Noncommutative cyclic isolated singularities

TL;DR

This paper investigates when noncommutative graded quotient singularities are isolated, focusing on cyclic actions on skew polynomial rings, and extends previous results by establishing a partial dichotomy theorem.

Contribution

It introduces a partial dichotomy theorem for isolatedness of noncommutative quotient singularities under cyclic actions, generalizing prior work in the field.

Findings

Established a partial dichotomy theorem for isolatedness.

Applied the theorem to cyclic actions on the (-1)-skew polynomial ring.

Extended previous results by Bao, He, and others.

Abstract

The question of whether a noncommutative graded quotient singularity $A^G$ is isolated depends on a subtle invariant of the $G$-action on $A$, called the pertinency. We prove a partial dichotomy theorem for isolatedness, which applies to a family of noncommutative quotient singularities arising from a graded cyclic action on the $(-1)$-skew polynomial ring. Our results generalize and extend some results of Bao, He and the third-named author and results of Gaddis, Kirkman, Moore and Won.

Equations

$$R\#G\to {\mathrm{End}}_{R^G}(R)$$

$$p(R,H):=\mathrm{GKdim}(R)-\mathrm{GKdim}(R\#H/(e_0))$$

$${\mathbb{k}}_{-1}[\mathbf{x}]:={\mathbb{k}}_{-1}[x_0,\dots ,x_{n-1}]$$

$$x_i{x}_j=(-1)x_j{x}_i$$

$$\sigma \ast x_i=x_{i+1}$$

$${\varphi }_2(n)=\{k\mid 0\le k\le n-1\text{with}\gcd (k,n)=2^w\text{for some w≥0}\}.$$

$$\begin{array}[]{|c|l|l|l|l|l|l|l|l|l|l|l|l|l|}\hline\cr n&2&3&4&5&6&7&8&9&10&11&12&13&14\\ \hline\cr p&2&2&4&4&5&7&8&8&9&11&\in[8,11]&13&14\\ \hline\cr\end{array}$$

$$2,2,4,4,5,7,8,8,9,\mathrm{11...}$$

$$\mathrm{GKdim}(R)=\underset{n\to \infty }{\mathrm{limsup}}{\log }_n(\sum _{i=0}^n{\dim }_{\mathbb{k}}{R}_i).$$

$${\mathbb{k}}_q[x_1,\dots ,x_m]:=\mathbb{k}⟨x_1,\dots ,x_m⟩/(x_i{x}_j-q{x}_j{x}_i\mid i<j)$$

$$\mathrm{GKdim}(M)=\underset{n\to \infty }{\mathrm{limsup}}{\log }_n(\sum _{i\le n}{\dim }_{\mathbb{k}}{M}_i).$$

$$\mathrm{GKdim}M\ge \mathrm{GKdim}M/cM\ge \mathrm{GKdim}M-1.$$

$$\sigma \ast x_i=x_{i+1},\forall i\in \underline{n}={\mathbb{Z}}_n.$$

$$(a\#g)(a^{\prime }\#g^{\prime })=ag(a^{\prime })\#g{g}^{\prime },$$

$$A\#C_n\cong \frac{\mathbb{k}⟨\mathbf{x},\sigma ⟩}{(x_i{x}_j+x_j{x}_i,{\sigma }^n,\sigma x_i-x_{i+1}\sigma )}.$$

$$A_1\cong \underset{\gamma =0}{\overset{n-1}{⨁}}{M}_{{\omega }^{\gamma }}$$

$$b_{\gamma }:=\frac{1}{n}\sum _{i=0}^{n-1}{\omega }^{i\gamma }{x}_i.$$

$$\sigma \ast b_{\gamma }=\frac{1}{n}\sum _{i=0}^{n-1}{\omega }^{i\gamma }{x}_{i+1}={\omega }^{-\gamma }{b}_{\gamma }.$$

$$e_{\alpha }:=\frac{1}{n}\sum _{i=0}^{n-1}({\omega }^{\alpha }\sigma {)}^i$$

$$[u,v]=uv-(-1{)}^{\mathrm{deg}(u)\mathrm{deg}(v)}vu.$$

$$A\#C_n\cong \frac{\mathbb{k}⟨\mathbf{b},\mathbf{e}⟩}{(e_{\alpha }{b}_{\gamma }-b_{\gamma }{e}_{\alpha -\gamma },e_i{e}_j-{\delta }_{ij}{e}_i,[b_0,b_k]-[b_l,b_{k-l}])}$$

$$r_{kl}=[b_0,b_k]-[b_l,b_{k-l}].$$

$$\iota :\mathbb{k}⟨\mathbf{b}⟩/(r_{kl})\to {\mathbb{k}}_{-1}[\mathbf{x}]$$

$$[b_{\gamma },b_{\delta }]=\frac{1}{n^2}\sum _{i,j=0}^{n-1}{\omega }^{i\gamma +j\delta }[x_i,x_j]=\frac{2}{n^2}\sum _{i=0}^{n-1}{\omega }^{i(\gamma +\delta )}{x}_i^2$$

$$n(|R_k/C_2|-1)=\left(\genfrac{}{}{0ex}{}{n}{2}\right).$$

$$\sum _{k\text{odd}}(|R_k/C_2|-1)+\sum _{k\text{even}}(|R_k/C_2|-1)=\frac{n}{2}\left(\frac{n}{2}-1\right)+\frac{n}{2}\left(\frac{n}{2}\right)=\left(\genfrac{}{}{0ex}{}{n}{2}\right).$$

$$\mathbb{k}{C}_n\cong \mathbb{k}⟨\mathbf{e}⟩/(e_i{e}_j-{\delta }_{ij}{e}_i)$$

$$e_{\alpha }{b}_{\gamma }=\frac{1}{n}\sum _{i=0}^{n-1}({\omega }^{\alpha }\sigma {)}^i{b}_{\gamma }=\frac{b_{\gamma }}{n}\sum _{i=0}^{n-1}{\omega }^{(\alpha -\gamma )i}{\sigma }^i=b_{\gamma }{e}_{\alpha -\gamma }.$$

$${\sigma }^i=\sum _{\alpha }{\omega }^{-\alpha i}{e}_{\alpha }$$

$$x_j=\sum _{\gamma }{\omega }^{-\gamma j}{b}_{\gamma }$$

Full text

0. Introduction

Auslander [Au] proved that if $G$ is a small finite subgroup of ${\text{GL}}_{n}({\mathbb{C}})$, acting linearly on the symmetric algebra over ${\mathbb{C}}$ (namely, the commutative polynomial ring) $R:={\mathbb{C}}[{\mathbb{C}}^{\oplus n}]$, with fixed subring $R^{G}$, then the natural map

[TABLE]

is an isomorphism of graded algebras. Here $R\#G$ denotes the skew group algebra associated to the $G$-action on $R$ and the hypothesis of $G$ being small means that $G$ does not contain any pseudo-reflections (e.g. $G$ is a finite subgroup of ${\text{SL}}_{n}({\mathbb{C}})$). This theorem plays an important role in the McKay correspondence, relating representations of $G$ and those of $R^{G}$; and in the special case of dimension two, further relating configuration of the exceptional fibers in the minimal resolution of $\operatorname{Spec}R^{G}$. The noncommutative version of this theorem of Auslander is an important ingredient in establishing a noncommutative McKay correspondence, see [CKWZ1, CKWZ2] for some recent developments. In [BHZ1, BHZ2], a numerical invariant was introduced for a semisimple Hopf algebra action on a (not necessarily commutative) algebra $R$ with finite Gelfand-Kirillov dimension (or GKdimension for short). The pertinency of a Hopf algebra $H$-action on $R$ [BHZ1, Definition 0.1] is defined to be

[TABLE]

where $(e_{0})$ is the two-sided ideal of the smash product $R\#H$ generated by the element $e_{0}:=1\#\textstyle\int$, where $\textstyle\int$ denotes an integral of $H$. One of the main results in [BHZ1, BHZ2] is the following.

