Birational invariants of toric orbifold surfaces

Abstract

We study birational invariants of toric orbifold surfaces.

1 Introduction

Let k be an algebraically closed field. Algebraic orbifolds over k are smooth separated irreducible Deligne-Mumford stacks of finite type over k, with trivial generic stabilizer. Birational invariants of algebraic orbifolds were introduced in [7]; they are invariant under birational projective morphisms of algebraic orbifolds. In this paper, we focus on toric Deligne-Mumford stacks, introduced in [3], and specialize to dimension 2, i.e., toric orbifold surfaces. The goal of this paper is to study their birational invariants over an algebraically closed field of characteristic 0.

2 Toric varieties

To fix notation, we recall basic definitions of toric varieties [6]. We fix a base field k, which will later be taken to be algebraically closed of characteristic 0, and a dimension n, which will later be taken to be 2. There are lattices

$$\begin{aligned} M:={\mathbb {Z}}^n\qquad \text {and}\qquad N:={{\,\mathrm{Hom}\,}}(M,{\mathbb {Z}}). \end{aligned}$$

The elements of M are characters on the algebraic torus

$$\begin{aligned} T:={{\,\mathrm{Spec}\,}}(k[M]). \end{aligned}$$

We view the lattice N as sitting inside its extension over the real numbers

$$\begin{aligned} N_{\mathbb {R}}:=N\otimes _{{\mathbb {Z}}}{\mathbb {R}}; \end{aligned}$$

we employ analogous notation, e.g., for an extension over ${\mathbb {Q}}$.

Cones in $N_{\mathbb {R}}$ are always strongly convex rational polyhedral cones: A fan in N is a finite collection of cones in $N_{\mathbb {R}}$, such that for every cone in the fan, its faces are also in the fan, and for every pair of cones in the fan, the intersection is a face of each. The support of a fan is the union of the cones.

A cone $\sigma$ determines an affine toric variety

$$\begin{aligned} U_\sigma :={{\,\mathrm{Spec}\,}}(k[\sigma ^\vee \cap M]), \end{aligned}$$

where $\sigma ^\vee$ denotes the dual cone in $M_{\mathbb {R}}$.

The toric variety determined by a fan $\Sigma$ is the union of $U_\sigma$ over all $\sigma \in \Sigma$, glued by identifying a common open subvariety $U_{\sigma \cap \tau }$ of $U_\sigma$ and $U_\tau$, for all $\sigma$, $\tau \in \Sigma$. This yields a normal algebraic vareity $X(\Sigma )$ over k. Every normal algebraic variety over k with T-action and equivariant open immersion of T is the toric variety of a uniquely determined fan in N.

Each $U_\sigma$ is T-invariant and has a unique closed T-orbit, the image of

$$\begin{aligned} {{\,\mathrm{Spec}\,}}(k[\sigma ^\perp \cap M])\rightarrow {{\,\mathrm{Spec}\,}}(k[\sigma ^\vee \cap M]), \end{aligned}$$

where $\sigma ^\perp \subset M_{\mathbb {R}}$ is defined by pairing to 0 with all elements of $\sigma$. The closure is a closed subvariety of $X(\Sigma )$, determined by $\sigma$. This is a point when $\sigma$ is an n-dimensional cone, and has codimension 1 when $\sigma$ is a ray (1-dimensional cone).

A cone is smooth if it is generated by part of a ${\mathbb {Z}}$-basis of N. We have $X(\Sigma )$ smooth if and only if all of the cones of $\Sigma$ are smooth.

For $X(\Sigma )$ to be projective, it is necessary and sufficient for the support of $\Sigma$ to be equal to $N_{\mathbb {R}}$ and $\Sigma$ to admit a piecewise linear convex support function. When $n=2$, the second condition is automatic, so projective toric surfaces correspond to fans with support equal to $N_{\mathbb {R}}$.

As well, when $n=2$ we profit from the elementary structure of low-dimensional cones. The cones are 0, rays ${\mathbb {R}}_{\ge 0} v$, and 2-dimensional cones ${\mathbb {R}}_{\ge 0} v+{\mathbb {R}}_{\ge 0} w$. The generator $v\in N_{{\mathbb {R}}}$ of a ray may be uniquely specified by the requirements to lie in N and not be a multiple of another generator in N; we call such $v\in N$ the primitive ray generator. For a 2-dimensional cone, primitive generators v, $w\in N$ are uniquely determined up to swapping.

