbe the congruence relation on PA⊕Q+Agenerated by the relation (eλ,λ0, eλ+λ0) ∼
(0, eλ+ eλ0).
Remark 5.27. By transitivity we have (eλ,λ0+ eλ+λ0,λ00, eλ+λ0+λ00) ∼ (0, eλ+ eλ0+ eλ00) since
(eλ,λ0+ eλ+λ0,λ00, eλ+λ0+λ00) ∼ (eλ,λ0, eλ+λ0+ eλ00)
∼ (0, eλ+ eλ0+ eλ00) .
Iterating this process we conclude that RP contains the relation (0, q) ∼ (f (q), em(q)) for any q ∈ Q+A.
Since RP is a congruence relation, it is symmetric and closed under addition. Hence we conclude that
RP contains the relation R0 defined by (p, q) ∼ (p0, q0) if m(q) = m(q0) and p + f (q) = p0+ f (q0). It is
clear that R0 is an equivalence relation and by Remark 5.22 it follows that R0 is a congruence relation. Note that R0 contains the relation (eλ,λ0, eλ+λ0) ∼ (0, eλ+ eλ0) and hence RP = R0.
Remark 5.28. We have a function
τ : PA⊕ Q+A→ QA
(p, q) 7→ (p + f (q), m(q)) and by Remark 5.22 we have
τ (p, q) + τ (p0, q0) = (p + p0+ f (q) + f (q0) + em(q),m(q0), m(q) + m(q0))
= (p + p0+ f (q + q0), m(q + q0)) = τ (p + p0, q + q0)
and hence τ is a homomorphism of monoids. Furthermore, we have τ (p, q) = τ (p0, q0) if and only if m(q) = m(q0) and p + f (q) = p0+ f (q0). This means that τ induces a morphism (PA⊕ Q+A)/RP → QA.
We also have a function
η : QA→ PA⊕ Q+A
(p, λ) 7→ (p, eλ)
which becomes a homomorphism when we quotient by RP. Furthermore, τ ◦ η = idQA and we conclude
that (PA⊕ Q+A)/RP → QA is an isomorphism.
Remark 5.29. By [Ton14, Lemma 4.5] PA → ϕA(PA) is the associated integral monoid and hence we
identify Pint
A with ϕA(PA). Similarly, the map QA→ ZA/he0i ; (p, λ) 7→ ϕA(p) + eλ is a homomorphism
and the image is the associated integral monoid Qint A ∼= hP int A , Q + Ai ⊂ Z A/he 0i. Also QintA ∼= P int A ×e(−,−)A.
Example 5.30. If A = Z/3Z, then we have Pint
A ⊂ QintA ⊂ Z2 with PAint= h(2, −1), (−1, 2)i ∼= N2 and
Qint
A = h(1, 0), (0, 1), (2, −1), (−1, 2)i. This is illustrated in Figure 1.
Figure 1. The red dots represents the elements of PAint ⊂ Q int
A and the blue dots
represents the elements of Qint A \ P
int
Definition 5.31. The group homomorphism
ZA/he0i → Z
defined on generators by sending eλ to 1 is written q 7→ |q| and we call |q| the value of q.
Lemma 5.32. 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.
Proof. Suppose that s + s0 = 0. Then v(s) + v(s0) = 0 and since v(s) ≥ 1 unless v = 0 we get that
s = s0= 0.
Lemma 5.33. The monoids Pint
A and QintA are fine and sharp.
Proof. Both PAint and QintA are finitely generated and integral (see Remark 5.29) and hence fine. It remains to show that the monoids are sharp and since PAint ⊆ Qint
A it is enough to show that Q int A is
sharp. But QintA and the homomorphism q 7→ |q| satisfies the condition of Lemma 5.32, and hence QintA
is sharp.
Lemma 5.34. The monoids PA and QA are quasi-fine and sharp.
Proof. The canonical morphism QA→ QintA sends a (p, λ) to ϕA(p) + eλ. Hence QA and the
homomor-phism (p, q) 7→ |ϕA(p) + eλ| satisfies the condition of Lemma 5.32, and hence QA is sharp. But then PA
is also sharp since it is a submonoid. Both PAand QA 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, |ϕA(p) + eλ| ≥ 1 unless (p, λ) = 0. If (p, λ) = (p, λ) + (p0, λ0) then
|ϕA(p0) + e0λ| = 0 and hence (p0, λ0) = 0. This completes the proof.