Theorem 0.1**.**

[BHZ1*, Theorem 0.3]**

Let $R$ be a noetherian, connected graded, Artin-Schelter regular, Cohen-Macaulay algebra of GKdimension at least 2. Let $H$ be a semisimple Hopf algebra acting on $R$ inner-faithfully and homogeneously. Then the following are equivalent:*

1. (1)

$\operatorname{{\sf p}}(R,H)\geq 2$. 2. (2)

The natural map $R\#H\to\operatorname{End}_{R^{H}}(R)$ is an isomorphism of graded algebras.

The above theorem is useful for studying quotient singularities $R^{H}$ and for connecting the representation theory of $H$ and that of $R^{H}$. Several groups of researchers have computed the pertinency $\operatorname{{\sf p}}(R,H)$ in different situations. A lower bound of the pertinency for the cyclic permutation action on the $(-1)$-skew polynomial rings and for the group actions on the universal enveloping algebra of some Lie algebras was given in [BHZ1, BHZ2]; in [GKMW], the authors computed the pertinency for many new examples; the authors in [HZ] introduced a new method of computing pertinency by using pertinent sequences; the paper [CKZ] provided a lower bound of the pertinency for group coactions on noetherian graded down-up algebras.

Although many of these ideas can be applied to the Hopf algebra setting, in this paper we only consider group actions, namely, $H$ is a group algebra over a finite group $G$. When $G$ is acting on an algebra $R$, we usually assume that this action is inner-faithful.

In algebraic geometry, singularities have been studied extensively. We recall the following basic result. When a small finite subgroup $G\subseteq{\text{GL}}_{n}({\mathbb{C}})$ acts naturally on the vector space $V:={\mathbb{C}}^{\oplus n}$, the quotient $V/G:={\text{Spec}}({\mathbb{C}}[V]^{G})$ has isolated singularities if and only if $G$ acts freely on $V\setminus\{0\}$, see [MSt, Lemma 2.1], [Fu, Corollary to Lemma 2] and [MU1, p.7359].

In noncommutative algebraic geometry, Ueyama gave the following definition of a graded isolated singularity [Ue, Definition 2.2]. Let $B$ be a noetherian connected graded algebra. Then $B$ is a graded isolated singularity if the associated noncommutative projective scheme ${\text{tails}}\;B$ (in the sense of [AZ]) has finite global dimension. Let $R$ be a noetherian Artin-Schelter regular algebra and $G$ a finite subgroup of the graded algebra automorphism group $\operatorname{Aut}_{gr}(R)$. Mori-Ueyama [MU1, Theorem 3.10] proved that if $\operatorname{{\sf p}}(R,G)\geq 2$, then $R^{G}$ is a graded isolated singularity if and only if $\operatorname{{\sf p}}(R,G)=\operatorname{GKdim}R$ (which is the largest possible). This result was extended to the Hopf algebra setting, namely, replacing $G$ by a semisimple Hopf algebra, in [BHZ1]. The first few examples of graded isolated singularities in the noncommutative setting were given in [Ue, Theorem 1.4, Examples 3.1, 4.7 and 5.5] by mimicking the commutative criterion of free action of $G$ on $V\setminus\{0\}$. More examples of graded isolated singularities were given in [CKWZ1, CKWZ2, BHZ2, GKMW]. One example of graded isolated singularities in dimension three was given in [CKZ, Lemma 2.11(1)]. A more interesting example is [Ue, Examples 5.4] or [KKZ1, Example 3.1], where the $G$-action on the degree one piece of the regular algebra $R$ is not free. We call such a graded isolated singularity non-conventional [Definition 10.1].

Since Mori-Ueyama’s condition of maximal pertinency is not easy to check in general, we only obtain some special examples of graded isolated singularities in high GKdimension [BHZ2]. It would be nice to understand exactly when the pertinency is maximal, but it seems extremely difficult to achieve this goal. The main object of this paper is to calculate a family of pertinencies all together, using induction. As a consequence, we obtain new examples of graded isolated singularities in arbitrarily large GKdimension.

We now fix some notation. Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let $n$ be an integer $\geq 2$. The algebra that we are interested in is the $(-1)$-skew polynomial ring

[TABLE]

that is generated by $\{x_{0},\ldots,x_{n-1}\}$ and subject to the relations

[TABLE]

for all $i\neq j$. Let $C_{n}$ be the cyclic group of order $n$ acting on $\Bbbk_{-1}[\mathbf{x}]$ by permuting the generators of the algebra cyclically, namely, $C_{n}$ is generated by $\sigma=(012\cdots n-1)$ of order $n$ that acts on the generators by

[TABLE]

for all $i\in{\mathbb{Z}}_{n}:={\mathbb{Z}}/n{\mathbb{Z}}$. We have two results which establish a partial dichotomy.

Theorem 0.2**.**

Let $A:=\Bbbk_{-1}[\mathbf{x}]$ and $G:=C_{n}$. If $n=2^{a}p^{b}$ for some prime $p\geq 7$ and integers $a,b\geq 0$, then $\operatorname{{\sf p}}(A,G)=\operatorname{GKdim}(A)=n$. As a consequence, $A^{G}$ is a graded isolated singularity.

Remark 0.3**.**

1. (1)

Theorem 0.2 is a generalization of [Ue, Examples 5.4] (when $n=2$) and [BHZ1, Theorem 5.7(4)] (when $n=2^{a}$ for some $a\geq 1$). 2. (2)

Although [Ue, Examples 5.4] and [BHZ1, Theorem 5.7(4)] have already provided examples of non-conventional graded isolated singularities of a similar type, Theorem 0.2 is still quite surprising and counter-intuitive.

Note that $\sigma\mid_{V}$ (where $V=\oplus_{i=0}^{n-1}\Bbbk x_{i}$) has eigenvalues $\{1,\xi,\xi^{2},\ldots,\xi^{n-1}\}$ where $\xi$ is a primitive $n$th root of unity. In particular, there is an eigenvalue of $\sigma$ on $V$ that is $1$ (which is not a primitive $n$th root of unity) with eigenvector $\sum_{i=0}^{n-1}x_{i}$ in $V$, or equivalently, the isolated singularity is non-conventional. 3. (3)

In fact, almost all graded isolated singularities considered in this paper will be non-conventional. One aim of this paper is to show that non-conventional graded isolated singularities are common in the noncommutative setting. 4. (4)

The proof of Theorem 0.2 is very involved, using several steps of reduction and induction. We hope to have a more conceptual proof in the future.

When $p=3$ or $5$, Theorem 0.2 fails.

Theorem 0.4**.**

Let $A:=\Bbbk_{-1}[\mathbf{x}]$ and $G:=C_{n}$. If either $3$ or $5$ divides $n$, then $\operatorname{{\sf p}}(A,G)<\operatorname{GKdim}(A)=n$. Consequently, $A^{G}$ is not a graded isolated singularity.

Combining the above two theorems, if $n=2^{a}p^{b}$ for some prime number $p$, then $A^{C_{n}}$ is a graded isolated singularity if and only if $p\neq 3,5$. It is not obvious to us why the primes 3 and 5 are different from other primes in this situation. Based on the above two results we make a conjecture.

Conjecture 0.5**.**

Let $A:=\Bbbk_{-1}[\mathbf{x}]$ and $G:=C_{n}$. Then $A^{G}$ is a graded isolated singularity if and only if $n$ is not divisible by $3$ and $5$.

The above conjecture holds for $n$ less than $77$ following Theorems 0.2 and 0.4.

Corollary 0.6**.**

If $n<77$, then Conjecture 0.5 holds.

Theorem 8.7 provides further evidence for Conjecture 0.5. For general $n$ we have the following lower bound. Let

[TABLE]

Theorem 0.7**.**

Let $A:=\Bbbk_{-1}[\mathbf{x}]$ and $G:=C_{n}$. Then $\operatorname{{\sf p}}(A,G)\geq|\phi_{2}(n)|$.