3 Orbifold toric surfaces

Toric Deligne-Mumford stacks were introduced in [3]. The idea is to consider only fans whose cones are simplicial, and in this case to glue the stack quotients of smooth affine toric varieties by finite groups to obtain a smooth separated irreducible Deligne-Mumford stack with trivial generic stabilizer, i.e., an orbifold. The construction requires the given fan $\Sigma$, with simplicial cones, to be supplemented by the additional data of a generator in N of every ray of $\Sigma$, to make a so-called stacky fan $\varvec{\Sigma }$.

We continue to work over a base field k, take $n=2$, and fix a prime number p, not equal to the characteristic of k. Let $\Sigma$ be a fan in N, such that every 2-dimensional cone is of the form $\sigma ={\mathbb {R}}_{\ge 0}v+{\mathbb {R}}_{\ge 0}w$ with primitive generators v, $w\in N$ such that the (well-defined) absolute value of the determinant of the $2\times 2$ integer matrix of coordinates of v and w, with respect to a ${\mathbb {Z}}$-basis of N, is equal to 1 or p. We consider a stacky fan $\varvec{\Sigma }$, where the additional data of a choice of generator of each ray $\varrho \in \Sigma$ is subject to the following conditions:

• The chosen generator is equal to or is p times the primitive generator of $\varrho$.

• When the generator is p times the primitive generator, the 2-dimensional cones containing $\varrho$ are smooth.

• No pair of rays, for which the chosen generator is p times the primitive generator, generates a cone of $\Sigma$.

The construction of the toric orbifold ${\mathcal {X}}(\varvec{\Sigma })$ of a stacky fan $\varvec{\Sigma }$ proceeds by suitably modifying the construction of $X(\Sigma )$; we only describe the construction in the present setting, where the conditions have been chosen to correspond precisely to toric orbifold surfaces ${\mathcal {X}}(\varvec{\Sigma })$, whose nontrivial stabilizer groups all have order p. For every 2-dimensional cone $\sigma ={\mathbb {R}}_{\ge 0}v+{\mathbb {R}}_{\ge 0}w$ with primitive vectors v, $w\in N$ and corresponding $2\times 2$ integer matrix with determinant of absolute value p, we have an index p sublattice of N generated by v and w, and by duality, a lattice $M'\subset M_{\mathbb {Q}}$ with $M\subset M'$ and $|M'/M|=p$. Then the image of

$$\begin{aligned} {{\,\mathrm{Spec}\,}}(k[M'/M])\rightarrow {{\,\mathrm{Spec}\,}}(k[M']) \end{aligned}$$

is a subgroup, isomorphic to $\mu _p$, of $T':={{\,\mathrm{Spec}\,}}(k[M'])$, which is the kernel of $T'\rightarrow T$. The stack quotient

$$\begin{aligned} {[}{{\,\mathrm{Spec}\,}}(k[\sigma ^\vee \cap M'])/\mu _p] \end{aligned}$$

is used in place of $U_\sigma$ in the construction; just with this modification, the result would be a smooth orbifold surface with finitely many points with $\mu _p$-stabilizer. This is further modified by passing to the pth root stack [4, Sect. 2] [1, App. B] of the union of divisors, that correspond to rays where p times the primitive ray generator is given in $\varvec{\Sigma }$. The resulting orbifold ${\mathcal {X}}(\varvec{\Sigma })$ has locus with $\mu _p$-stabilizer equal to the union of finitely many points and finitely many pairwise disjoint rational curves.

The smooth Deligne-Mumford stack ${\mathcal {X}}(\varvec{\Sigma })$ has coarse moduli space $X(\Sigma )$. We describe an orbifold as projective, when it has projective coarse moduli space. But $n=2$, so ${\mathcal {X}}(\varvec{\Sigma )}$ is projective if and only if the support of $\Sigma$ is equal to $N_{\mathbb {R}}$.

4 Orbifold Burnside group

The Burnside group of stacks $\overline{\mathrm{Burn}}_n$ and its weight module $\overline{{\mathcal {B}}}=\bigoplus _{j\ge 0}\overline{{\mathcal {B}}}_j$ were introduced in [7]. The weight module is an abelian group, defined by explicit generators and relations.