Flat Kummer homomorphisms. Now consider the following situation. Suppose that we have a flat Kummer homomorphism (Definition 4.9 and 4.13) of quasi-fine (Definition 4.1) and sharp monoids γ : P → Q. Note that A = Q/P is a finite abelian group. For q, q0 ∈ Q we write q ≤ q0 if there exists a
p ∈ P such that q0 = q + p. The relation ≤ is a partial order since Q and P are quasi-integral and sharp. Proposition 5.35. Let P ,→ Q be a flat Kummer morphism of quasi-fine and sharp monoids. Put A = Q/P and let m : Q → A be the quotient. The set Qλ= m−1(λ) has a unique minimal element ι(λ).
Moreover, Qλ= ι(λ) + P .
Proof. If m(q1) = m(q2), then there exists p1 and p2such that q1+ p1= q2+ p2and since P → Q is flat
there exists q0 ∈ Q and p0
1, p02∈ P such that q1= q0+ p01, q2 = q0+ p02, and p1+ p01 = p2+ p02. Hence
q0 ≤ q1 and q0 ≤ q2. Since Q is finitely generated and m−1(λ) is partially ordered and positive (i.e.,
0 ≤ q for all q ∈ Q), we get that m−1(λ) has a unique minimal element [EKM+01, Corollary 1.2]. Corollary 5.36. 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 5.35 we need only show that the map P × A → Q sending (p, λ) to p + ι(λ) is injective. Suppose that p1+ ι(λ) = p2+ ι(λ). Then the flatness hypothesis says that there exists a q ∈ Q
and p0 ∈ P such that q + p0 = ι(λ) and p0+ p
1 = p0+ p2. But q ∈ m−1(λ) and q ≤ ι(λ) implies that
q = ι(λ) by minimality and unicity of ι(λ). Hence ι(λ) + p0 = ι(λ) so p0 = 0 since Q is quasi-integral.
Hence p1= p2. This proves the claim.
Proposition 5.37. 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 QA → Q and PA → P such that
the following diagram commutes:
PA P
QA Q
A .
Proof. This follows from Proposition 5.5, Remark 5.11, and Corollary 5.36. Remark 5.38. Proposition 5.37 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 chart
PA P
QA Q
inducing an isomorphism of root stacks SQA/PA ' SQ/P. This also works when we have sheaves of
monoids instead of constant monoids (see Definition 7.3).
6. Deligne–Faltings data from ramified D(A)-covers
Let A be a finite abelian group. We will freely switch between the description of QAas PA×e(−,−)A
and as (PA⊕ Q+A)/RP (Definition 5.26). Recall that every D(A)-cover f : X → S comes with a canonical
splitting
f∗OX ∼=
M
λ∈A
Lλ
and multiplication morphisms Lλ⊗ Lλ0 → Lλ+λ0 for every λ, λ0∈ A, which we think of as global sections
sλ,λ0 ∈ Γ(S, L−1λ ⊗ L−1λ0 ⊗ Lλ+λ0) .
Remark 6.1. We have that f : X → S is a D(A)-torsor if and only if every sλ,λ0 is invertible.
The quotient stackX = [X/D(A)] has a canonical D(A)-torsor p: X → X and we have a canonical splitting of p∗OX indexed by the elements of A.
Definition 6.2. With the notation above we write OX[λ] for the line bundle which is the direct summand of p∗OX of weight λ, so that
p∗OX∼=
M
λ∈A
OX[λ] .
We explain the notation in the following remark:
Remark 6.3. Note first that OX can be thought of as f∗OX 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) → Gm gives a morphism λ : BD(A) → BGm. The stack BGm has a canonical
Gm-torsor Spec Z → BGmwhich is the relative spectrum of a Z-graded BGm-algebra. The graded piece
of weight 1 is just OBGm[1], that is, OBGm shifted by 1. This means that OBGm[1] is the Z-graded Z-module which is 0 in every degree except in degree −1 where it is Z. Pulling back OBGm[1] along
λ : BD(A) → BGmwe get the line bundle OBD(A)[λ], which is OBD(A)shifted by λ. If we now pull back
Remark 6.4. Let π :X = [X/D(A)] → S be the structure morphism. First note that π∗OX[λ] = Lλ.