Note that Theorem 0.7 is an improvement of [BHZ1, Theorem 5.7] when $n$ is even. Combining Theorems 0.2, 0.4, 0.7 and further analysis, we have the following table of pertinencies.

Proposition 0.8**.**

Let $p=\operatorname{{\sf p}}(A,C_{n})$. Then

[TABLE]

where the notation $\in[8,11]$ means that $8\leq p\leq 11$.

By Proposition 0.8, the integer 12 is the smallest $n$ where that the exact value of $\operatorname{{\sf p}}(A,C_{n})$ is unknown. It would be nice to have exact values of $\operatorname{{\sf p}}(A,C_{n})$ for all $n$. In particular, we ask:

Question 0.9**.**

Retain the above notation.

1. (1)

If $n$ is divisible by either $3$ or $5$, what is the exact value of $\operatorname{{\sf p}}(A,C_{n})$? 2. (2)

Does the sequence, from Proposition 0.8,

[TABLE]

match up with any other sequences in literature? The On-Line Encyclopedia of Integer Sequences website

https://oeis.org/

does not give any sequences that match up with (E0.9.1)

Graded isolated singularities have various special properties. Ueyama and Mori-Ueyama investigated certain properties of graded isolated singularities from the viewpoint of derived categories and representation theory. As an immediate consequence of [Ue, MU1, MU2], we have the following. We refer to [Ue, MU1, MU2] for undefined terms in the next corollary.

Corollary 0.10**.**

Suppose $n=2^{a}p^{b}$ for some prime $p\geq 7$ and integers $a,b\geq 0$. Then the following hold.

1. (1)

${\text{tails}}\;A^{G}\cong{\text{tails}}\;A\#G$. 2. (2)

$A$* is a $(n-1)$-cluster tilting object in the category of graded maximal Cohen-Macaulay modules over $A^{G}$.* 3. (3)

The derived category $D^{b}({\text{tails}}\;A^{G})$ has a tilting object. 4. (4)

The derived category $D^{b}({\text{tails}}\;A^{G})$ has a Serre functor.

This paper is organized as follows. We provide background material in Section 1. Theorem 0.7 is proven in Section 2. In Section 3, we give some preliminary results and Theorem 0.4 is proven in Section 4. We continue some preparation in Sections 5 and 6. The main result, Theorem 0.2, is proven in Section 7. In Section 8, we discuss some partial results when $n=p_{1}p_{2}$. Proposition 0.8 is proven in Section 9. In Section 10, we construct more examples of non-conventional graded isolated singularities. The final section contains some questions and comments.

Acknowledgments

The authors thank Jason Bell, Ken Goodearl, Zheng Hua, Lance Small, Agata Smoktunowicz and Robert Won for many useful conversations on the subject and thank Jason Bell for the proof of Lemma 10.6. J.J. Zhang was partially supported by the US National Science Foundation (No. DMS-1700825).

1. Preliminaries

Throughout let $\Bbbk$ be a base field that is algebraically closed of characteristic zero. All objects are $\Bbbk$-linear.

An algebra $R$ is called connected graded if $R=\bigoplus_{n\geq 0}R_{n}$ satisfying $R_{i}R_{j}\subseteq R_{i+j}$ for all $i,j$ and $1\in R_{0}=\Bbbk$. We say $R$ is locally finite if $\dim_{\Bbbk}R_{n}<\infty$ for all $n$. In this paper all connected graded algebras will be locally finite.

We refer to [KL, MR] for the definition of the Gelfand-Kirillov dimension (or GKdimension) of an algebra or a module. When $R$ is connected graded and finitely generated, its GKdimension is equal to

[TABLE]

Observe that $\operatorname{GKdim}(R)=0$ if and only if $\dim_{\Bbbk}R<\infty$. For $q\in\Bbbk^{\times}$, the $q$-polynomial ring

[TABLE]

has GKdimension $m$ (equal to the number of generators). If $B$ is either a subalgebra or a homomorphic image of an algebra $R$, then $\operatorname{GKdim}(B)\leq\operatorname{GKdim}(R)$.

Let $B$ be a noetherian connected graded algebra. If $M$ is a finitely generated graded right $B$-module, then we have a formula similar to (E1.0.1), see [SZ, p.1594],

[TABLE]

Let $c$ be a homogenous central element of $B$ of positive degree. If $M$ is a finitely generated left graded $B$-module, it follows from (E1.0.2) that

[TABLE]

Definitions of other standard concepts such as Artin-Schelter regularity, Auslander regularity, Cohen-Macaulay property are omitted as these can be found in many papers such as [Le, CKWZ1, MSm].

For the first nine sections we consider noncommutative cyclic singularities arising from the action of the cyclic group on the $(-1)$-skew polynomial ring as follows.

Let $n$ be a fixed integer $\geq 2$. Let $\underline{n}:=\{0,\ldots,n-1\}$. Note that $\underline{n}$ can be identified with the additive group ${\mathbb{Z}}_{n}$. Let $\mathbf{x}$ be the set $\{x_{0},\ldots,x_{n-1}\}$ or $\{x_{i}\mid i\in{\mathbb{Z}}_{n}\}$ and $A$ be the $(-1)$-skew polynomial ring $\Bbbk_{-1}[\mathbf{x}]$ as defined in the introduction. Then $A$ is a graded $\Bbbk$-algebra with $\mathrm{deg}(x_{i})=1$ for each $i$ and we denote $A_{j}$ the $\Bbbk$-subspace of degree $j$ elements of $A$. It is well-known that $A$ is noetherian, Artin-Schelter regular, Auslander regular and Cohen-Macaulay of global dimension and GKdimension $n$. Let $\sigma$ be the cycle $(012\cdots n-1)$ which generates the cyclic group $C_{n}$ of order $n$ as a subgroup of the symmetric group $S_{n}$ (considering $S_{n}$ as a set of bijections of $\underline{n}:=\{0,\ldots,n-1\}$). As abstract groups, we have ${\mathbb{Z}}_{n}\cong C_{n}$. The action of $C_{n}$ on $A$ is determined by its action on generators

[TABLE]

The skew group algebra $A\#C_{n}$ with respect to this action consists of all linear combinations of elements $a\#g$ with $a\in A$ and $g\in C_{n}$, with multiplication given by

[TABLE]

extended linearly to all of $A\#C_{n}$. We omit $\#$ if no confusion occurs.

The skew group algebra can be presented in the standard way,

[TABLE]

We now describe a different presentation of the above skew group algebra, using eigenvectors of the $\sigma$-action, which we will use for the rest of the paper.

Since the action of $C_{n}$ on $A$ is graded, the generating subspace $A_{1}$ is a $C_{n}$-module. Let $\omega$ be a primitive $n$th root of unity and $M_{\omega^{j}}$ be the simple (hence $1$-dimensional) $C_{n}$-module where $\sigma$ acts by multiplication by $\omega^{j}$. The $\sigma$-action on $A_{1}$ has minimal polynomial $p(X)=X^{n}-1$, so we can decompose $A_{1}$ as a $C_{n}$-module as follows

[TABLE]

For $\gamma=0,\ldots,n-1$, define the following elements of $A_{1}\subseteq A\#C_{n}$

[TABLE]

The following calculation shows that $b_{\gamma}$ is a $\omega^{-\gamma}$-eigenvector of $\sigma$,

[TABLE]

In other words, we have $\Bbbk b_{\gamma}\cong M_{-\gamma}$ as $C_{n}$-modules, so the basis $\{b_{0},\ldots,b_{n-1}\}$ gives the $C_{n}$-module decomposition of $A_{1}$ in (E1.0.4). We also define the following idempotent elements

[TABLE]

in $\Bbbk C_{n}\subseteq A\#C_{n}$. Let $\mathbf{b}:=(b_{0},\ldots,b_{n-1})$ and $\mathbf{e}:=(e_{0},\ldots,e_{n-1})$. Define the graded commutator, denoted by $[\cdot,\cdot]$, for any homogeneous elements $u,v\in A\#C_{n}$ (or $u,v$ in another graded algebra) by

[TABLE]

We have the following Lemma.