We suppose that the base field k is algebraically closed of characteristic 0. Then a projective orbifold ${\mathcal {X}}$ of dimension n determines a class in an abelian group $\overline{\mathrm{Burn}}_n$, generated by pairs $(K,\alpha )$ with K a finitely generated field over k, of some transcendence degree $d\le n$, and $\alpha \in \overline{{\mathcal {B}}}_{n-d}$. With suitable relations, this class is invariant under birational projective morphisms of projective orbifolds. In other words, $[{\mathcal {X}}]=[{\mathcal {X}}']$ in $\overline{\mathrm{Burn}}_n$ if there exists an orbifold ${\mathcal {Y}}$ with birational projective morphisms

$$\begin{aligned} {\mathcal {Y}}\rightarrow {\mathcal {X}}\qquad \text {and}\qquad {\mathcal {Y}}\rightarrow {\mathcal {X}}'. \end{aligned}$$

Symbols with $K=k$ generate a subgroup

$$\begin{aligned} \overline{{\mathcal {B}}}_n\subset \overline{\mathrm{Burn}}_n. \end{aligned}$$

(1)

We describe the group $\overline{{\mathcal {B}}}_2$, or rather a particular subgroup $\overline{{\mathcal {B}}}_2^{[p]}$, for which we have

$$\begin{aligned} {[}{\mathcal {X}}(\varvec{\Sigma })]\in \overline{{\mathcal {B}}}_2^{[p]}, \end{aligned}$$

in the subgroup (1). The locus with $\mu _p$-stabilizer of ${\mathcal {X}}(\varvec{\Sigma })$, as mentioned above, consists of finitely many divisors and isolated points, which we call $\mu _p$-divisors and $\mu _p$-points, respectively. Their complement is a quasi-projective variety.

The abelian group $\overline{{\mathcal {B}}}_2^{[p]}$ is generated by symbols $(\mathrm{triv},(0,0))$, $({\mathbb {Z}}/p{\mathbb {Z}},(0,1))$, and $({\mathbb {Z}}/p{\mathbb {Z}},(1,a))$ for $a\in ({\mathbb {Z}}/p{\mathbb {Z}})^\times$. These are subject to the relations

$$\begin{aligned} ({\mathbb {Z}}/p{\mathbb {Z}},(0,1))&=({\mathbb {Z}}/p{\mathbb {Z}},(1,1)), \\ (\mathrm{triv},(0,0))&=({\mathbb {Z}}/p{\mathbb {Z}},(1,p-1)), \\ ({\mathbb {Z}}/p{\mathbb {Z}},(1,a))&=({\mathbb {Z}}/p{\mathbb {Z}},(1,a^{-1})), \\ ({\mathbb {Z}}/p{\mathbb {Z}},(1,a))&=({\mathbb {Z}}/p{\mathbb {Z}},(1,a-1))+({\mathbb {Z}}/p{\mathbb {Z}},(1,a^{-1}-1))-(\mathrm{triv},(0,0)) \\&\qquad \qquad \text {for}\quad a\in \{2,\dots ,p-1\}, \end{aligned}$$

where $a^{-1}$ denotes the multiplicative inverse to a mod p. The class $[{\mathcal {X}}(\varvec{\Sigma })]$ is the sum of contributions from the $\mu _p$-divisors, the $\mu _p$-points, and the quasi-projective variety that is the complement of the $\mu _p$-divisors and $\mu _p$-points. We let $n_1$ denote the number of $\mu _p$-divisors, and $n_2$, the number of $\mu _p$-points.

The quasi-projective variety contributes

$$\begin{aligned} (1-n_1-n_2)(\mathrm{triv},(0,0)). \end{aligned}$$

The $\mu _p$-divisors contribute

$$\begin{aligned} n_1({\mathbb {Z}}/p{\mathbb {Z}},(0,1)). \end{aligned}$$

The contribution from the $\mu _p$-points is the sum of individual contributions

$$\begin{aligned} ({\mathbb {Z}}/p{\mathbb {Z}},(1,a)) \end{aligned}$$

from each $\mu _p$-point. At a $\mu _p$-point, an identification of the stabilizer with $\mu _p$ is made, so that one of the weights of the action on the tangent space is 1; then a is the second weight. But this involves a choice: if we reverse the roles of the two weights then the resulting weight would be $a^{-1}$. Thanks to the corresponding relation in $\overline{{\mathcal {B}}}_2^{[p]}$, the contribution $({\mathbb {Z}}/p{\mathbb {Z}},(1,a))\in \overline{{\mathcal {B}}}_2^{[p]}$ of a $\mu _p$-point is well-defined.