We have a counit
ελ: π∗π∗OX[λ] → OX[λ]
which is just the A-graded morphism
Lλ⊗OS OX → OX[λ]
which is sλ,λ0 in degree λ0∈ A. We call the corresponding generalized Cartier divisor
(Eλ, ελ) = ((π∗π∗OX[λ])∨⊗ OX[λ], ελ) ,
the universal divisor associated to the character λ ∈ A and we call Eλthe universal line bundle associated
to the character λ ∈ A.
Similarly, for every pair of universal line bundles Eλ and Eλ0 with characters λ, λ0 ∈ A, we have a
morphism
OS ∼= π∗Eλ⊗ π∗Eλ0 → π∗(Eλ⊗ Eλ0) ∼= L∨λ⊗ L∨λ⊗ Lλ+λ0,
which is nothing but sλ,λ0. Let (Lλ,λ0, sλ,λ0) := (L∨
λ⊗ L∨λ⊗ Lλ+λ0, sλ,λ0) be the corresponding generalized
Cartier divisor. If we pull back via π∗, we a get a generalized Cartier divisor (π∗L
λ,λ0, π∗sλ,λ0). Note
that we have a canonical isomorphism
Eλ⊗ Eλ0⊗ Eλ+λ∨ 0 = (π∗π∗OX[λ])∨⊗ OX[λ] ⊗ (π∗π∗OX[λ0])∨⊗ OX[λ0] ⊗ (π∗π∗OX[λ + λ0]) ⊗ OX[λ + λ0]∨
∼
= (π∗π∗OX[λ])∨⊗ (π∗π∗OX[λ0])∨⊗ (π∗π∗OX[λ + λ0])
∼
= π∗(L∨λ⊗ L∨λ ⊗ Lλ+λ0) ,
which we denote by α−1λ,λ0 and we denote it’s inverse by αλ,λ0. We write
(Eλ,λ0, ελ,λ0) := (Eλ⊗ Eλ0⊗ Eλ+λ∨ 0, αλ,λ0(π∗sλ,λ0)) .
We are now ready to define the Deligne–Faltings datum associated to a ramified D(A)-cover X → S: Definition 6.5. Let
LX: PA→Div S
be the symmetric monoidal functor obtained by sending a generator eλ,λ0 to (Lλ,λ0, sλ,λ0). We write
p 7→ L(p) and p 7→ s(p). Let
EX: QA→Div X
be the symmetric monoidal functor defined as follows: Note that QA= (PA⊕ Q+A)/RP. We define EX|Q+A
by sending a generator eλ ∈ Q+A to (Eλ, ελ) and we define EX|PA by sending a generator eλ,λ0 ∈ PA to
(Eλ,λ0, ελ,λ0) (EX|P
Ais well-defined by Remark 6.4). We write q 7→ E (q) and q 7→ ε(q). This is well-defined
by Remark 6.7 below.
Let αX: π∗LX → EX ◦ γA be the isomorphism defined on generators by αX(eλ,λ0) = αλ,λ0. With
notation as in [BV12, Definition 2.1], the isomorphism µEe
λ,λ0,eλ+λ0: E (eλ,λ0) ⊗ E (eλ+λ0) ∼= E (eλ+ eλ0)
is obtained via the projection formula and the canonical isomorphism OX[λ] ⊗ OX[λ0] ∼= OX[λ + λ0]. Remark 6.6. As we saw in Remark 6.4, the global section ελ∈ Γ(X , Eλ) corresponds to the morphism
Lλ⊗OS OX → OX[λ]
which is sλ,λ0 in degree λ0. Similarly, the section ελ⊗ελ0 ∈ Γ(X , Eλ⊗Eλ0) corresponds to the composition
Lλ⊗ OX ⊗ Lλ0⊗ OX
sλ,(−)⊗id(Lλ0⊗OX )
−−−−−−−−−−−−−→ OX[λ] ⊗ Lλ0⊗ OX
sλ+(−),λ0
−−−−−−→ OX[λ] ⊗ OX[λ0] where sλ,(−) is sλ,λ00 in degree λ00 and sλ+(−),λ0 is sλ+λ00,λ0 in degree λ00. Hence it makes sense to say
that ελ⊗ ελ0 is sλ,λ00sλ+λ00,λ0 in degree λ00. Iterating this process we get a description for ε(q) for any