Lemma 1.1**.**

Suppose $\deg(b_{i})=1$ and $\deg(e_{i})=0$ for all $i\in{\mathbb{Z}}_{n}$. The graded algebra $A\#C_{n}$ can be presented as follows

[TABLE]

where $\delta_{ij}$ is the Kronecker delta and indices are taken modulo $n$.

Proof.

Let $b_{i}$ be defined as in (E1.0.5) and let

[TABLE]

We first show that the map

[TABLE]

is well-defined and is an isomorphism. By (E1.0.6) the elements $b_{0},\dots,b_{n-1}$ are eigenvectors for the $\sigma$-action on $A_{1}$ with distinct eigenvalues, hence this is a basis for $A_{1}$, so $\iota$ is an isomorphism in degree $1$. To see that $\iota$ is well-defined as an algebra map, note that the graded commutator of $b_{\gamma}$ and $b_{\delta}$ depends only on the sum of $\gamma$ and $\delta$,

[TABLE]

So the relations $r_{kl}$ go to zero in $\Bbbk_{-1}[\mathbf{x}]$. To show that $\iota$ is an algebra isomorphism, we count the number of independent quadratic relations in $\mathbf{b}$ and show that this number is equal to $\binom{n}{2}$.

For any fixed $k$, the only linear relations among $R_{k}:=\{r_{k0},r_{k1},\ldots,r_{k,n-1}\}$ are $r_{k0}=r_{kk}=0$ and $r_{kl}=r_{k,k-l}$. Define a $C_{2}$-action on $R_{k}$ by $r_{kl}\mapsto r_{k,k-l}$. Then the number of independent relations in $R_{k}$ is equal to $|R_{k}/C_{2}|-1$.

Case 1: For odd $n$, the $C_{2}$-action has exactly one fixed point $r_{kl}$ where $2l=k\mod n$. Therefore $|R_{k}/C_{2}|=(n+1)/2$. The relations in $R_{k}$ are independent from the relations in $R_{k^{\prime}}$ for distinct $k,k^{\prime}$. Since $k$ ranges from [math] to $n-1$, the total number of independent relations is equal to

[TABLE]

Case 2: Let $n$ be even. For odd $k$, the $C_{2}$-action has no fixed points. Therefore $|R_{k}/C_{2}|=n/2$. If $k$ is even, then the $C_{2}$-action has two fixed points, coming from the two solutions of $2l=k\mod n$. Therefore $|R_{k}/C_{2}|=n/2+1$. By considering the odd and even cases separately, we get that the total number of independent relations is equal to

[TABLE]

Therefore $\iota$ is an algebra isomorphism.

The isomorphism

[TABLE]

is well-known. The relations between $\mathbf{b}$ and $\mathbf{e}$ are obtained as follows

[TABLE]

By using the facts

[TABLE]

and

[TABLE]

for $i,j\in{\mathbb{Z}}_{n}$, it is easy to check that the set of relations

[TABLE]

is equivalent to the set of relations

[TABLE]

This completes the proof. ∎

We define the elements

[TABLE]

for all $j\in{\mathbb{Z}}_{n}$. Equation (E1.1.1) shows that the definition of $c_{j}$ does not depend on $k$, and while they are central elements of $A$, they are not central in $A\#C_{n}$. By the relations in Lemma 1.1, we have

[TABLE]

for all $\alpha,j\in{\mathbb{Z}}_{n}$. As above, we denote by $\mathbf{c}=(c_{0},c_{1},\dots,c_{n-1})$. Recall that, for a vector $\mathbf{i}=(i_{0},i_{1},\ldots,i_{n-1})\in{\mathbb{N}}^{n}$,

[TABLE]

We will use the following notation

[TABLE]

Proposition 1.2**.**

For each $r\geq 0$, the set

[TABLE]

is a $\Bbbk$-linear basis for $A_{r}$.

Proof.

The generating function for $\mathcal{B}_{r}$, namely, $g(t)=\sum_{r\geq 0}|\mathcal{B}_{r}|t^{r}$ is

[TABLE]

which agrees with the Hilbert series of $A$. It remains to show that $\mathcal{B}_{r}$ spans $A_{r}$ for each $r$.

Since $\mathbf{b}$ generates $A$, the set \{b_{t_{1}}\cdots b_{t_{r}}\mid{\text{ for different t_{s}}}\} spans $A_{r}$. Using the relation $c_{j}=[b_{k},b_{j-k}]$ and the fact that $c_{j}$ are central, we can ensure that $b_{t_{1}}\cdots b_{t_{r}}$ is in the linear span of $\mathcal{B}_{r}$, as required. ∎

We can extend the above basis for $A_{r}$ to a basis for $(A\#C_{n})_{r}$ by adjoining the $n$ idempotent elements coming from $\Bbbk C_{n}$. Therefore

[TABLE]

and

[TABLE]

are both $\Bbbk$-linear bases for $(A\#C_{n})_{r}$. The following is an immediate consequence of Proposition 1.2.

Corollary 1.3**.**

Retain the above notation.

1. (1)

The union $\mathcal{B}=\bigcup_{r\in\mathbb{N}}\mathcal{B}_{r}$ is a $\Bbbk$-linear basis for $A$. 2. (2)

Both $\mathbf{e}\times\mathcal{B}$ and $\mathcal{B}\times\mathbf{e}$ are $\Bbbk$-linear bases for $A\#C_{n}$. 3. (3)

$A\#C_{n}$* is a finitely generated left and right module over the commutative subring $\Bbbk[\mathbf{c}]\subseteq A$.*

Let $(e_{0})\subset A\#C_{n}$ denote the two sided ideal generated by the idempotent $e_{0}$. We will be concerned with computing the GKdimension of the quotient algebra

[TABLE]

Since $e_{0}$ is the integral of the group algebra $\Bbbk C_{n}$, we obtain that

[TABLE]

Let

[TABLE]

The following lemma is easy.

Lemma 1.4**.**

Retain the above notation.

1. (1)

Let $\overline{C}$ be the quotient ring $\Bbbk[\mathbf{c}]/(c_{k}^{N_{k}};k\in\Phi_{n})$. Then $\operatorname{GKdim}\overline{C}\leq n-|\Phi_{n}|$. 2. (2)

The algebra $E$ is a finitely generated right module over $\overline{C}$. As a consequence,

[TABLE] 3. (3)

$k\in\Phi_{n}$* if and only if, for each $\alpha$, $e_{\alpha}c_{k}^{N}\in(e_{0})$ for $N\gg 0$.*

Proof.

(1) This is true because $\{c_{k}^{N_{k}}\mid k\in\Phi_{n}\}$ is a regular sequence of $\Bbbk[\mathbf{c}]$.

(2) The first assertion follows from Proposition 1.2 (or Corollary 1.3(2)). The consequence follows from [MR, Proposition 8.3.2].

(3) If $c_{k}^{N}\in(e_{0})$, then clearly $e_{\alpha}c_{k}^{N}\in(e_{0})$ for all $\alpha$. The converse follows from the fact $1=\sum_{\alpha}e_{\alpha}$. ∎

In the next few sections we provide upper and lower estimates for $\operatorname{GKdim}E$.

2. An upper bound on $\operatorname{GKdim}E$

This section is a warm-up for more complicated computations to be done in later sections. Fix $n\in\mathbb{N}$, define the following functions on $\mathbb{Z}_{n}$. Let $k$ be in ${\mathbb{Z}}_{n}$. For every $\alpha\in{\mathbb{Z}}_{n}$,

[TABLE]

Let $S_{k}$ be the multiplicative semigroup of $\operatorname{End}_{\mathbb{Z}}(\mathbb{Z}_{n})$ generated by $f_{k}$ and $g_{k}$.