The quotient group

$$\begin{aligned} \overline{{\mathcal {B}}}_2^{[p]}/\langle (\mathrm{triv},(0,0)), ({\mathbb {Z}}/p{\mathbb {Z}},(0,1)) \rangle \end{aligned}$$

(2)

is the group denoted by $\overline{{\mathcal {B}}}_2^{[p]}/ ({\mathcal {C}}\cap \overline{{\mathcal {B}}}_2^{[p]})$ in [7, Sect. 4]. In [7, Lemma 5.3], an explicit isomorphism is given from this group to a certain quotient of the group $H_1(X_0(p)_{\mathrm{orb}}, {\mathbb {Z}})$, the first orbifold homology of the modular curve $X_0(p)$. This group (2) is also mentioned as providing an obstruction to being linked by blow-ups of points to an orbifold surface without $\mu _p$-points. For the group (2) the following isomorphism types were found:

$$\begin{aligned} p=2:\,0,\quad p=3:\,0,\quad p=5:\,{\mathbb {Z}}/2{\mathbb {Z}}, \quad p=7:\,0,\quad p=11:\,{\mathbb {Z}}. \end{aligned}$$

As mentioned in [7, Sect. 4], the vanishing for $p=2$ and $p=3$ is the $n=2$ case of a more general phenomenon, related to destackification [2]. By [7, Prop. 4.4], the group (2) is nonzero for all $p\ge 5$, except $p=7$.

5 Refined invariant

We continue to work over an algebraically closed base field k of characteristic 0. A natural class of birational projective morphisms of projective toric Deligne-Mumford stacks is the class of morphisms associated with the stacky star subdivisions (of toric Deligne-Mumford stacks) of [5]. All toric stacks will be projective toric orbifold surfaces with $\mu _p$ as the only possible nontrivial stabilizer group. We emphasize that the torus T is regarded as fixed. All stacky fans will satisfy the corresponding conditions, stated in Sect. 3, and have support equal to $N_{\mathbb {R}}$.

A stacky star subdivision is an operation that, given a stacky fan $\varvec{\Sigma }$ and a chosen 2-dimensional cone $\sigma \in \Sigma$, yields a new stacky fan $\varvec{\Sigma }_\sigma$ and a T-equivariant birational projective morphism

$$\begin{aligned} f_\sigma :{\mathcal {X}}(\varvec{\Sigma }_\sigma )\rightarrow {\mathcal {X}}(\varvec{\Sigma }). \end{aligned}$$

Let us write $\sigma ={\mathbb {R}}_{\ge 0}v+{\mathbb {R}}_{\ge 0}w$ with primitive vectors v, $w\in N$; then the stacky fan $\varvec{\Sigma }_\sigma$ is constructed by deleting the cone $\sigma$, adding the cones

$$\begin{aligned} {\mathbb {R}}_{\ge 0}(v+w),\quad {\mathbb {R}}_{\ge 0}v+{\mathbb {R}}_{\ge 0}(v+w), \quad {\mathbb {R}}_{\ge 0}(v+w)+{\mathbb {R}}_{\ge 0}w, \end{aligned}$$

and endowing the ray ${\mathbb {R}}_{\ge 0}(v+w)$ with the choice of generator $v+w$. Geometrically, $f_\sigma$ is the blow-up of ${\mathcal {X}}(\varvec{\Sigma })$ at the point, corresponding to $\sigma$; this is a point of the scheme locus when $\sigma$ is a smooth cone whose rays come with integer multiple 1, a $\mu _p$-point when $\sigma$ is not smooth, a point on a $\mu _p$-divisor when $\sigma$ is smooth and one of its rays comes with multiple p.

By the classical factorization of birational projective morphisms of smooth surfaces as compositions of blow-ups, for any T-equivariant birational projective morphism

$$\begin{aligned} f:{\mathcal {X}}(\varvec{\Sigma }')\rightarrow {\mathcal {X}}(\varvec{\Sigma }), \end{aligned}$$

we obtain $\varvec{\Sigma }'$ from $\varvec{\Sigma }$ by performing a finite sequence of stacky star subdivisions, and f is the composite of the corresponding T-equivariant birational projective morphisms.