Lemma 2.1**.**

For each $s\in S_{k}$ we have $e_{\alpha}c_{k}^{N}\in(e_{0})+(e_{s(\alpha)})$ for $N\gg 0$.

Proof.

We have two simple calculations

[TABLE]

which imply that $e_{\alpha}c_{k}^{N}\in(e_{f_{k}(\alpha)})$ and that $e_{\alpha}c_{k}^{N}\in(e_{0})+(e_{g_{k}(\alpha)})$. Since $s$ is generated by $f_{k}$ and $g_{k}$, the claim follows. ∎

For fixed $\alpha$, it is easy to see that

[TABLE]

Lemma 2.2**.**

Let $k\in{\mathbb{Z}}_{n}$ be fixed. If $0\in S_{k}(\alpha)$ for every $\alpha$, then $k\in\Phi_{n}$.

Proof.

For each $\alpha$, pick $s\in S_{k}$ so that $s(\alpha)=0$. By Lemma 2.1, $e_{\alpha}c_{k}^{N}\in(e_{0})+(e_{s(\alpha)})=(e_{0})$. The assertion follows by Lemma 1.4(3). ∎

Recall from (E0.6.1) that

[TABLE]

Proposition 2.3**.**

Retain the above notation.

1. (1)

If $k=2^{w}q<n$ such that $q$ is odd and $(n,q)=1$, then $k\in\Phi_{n}$. Equivalently, $\phi_{2}(n)\subseteq\Phi_{n}$. 2. (2)

$|\Phi_{n}|\geq|\phi_{2}(n)|$. 3. (3)

$\operatorname{GKdim}(E)\leq n-|\phi_{2}(n)|$. As a consequence,

1. (3a)

If $n=2^{j}$, then $\operatorname{GKdim}E=0$. 2. (3b)

If $n$ is an odd prime, then $\operatorname{GKdim}E\leq 1$.

Proof.

Let $n$ be a positive integer such that $n=2^{m}p$ where $p$ is odd. Then let $|n|_{2}=m$.

(1) By Lemma 2.2, we need to show that, for every $\alpha$, there is an $s\in S_{k}$ such that $s(\alpha)=0$. Write $\alpha=2^{r}\beta$ where $r=|\alpha|_{2}$. Recall that $k=2^{w}q<n$ such that $(p,q)=1$ where $w=|k|_{2}$. We have two cases, depending on the relative magnitudes of $r$ and $w$.

Case 1: If $r\geq w$, then there exists $j$ such that $\alpha=jk$ in $\mathbb{Z}_{n}$ ($j=2^{r-w}q^{-1}\beta$ where $q^{-1}$ exists in ${\mathbb{Z}}_{n}$), so

[TABLE]

So we take $s=f^{j}_{k}$.

Case 2: If $r<w$, then

[TABLE]

hence $|g_{k}^{w-r}(\alpha)|_{2}\geq w$. This reduces to the first case.

Hence, in both cases, there is an $s\in S_{k}$ such that $s(\alpha)=0$ as required.

(2) This is an immediate consequence of part (1).

(3) The main assertion follows from part (2) and Lemma 1.4(2). Two consequences are special cases of the main assertion. ∎

It is easy to see that Theorem 0.7 is equivalent to Proposition 2.3(3).

3. Preparation, part one

Recall that $E=(A\#C_{n})/(e_{0})$. In this section, we reduce the problem of computing $\operatorname{GKdim}E$ to that of a right quotient module of $A$. Let $\bar{e}_{k}$ denote the image of the idempotent $e_{k}$ in $E$. This gives a right module decomposition

[TABLE]

and it follows that

[TABLE]

For each $j$, we have the following isomorphism of right $A\#C_{n}$-modules

[TABLE]

Using the basis $\mathbf{e}\times\mathcal{B}$ for $A\#C_{n}$ we obtain immediately the right $A$-module isomorphism $e_{j}(A\#C_{n})\cong A$ by $e_{j}a\mapsto a$ with inverse given by $a\mapsto e_{j}a$. We will use this isomorphism to identify $e_{j}(A\#C_{n})$ with $A$ below.

In the following, it will be useful to decompose $A$ according to the characters of the $C_{n}$-action, or equivalently, as modules over the invariant subring $A^{C_{n}}$. Let $R_{j}$ be the $\Bbbk$-subspace of $A$ spanned by the basis consisting of the elements $\mathbf{b}^{\mathbf{i}}\mathbf{c}^{\mathbf{j}}$ where $(\mathbf{i}+\mathbf{j})\cdot\mathbf{v}=j\mod n$ and $\mathbf{v}:=(0,1,\ldots,n-1)$. Since $\mathcal{B}$ is an eigenbasis with respect to the $\sigma$-action, we have that $R_{0}$ is the invariant subring $A^{C_{n}}$ and $R_{j}$ is the $M_{\omega^{-j}}$-isotypic component of the $C_{n}$-action on $A$. This gives an $R_{0}$-module decomposition

[TABLE]

We next find a finite generating set for $R_{j}$.

Lemma 3.1**.**

For each $j=1,\dots,n-1$, define $B_{j}$ to be the set of elements $\mathbf{b}^{\mathbf{i}}\mathbf{c}^{\mathbf{j}}$ satisfying

1. (i)

$(\mathbf{i}+\mathbf{j})\cdot\mathbf{v}=j\mod n$, and 2. (ii)

for each nontrivial $\mathbf{b}^{\mathbf{i}^{\prime}}\mathbf{c}^{\mathbf{j}^{\prime}}$ with $\mathbf{i}^{\prime}\leq\mathbf{i}$ and $\mathbf{j}^{\prime}\leq\mathbf{j}$ we have $\mathbf{b}^{\mathbf{i}^{\prime}}\mathbf{c}^{\mathbf{j}^{\prime}}\not\in R_{0}$.

Then $B_{j}$ generates $R_{j}$ as a right $R_{0}$-submodule of $A$.

Proof.

By definition, the elements $\mathbf{b}^{\mathbf{i}}\mathbf{c}^{\mathbf{j}}$ satisfying (i) generate $R_{j}$. Now suppose $\mathbf{b}^{\mathbf{i}}\mathbf{c}^{\mathbf{j}}$ satisfies (i) but not (ii), that is, there exist some nontrivial $\mathbf{i}^{\prime}\leq\mathbf{i}$ and $\mathbf{j}^{\prime}\leq\mathbf{j}$ such that $\mathbf{b}^{\mathbf{i}^{\prime}}\mathbf{c}^{\mathbf{j}^{\prime}}\in R_{0}$. If $\mathbf{i}^{\prime}=0$, then we can write it as $\mathbf{b}^{\mathbf{i}}\mathbf{c}^{\mathbf{j}-\mathbf{j}^{\prime}}\mathbf{c}^{\mathbf{j}^{\prime}}$. If $\mathbf{i}^{\prime}\neq 0$, then using the commutation relations (E1.1.2), we can move the $\mathbf{b}^{\mathbf{}\mathbf{i}^{\prime}}$ terms, one at a time, to the right side of the expression so that

[TABLE]

where each $\mathbf{k}$ in the summation above satisfies $\mathbf{k}<\mathbf{i}$ and $\lambda_{\mathbf{k},\mathbf{l}}\in\Bbbk$. In particular, we have expressed $\mathbf{b}^{\mathbf{i}}\mathbf{c}^{\mathbf{j}}$ as an $R_{0}$-linear combination of terms in $\mathcal{B}$ whose $\mathbf{b}$-exponent vector is strictly less than $\mathbf{i}$. By induction on the $\mathbf{b}$-exponent vector, we obtain the result. ∎

Lemma 3.2**.**

Retain the above notation. Suppose $1\leq j\leq n-1$ and $0\leq k\leq n-1$.

1. (1)

The intersection $e_{j}(A\#C_{n})\cap(e_{0})$ considered as a right ideal in $A$ is generated by $B_{j}$. 2. (2)

For $N\geq 0$, $c_{k}^{N}\in B_{j}A$ if and only if $e_{j}c_{k}^{N}\in(e_{0})$. 3. (3)

$k\in\Phi_{n}$* if and only if, for each $j$, $e_{j}c_{k}^{N}\in(e_{0})$ for some $N\gg 0$; and if and only if, for each $j$, $c_{k}^{N}\in B_{j}A$ for some $N\gg 0$.*

Proof.

(1) Using the fact $A\#C_{n}=\sum_{i}Ae_{i}=\sum_{i}e_{i}A$, one sees that every element $f\in(e_{0}):=(A\#C_{n})e_{0}(A\#C_{n})$ can be written as a linear combination of terms $ue_{0}v$ where $u,v\in\mathcal{B}$. Without loss of generality let $f=ue_{0}v$ where $u,v\in\mathcal{B}$. If, in addition, $f\in e_{j}(A\#C_{n})$, then

[TABLE]

where $u\in R_{\gamma}$. Hence we can assume that $j=\gamma$ and $u\in R_{j}=B_{j}R_{0}$ by Lemma 3.1. Since elements of $R_{0}$ commute with $e_{0}$, we can actually assume that $u\in B_{j}$. Finally, $ue_{0}v=e_{j}uv$ since $u\in B_{j}$.

(2) This follows from part (1).

(3) This follows from part (2) and Lemma 1.4(3). ∎

Identify $e_{j}(A\#C_{n})$ with $A$ and combining Lemma 3.2 and (E3.0.2), we get

[TABLE]

We can say more: Lemma 3.4 below finds a sufficient condition for when these quotients are isomorphic.

Definition 3.3**.**

Let $\lambda\in{\mathbb{Z}}$ be an integer with $\gcd(\lambda,n)=1$. Let $f_{\lambda}:A\longrightarrow A$ be the algebra map determined by

[TABLE]

for all $i\in{\mathbb{Z}}_{n}$. To see this is an algebra homomorphism, note that

[TABLE]

Since $\lambda$ is invertible in ${\mathbb{Z}}_{n}$, $f_{\lambda}$ is an algebra automorphism of $A$. It is easy to check that $f_{\lambda}(x_{i})=x_{ai}$ where $a=\lambda^{-1}$ in ${\mathbb{Z}}_{n}$.

Lemma 3.4**.**

For any positive integer $\lambda$ with $\gcd(\lambda,n)=1$, we have the following isomorphism of $\Bbbk$-vector spaces

[TABLE]

In particular, if $n$ is prime, then for each $j=2,\ldots,n-1$, we have $\bar{e}_{1}E\cong\bar{e}_{j}E$.

Proof.

Let $f_{\lambda}:A\longrightarrow A$ be the algebra isomorphism defined in Definition 3.3. Now

[TABLE]

hence

[TABLE]

as $\Bbbk$-vector spaces. ∎

Since GKdimension of a finitely generated $A$-module is only dependent on its Hilbert series (E1.0.2), we have the following immediate consequences.

Corollary 3.5**.**

Retain the above notation.

1. (1)

For any $0<j<n$, we have the following lower bound for $\operatorname{GKdim}(E)$

[TABLE] 2. (2)

We have

[TABLE]

where $j$ ranges over positive integers less than $n$ that divide $n$. 3. (3)

If $n$ is prime, then

[TABLE]

For the rest of this section we will consider two distinct values of $n$, with one a factor of the other, and arguments will involve two particular natural algebra homomorphisms between the $(-1)$-skew polynomial rings of these different dimensions.

We fix two integers $m$ and $n$ such that $m$ divides $n$. Let $A$ (resp. $\tilde{A}$) denote the $(-1)$-skew polynomial ring of dimension $n$ (respectively, $m$). Usually we use $\tilde{\quad}$ to denote the corresponding notation for the algebra $\tilde{A}$. For example, since we use $\mathbf{b}$ for the generating set for $A$ (see (E1.0.5)), then we use $\tilde{\mathbf{b}}$ to denote the corresponding generating set for $\tilde{A}$. Recall from the proof of Lemma 1.1, the algebra $A$ is determined completely by the set of relations of the form

[TABLE]

for all $k,l\in{\mathbb{Z}}_{n}$. Similarly for the algebra $\tilde{A}$.

Definition 3.6**.**

Suppose $m$ divides $n$. There is a surjective homomorphism

[TABLE]

determined by sending $b_{j}\mapsto\tilde{b}_{j}$, where in the $\tilde{\mathbf{b}}$ variables the indices are taken modulo $m$. Since $m$ divides $n$, $\pi_{n,m}$ maps any relation of $A$ of the form (E3.5.1) to a relation of $\tilde{A}$. Therefore $\pi_{n,m}$ is an algebra homomorphism. The surjectivity of $\pi_{n,m}$ follows from the fact that it is surjective in degree 1.

Lemma 3.7**.**

Suppose that $m$ divides $n$. Then $\operatorname{GKdim}(E)\geq\operatorname{GKdim}(\tilde{E})$.

Proof.

Let $\tilde{R}_{j}\subset\tilde{A}$, for $j=0,\ldots,m-1$, be defined as in the beginning of Section 3 for the algebra $\tilde{A}$ with $m$ variables. Since $\pi_{n,m}(R_{j})\subset\tilde{R}_{j}$, for $j=0,\ldots,m-1$, we get a surjective homomorphism $A/R_{j}A\to\tilde{A}/\tilde{R_{j}}\tilde{A}$. Then

[TABLE]

∎

Lemma 3.7 will be used in the proof of Theorem 0.4.

For the proof the main result (Theorem 0.2), we need to consider another homomorphism. As before, let $m$ and $n$ be two integers such that $m$ divides $n$. Write $q=n/m$.

Definition 3.8**.**

Suppose $m$ divides $n$ and write $q=n/m$. Let

[TABLE]

be an algebra homomorphism determined by $\theta_{m,n}(\tilde{b}_{i})=b_{qi}$ for all $i\in{\mathbb{Z}}_{m}$. Since $\theta_{m,n}$ maps the relation $\tilde{r}_{kl}$ of $\tilde{A}$ of the form (E3.5.1) to $r_{qk,ql}$ of $A$, $\theta_{m,n}$ is an algebra homomorphism.

We have the following easy lemma.

Lemma 3.9**.**

Suppose that $m$ divides $n$ and write $q=n/m$. Let $N$ be a positive integer. Then

1. (1)

$\theta_{m,n}(\tilde{R}_{j})\subseteq R_{qj}$* for all $j\in{\mathbb{Z}}_{m}$.* 2. (2)

$\theta_{m,n}(\tilde{c}_{j})=c_{qj}$* for all $j\in{\mathbb{Z}}_{m}$.* 3. (3)

If $\tilde{c}^{N}_{i}\in\tilde{R}_{j}\tilde{A}$ for some $i,j\in{\mathbb{Z}}_{m}$ and $N\geq 0$, then $c^{N}_{qi}\in R_{qj}A$.

4. Proof of Theorem 0.4

We first show that for $n=3,5$ the GKdimension of $A/B_{1}A$ is equal to $1$. Hence the GKdimension of $E$ is also equal to $1$ by Corollary 3.5(3). It turns out that we can use these two cases to infer that the GKdimension of $E$ is positive whenever $3$ or $5$ divides $n$.

Proposition 4.1**.**

Let $n=3$.

1. (1)

$B_{1}A\cong(b_{1}A+c_{1}A)$* as right $A$-modules.* 2. (2)

$\operatorname{GKdim}(A/B_{1}A)=1$. 3. (3)

$\operatorname{GKdim}(E)=1$.

Proof.

(1) Recall that the definition of $B_{j}$ is given in Lemma 3.1. By definition one can easily check that $B_{1}=\{b_{1},c_{1},c_{2}^{2},b_{2}c_{2}\}$. Note that $c_{2}=2b_{1}^{2}$ so $c_{2}^{2}=4b_{1}^{4}$ and $b_{2}c_{2}=2b_{1}^{2}b_{2}$, hence $B_{1}A=b_{1}A+c_{1}A$.