Inspired by [2, Example 4.2] and [7, Section 4], we might ask: For a given pair of toric orbifold surfaces ${\mathcal {X}}(\varvec{\Sigma })$ and ${\mathcal {X}}(\varvec{\Sigma }')$, does there exist a toric orbifold surface ${\mathcal {X}}(\varvec{\Sigma }'')$ with T-equivariant birational projective morphisms

$$\begin{aligned} {\mathcal {X}}(\varvec{\Sigma }'')\rightarrow {\mathcal {X}}(\varvec{\Sigma })\quad \text {and} \quad {\mathcal {X}}(\varvec{\Sigma }'')\rightarrow {\mathcal {X}}(\varvec{\Sigma }')\,? \end{aligned}$$

When the answer is affirmative, we say that ${\mathcal {X}}(\varvec{\Sigma })$ and ${\mathcal {X}}(\varvec{\Sigma }')$ are equivalent under T-equivariant birational projective morphisms.

As mentioned in Sect. 4, the group $\overline{{\mathcal {B}}}_2^{[p]}/\langle (\mathrm{triv},(0,0)), ({\mathbb {Z}}/p{\mathbb {Z}},(0,1))\rangle$ supplies an obstruction to the equivalence under T-equivariant birational projective with a toric orbifold surface whose coarse moduli space is smooth.

We record the observation, that in the toric setting the abelian group $\overline{{\mathcal {B}}}_2^{[p]}$ admits a refinement.

Definition 1

The refined toric orbifold surface weight group, for $\mu _p$-stabilizers, is the abelian group

$$\begin{aligned} \overline{{\mathcal {T}}}_2^{[p]}, \end{aligned}$$

generated by $(\mathrm{triv},(0,0))$, $({\mathbb {Z}}/p{\mathbb {Z}},(0,1))$, and $({\mathbb {Z}}/p{\mathbb {Z}},(1,a))$ for $a\in ({\mathbb {Z}}/p{\mathbb {Z}})^\times$, with relations

$$\begin{aligned} ({\mathbb {Z}}/p{\mathbb {Z}},(0,1))&=({\mathbb {Z}}/p{\mathbb {Z}},(1,1)),\\ (\mathrm{triv},(0,0))&=({\mathbb {Z}}/p{\mathbb {Z}},(1,p-1)),\\ ({\mathbb {Z}}/p{\mathbb {Z}},(1,a))&=({\mathbb {Z}}/p{\mathbb {Z}},(1,a-1)) +({\mathbb {Z}}/p{\mathbb {Z}},(1,(a^{-1}-1)^{-1}))-(\mathrm{triv},(0,0)) \\&\qquad \qquad \text {for}\quad a\in \{2,\dots ,p-1\}. \end{aligned}$$

With the additional relations $({\mathbb {Z}}/p{\mathbb {Z}},(1,a))=({\mathbb {Z}}/p{\mathbb {Z}},(1,a^{-1}))$ for $a\in ({\mathbb {Z}}/p{\mathbb {Z}})^\times$, we recover the presentation of $\overline{{\mathcal {B}}}_2^{[p]}$. So, we have a canonical surjective homomorphism

$$\begin{aligned} \overline{{\mathcal {T}}}_2^{[p]}\rightarrow \overline{{\mathcal {B}}}_2^{[p]}. \end{aligned}$$

(3)

Proposition 2

The assignment, to ${\mathcal {X}}(\varvec{\Sigma })$, of the class

$$\begin{aligned} (1-n_1-n_2)(\mathrm{triv},(0,0))+n_1({\mathbb {Z}}/p{\mathbb {Z}},(0,1))+ \sum _{\begin{array}{c} \mathrm{non}\text {-}\mathrm{smooth}\ \sigma \in \Sigma \\ \sigma ={\mathbb {R}}_{\ge 0}v+{\mathbb {R}}_{\ge 0}w\\ v_1w_2-w_1v_2=p \end{array}}({\mathbb {Z}}/p{\mathbb {Z}},(1,-w_i^{-1}v_i)) \end{aligned}$$

in $\overline{{\mathcal {T}}}_2^{[p]}$, where in the sum we have $v=(v_1,v_2)$, $w=(w_1,w_2)\in N$, and $i\in \{1,2\}$ is chosen so that $p\not \mid w_i$, gives rise to an invariant of toric orbifold surfaces with $\mu _p$ as only possible nontrivial stabilizer group, under torus-equivariant birational projective morphisms. The assignment is compatible with the homomorphism (3).