(2) Since $c_{1}$ is central in $A$ the quotient $W=A/(c_{1})$ has the structure of a $\Bbbk$-algebra with $\operatorname{GKdim}(W)=\operatorname{GKdim}(A)-1=2$. Then $A/B_{1}A\cong W/b_{1}W$, so

[TABLE]

On the other hand, $b_{1}W\supseteq c_{2}W$. Then

[TABLE]

The assertion follows.

(3) The assertion follows from Corollary 3.5(3) and part (2). ∎

Proposition 4.2**.**

Let $n=5$.

1. (1)

$B_{1}A\subseteq I$* where $I=(b_{1}A+b_{2}A+c_{1}A+c_{2}A+c_{3}A+c_{4}A)$.* 2. (2)

$\operatorname{GK}(A/I)=1$. 3. (3)

$\operatorname{GKdim}(E)=\operatorname{GKdim}(A/B_{1}A)=1$.

Proof.

(1) An element $\mathbf{b}^{\mathbf{i}}\mathbf{c}^{\mathbf{j}}\in B_{1}$ with $\mathbf{j}\neq 0$ is clearly in $I$. To verify the inclusion, it suffices to show that if $\mathbf{b}^{\mathbf{i}}\in B_{1}$ with $\mathbf{i}\in\{0,1\}^{n}$ then $\mathbf{b}^{\mathbf{i}}\in I$. There are two such elements $b_{1}$ and $b_{2}b_{4}$ and these are both in $I$.

(2) Let $J$ be the two sided ideal of $A$ generated by central elements $c_{1},c_{2},c_{3},c_{4}$. Let $\beta_{j}$ (resp. $\gamma_{0}$) denote the image of $b_{j}$ (resp. $c_{0}$) in $A/J$. Then $A/J$ is a finitely generated left $\Bbbk[b_{0}]$-module. Since $A/J$ has no $\beta_{0}$-torsion, it is actually a free module over $\Bbbk[\beta_{0}]$. Moreover, $\beta_{0}$ skew-commutes with the other $\beta_{i}$’s, so a $\Bbbk[\beta_{0}]$-basis for $A/J$ is given by squarefree monomials (with respect to the lexicographical ordering) in $\beta_{1},\dots,\beta_{4}$. Using this basis, we see that

[TABLE]

Hence $\operatorname{GK}(A/I)=1$.

(3) By part (1), the map $A/B_{1}A\rightarrow A/I$ is surjective, so $\operatorname{GKdim}(A/B_{1}A)\geq 1$. By Proposition 2.3(3b), $\operatorname{GKdim}(E)\leq 1$. Combining with Corollary 3.5(3), we have $\operatorname{GKdim}(E)=\operatorname{GKdim}(A/B_{1}A)=1$. ∎

Now we are ready to prove Theorem 0.4.

Proof of Theorem 0.4.

Retain notation as in Lemma 3.7. Take $m=3$ or $5$ (two different cases). By Proposition 4.1(3) and 4.2(3), $\operatorname{GKdim}(\tilde{E})=1$. By Lemma 3.7, $\operatorname{GKdim}(E)\geq 1$. Hence $\operatorname{{\sf p}}(A,G)\leq n-1$. By [MU1, Theorem 3.10], $A^{G}$ is not a graded isolated singularity. ∎

5. Preparation, part two

In Section 7 we will prove Theorem 0.2. We need to do several reduction steps, some of which are given in this section. First we fix some convention throughout the rest of the paper.

Convention 5.1**.**

Let $n$ denote a fixed integer $\geq 2$. Letters such as $i,j,k$ denote elements in ${\mathbb{Z}}_{n}$. Usually these take values in $[0,1,2,\ldots,n-1]$. However [math] is identified with $n$. If we use induction, the induction process starts with $1$ and ends with $n$ (then $n$ is identified with [math]). So, when we use induction on the integer $i$ it will take values in $[1,2,\ldots,n]$.

In Section 5, we only use $i$ and $j$.

Some ideas in this section have appeared in previous sections, but we will do finer analysis. In order to prove Theorem 0.2, we seek to show that for every $j$ with $1\leq j<n$, the $\Bbbk$-vector space $A/B_{j}A$ is finite dimensional. It is necessary and sufficient to show that for every $i$, the element $c_{i}^{N}\in B_{j}A$ for some $N\geq 0$.

Definition 5.2**.**

Retain notation above.

1. (1)

We say $c_{i}$ is nilpotent in $A/B_{j}A$ if $c_{i}^{N}\in B_{j}A$ for some $N\geq 0$. In this case we write $i\in\Psi^{[n]}_{j}$. 2. (2)

We say $n$ is admissible if, for every all $i$ and $j$, $i\in\Psi^{[n]}_{j}$, or equivalently, $\operatorname{GKdim}E=0$, see Lemma 5.3(1) below.

Note that it is automatic that $i\in\Psi^{[n]}_{0}$. Therefore usually we only consider the case when $1\leq j\leq n-1$. We start with some initial analysis and easy reductions.

Lemma 5.3**.**

Retain notation above.

1. (1)

$n$* is admissible if and only if $\operatorname{GKdim}(E)=0$. In this case, $\operatorname{{\sf p}}(A,G)=n$.* 2. (2)

If $i\in\Psi^{[n]}_{j}$ for all $i$ and all divisors $j\mid n$ with $1\leq j<n$, then $n$ is admissible. 3. (3)

If $m$ is a factor of $n$ and $n$ is admissible, then $m$ is admissible.

Proof.

(1) The assertion follows from (E3.0.1), (E3.2.1) and the fact that

[TABLE]

(2) This is Corollary 3.5(2).

(3) This follows from part (1) and Lemma 3.7. ∎

Lemma 5.4**.**

Retain notation above.

1. (1)

If $n=mq$ and $i\in\Psi^{[m]}_{j}$, then $iq\in\Psi^{[n]}_{jq}$. 2. (2)

Let $j$ be a divisor of $n$. If $\gcd(i,n)=\gcd(i,j)$, or $\gcd(i,n)|j$, then $i\in\Psi^{[n]}_{j}$.

Proof.

(1) This is Lemma 3.9(3).

(2) Let $q=\gcd(i,n)$. Then $q=\gcd(i,j)=\gcd(i,j,n)$. By part (1), we might assume that $q=1$. In this case, $i$ is invertible in ${\mathbb{Z}}_{n}$. Let $s$ be the inverse of $i$ in ${\mathbb{Z}}_{n}$. Then there is a $t:=js$ such that $j=ti$ in ${\mathbb{Z}}_{n}$. In this case $c_{i}^{t}\in R_{j}=B_{j}A$ as desired. ∎

Parts (1) to (3) of the next lemma are in fact a slightly different version of Lemma 2.1.

Lemma 5.5**.**

Retain notation above.

1. (1)

If $i\in\Psi^{[n]}_{2j-i}$, then $i\in\Psi^{[n]}_{j}$. 2. (2)

If $i\in\Psi^{[n]}_{j-i}$, then $i\in\Psi^{[n]}_{j}$. 3. (3)

If $i\in\Psi^{[n]}_{2^{s}j+ti}$ for some integers $s,t\geq 0$, then $i\in\Psi^{[n]}_{j}$. 4. (4)

Suppose that every proper divisor of $n$ is admissible. If $\gcd(i,n)$ is even, then $i\in\Psi^{[n]}_{j}$.

Proof.

(1) If $c_{i}^{N}\in B_{2j-i}A$, then we can show $c_{i}^{N+1}\in B_{j}A$ as follows:

[TABLE]

(2) By definition, we have $c_{i}^{N}\in B_{j-i}A$ for some $N>0$. Then

[TABLE]

The assertion follows.

(3) Applying the statement of (2) multiple times, we have that $i\in\Psi^{[n]}_{2^{s}j-i}$. By part (1), we have $i\in\Psi^{[n]}_{2^{s-1}j}$. The assertion follows by induction on $s$.