Proof

From the explicit description of stacky star subdivision, it is easy to check that $\varvec{\Sigma }$ and any stacky star subdivision $\varvec{\Sigma }_\sigma$ have the same class in $\overline{{\mathcal {T}}}_2^{[p]}$. $\square$

Example

We may form a quotient

$$\begin{aligned} \overline{{\mathcal {T}}}_2^{[p]}/\langle (\mathrm{triv},(0,0)),({\mathbb {Z}}/p{\mathbb {Z}},(0,1))\rangle , \end{aligned}$$

analogous to (2). Isomorphism types for some p are:

$$\begin{aligned} p=2:\,0,\quad p=3:\,0,\quad p=5:\,{\mathbb {Z}}/2{\mathbb {Z}}, \quad p=7:\,{\mathbb {Z}}/3{\mathbb {Z}},\quad p=11:\,{\mathbb {Z}}^2. \end{aligned}$$

In particular, we obtain a nontrivial refinement for $p=7$ of the trivial obstruction group mentioned in Sect. 4.

Comparison with the isomorphism types of $H_1(X_0(p)_{\mathrm{orb}}, {\mathbb {Z}})$ from [7, Sect. 5] suggests that this refinement reproduces the full first orbifold homology of which (2) is a quotient¹. As recalled in [7, Lemma 5.2], the group $H_1(X_0(p)_{\mathrm{orb}}, {\mathbb {Z}})$ is generated by Manin symbols

$$\begin{aligned} \Big \{ 0, \frac{1}{a} \Big \}, \quad 2 \le a \le p-2, \end{aligned}$$

with relations

$$\begin{aligned}&\Big \{ 0, \frac{1}{a}\Big \} + \Big \{ 0, \frac{1}{p-a^{-1}}\Big \} = 0, \end{aligned}$$

(4)

$$\begin{aligned}&\Big \{ 0, \frac{1}{(p-1)/2} \Big \} + \Big \{ 0, \frac{1}{p-2} \Big \} = 0, \end{aligned}$$

(5)

$$\begin{aligned}&\Big \{ 0, \frac{1}{a} \Big \} + \Big \{ 0, \frac{1}{a'} \Big \} +\Big \{ 0, \frac{1}{a''} \Big \} = 0 \quad (a \notin \{ p-2, (p-1)/2 \}), \end{aligned}$$

(6)

where

$$\begin{aligned} a' \equiv -a^{-1} -1 \, \mathrm{mod} \, p, \quad a'' \equiv -(a+1)^{-1} \, \mathrm{mod} \, p. \end{aligned}$$

Theorem 3

There is an isomorphism

$$\begin{aligned} \overline{{\mathcal {T}}}_2^{[p]}/\langle (\mathrm{triv},(0,0)), ({\mathbb {Z}}/p{\mathbb {Z}},(0,1))\rangle \rightarrow H_1(X_0(p)_{\mathrm{orb}}, {\mathbb {Z}}) \end{aligned}$$

sending $({\mathbb {Z}}/p{\mathbb {Z}},(1,a))$ to the Manin symbol $\{ 0, 1/a\}$ for $2 \le a \le p-2$.

Proof

We are working in the quotient $\overline{{\mathcal {T}}}_2^{[p]}/\langle (\mathrm{triv},(0,0)), ({\mathbb {Z}}/p{\mathbb {Z}},(0,1))\rangle$. We start by obtaining the relation

$$\begin{aligned} ({\mathbb {Z}}/p{\mathbb {Z}},(1,-b)) = -({\mathbb {Z}}/p{\mathbb {Z}},(1,b^{-1})). \end{aligned}$$

(7)

The relation follows by induction on b, starting with the base case $b = 2$, which is a consequence of the last relation from Definition 1 for $a = 2^{-1}$ and $a = p-1$. The inductive step uses the same relation with $a = b^{-1}$ and $a = p+1-b$. This recovers (4). Relation (5) is recovered by once again using the last relation from Definition 1 with $a = p-1$. Using (7) we may rewrite the last relation from Definition 1 in the form

$$\begin{aligned} 0 = ({\mathbb {Z}}/p{\mathbb {Z}},(1,-a^{-1})) + ({\mathbb {Z}}/p{\mathbb {Z}},(1,a-1)) + ({\mathbb {Z}}/p{\mathbb {Z}},(1,((a^{-1}-1)^{-1}))). \end{aligned}$$

Replacing a by $a+1$ we reproduce (6). $\square$

The isomorphism of Theorem 3 is compatible with that of [7, Lemma 5.3].