(4) Let $i=2i^{\prime}$ and $n=2n^{\prime}$. Since $n^{\prime}$ is admissible, $i^{\prime}\in\Psi^{[n^{\prime}]}_{j-i^{\prime}}$. By Lemma 5.4(1), $i\in\Psi^{[n]}_{2j-i}$. The assertion follows from part (1). ∎

6. Preparation, part three

Recall from (E1.3.1) that

[TABLE]

For each $k\in\Phi_{n}$, there exists $N_{k}\geq 0$ such that $c_{k}^{N_{k}}=0$ in $E=(A\#C_{n})/(e_{0})$. It is easy to see that the set $\Phi_{n}$ satisfies the condition in the following definition.

Definition 6.1**.**

A subset of $\Phi\subseteq{\mathbb{Z}}_{n}$ is called special if $k\in\Phi$ if and only if $\lambda k\in\Phi$ for all invertible elements $\lambda\in{\mathbb{Z}}_{n}$. In this case, the ideal $c_{\Phi}:=\langle c_{k}\mid k\in\Phi\rangle$ of $A$ is called the special ideal of $A$ associated to $\Phi$.

Here are some examples of special subsets:

1. (1)

$\Phi=\emptyset$ (in which case, $c_{\Phi}=0$). 2. (2)

$\Phi=\Phi_{n}$ as in (E1.3.1). 3. (3)

$\Phi=\phi_{2}(n)$ as in (E0.6.1). 4. (4)

$\Phi=\{1,2,\ldots,n-1\}$. 5. (5)

$\Phi=\{0,1,2,\ldots,n-1\}$ (in which case, $c_{\Phi}=\{c_{k}\mid 0\leq k\leq n-1\}$).

Fix one special ideal $c_{\Phi}$ of $A$, and write $\overline{A}=A/c_{\Phi}$. Clearly, $C_{n}$ acts on $\overline{A}$. Let $\overline{E}$ be the algebra $(\overline{A}\#C_{n})/(e_{0})$. The following lemma shows that it is useful to pass into the quotient rings.

Lemma 6.2**.**

Retain the notation above and suppose that $\Phi=\Phi_{n}$. Then

[TABLE]

Proof.

Since $E$ is noetherian,

[TABLE]

where the $\max$ runs over all prime ideals ${\mathfrak{p}}$ of $E$. Since $c_{k}$, for each $k\in\Phi_{n}$, is normal and nilpotent in $E$, we have $c_{k}\in{\mathfrak{p}}$ for each prime ${\mathfrak{p}}$. Hence $E/{\mathfrak{p}}$ is annihilated by the ideal $c_{\Phi}$. As a consequence,

[TABLE]

This implies that $\operatorname{GKdim}E\leq\operatorname{GKdim}\overline{E}$. It is clear that $\operatorname{GKdim}E\geq\operatorname{GKdim}\overline{E}$. The assertion follows. ∎

Next we repeat some arguments in Section 3. Going back to a general fixed special ideal (not necessarily associated to $\Phi_{n}$), by abuse of notation, let $\bar{e}_{k}$ also denote the image of the idempotent $e_{k}$ in $\overline{E}$. Then we have a right $\overline{E}$-module decomposition

[TABLE]

and it follows that

[TABLE]

For any $j$, we have the following isomorphism of right $\overline{A}\#C_{n}$-modules

[TABLE]

We recycle the letters $x_{i},b_{i},c_{i}$ for $\overline{A}$ (with some of $c_{i}=0$ in $\overline{A}$). There is a right $\overline{A}$-module isomorphism $e_{j}(\overline{A}\#C_{n})\cong\overline{A}$ by $e_{j}a\mapsto a$ with inverse given by $a\mapsto e_{j}a$. So we will identify $e_{j}(\overline{A}\#C_{n})$ with $\overline{A}$ below.

Let $\overline{\mathcal{B}}_{j}$ (respectively, $\overline{\mathcal{B}}$, $\overline{B}_{j}$) be defined as in Proposition 1.2 (respectively, Corollary 1.3, Lemma 3.1) after removing all $\{c_{k}\mid k\in\Phi\}$. Let $\overline{R}_{j}$ be the $M_{\omega^{-j}}$-isotypic component of the $C_{n}$-action on $\overline{A}$. We have an $R_{0}$-module decomposition

[TABLE]

where $\overline{R}_{0}=(\overline{A})^{C_{n}}$. The following is an $\overline{A}$-version of Lemma 3.2.

Lemma 6.3**.**

Retain the notation above. We are working in the algebra $\overline{A}\#C_{n}$.

1. (1)

The intersection $e_{j}(\overline{A}\#C_{n})\cap(e_{0})$ considered as a right ideal in $\overline{A}$ is generated by $\overline{B}_{j}$. 2. (2)

For $N\geq 0$, we have $c_{k}^{N}\in\overline{B}_{j}\overline{A}$ if and only if $e_{j}c_{k}^{N}\in(e_{0})$. 3. (3)

If there is an integer $N\geq 0$ such that, for each $j$, we have $e_{j}c_{k}^{N}\in(e_{0})$, then $c_{k}^{N}=0$ in $\overline{E}$. If, in addition, we have $\Phi=\Phi_{n}$, then $c_{k}^{N}=0$ in $E$, or equivalently, $c_{k}=0$ in $\overline{E}$.

Proof.

For (1) and (2), see the proof of Lemma 3.2.

(3) Since $1=\sum e_{j}$, we have $c_{k}^{N}\in(e_{0})$. This means that $c_{k}^{N}=0$ in $\overline{E}$.

Now assume $\Phi=\Phi_{n}$. Since $c_{k}$ is normal, $c_{k}\in{\mathfrak{q}}$ for every prime ideal ${\mathfrak{q}}$ of $\overline{E}$. By the proof of Lemma 6.2, every prime quotient $E/{\mathfrak{p}}$ of $E$ is isomorphic to $\overline{E}/{\mathfrak{q}}$ for some prime ideal ${\mathfrak{q}}$ of $\overline{E}$. This implies that $c_{k}$ is zero in $E/{\mathfrak{p}}$, consequently, $c_{k}$ is nilpotent in $E$, or $c_{i}^{N^{\prime}}=0$ in $E$ for some $N^{\prime}$. The assertion follows. ∎

We also have the $\overline{A}$-versions of Lemma 3.4 and Corollary 3.5. The statements are the following and proofs are omitted.

Lemma 6.4**.**

For any positive integer $\lambda$ with $\gcd(\lambda,n)=1$, we have the following isomorphism of $\Bbbk$-vector spaces

[TABLE]

In particular, if $n$ is prime, then for each $j=2,\ldots,n-1$, we have $\bar{e}_{1}\overline{E}\cong\bar{e}_{j}\overline{E}$.

Lemma 6.5**.**

Retain the above notation.

1. (1)

For any $0<j<n$, we have the following lower bound for $\operatorname{GKdim}\overline{E}$

[TABLE] 2. (2)

We have

[TABLE]

where $j$ ranges over positive integers less than $n$ that divide $n$. 3. (3)

If $n$ is prime, then

[TABLE]

One advantage of working with $\overline{A}$ is that

[TABLE]

for all $k\in\Phi$.

Similar to Definition 5.2(1), we say $c_{i}\in\overline{A}$ is nilpotent in $\overline{A}/\overline{B}_{j}\overline{A}$ if $c_{i}^{N}\in\overline{B}_{j}\overline{A}$ for some $N\geq 0$. In this case we write $i\in\overline{\Psi}^{[n]}_{j}$. Now we are ready to take care of Theorem 0.2 when $n$ is prime.

Proposition 6.6**.**

Suppose $n\geq 2$ is neither 3 nor 5.

1. (1)

Suppose $\Phi\supseteq\{1,\ldots,n-1\}$. Then $0\in\overline{\Psi}^{[n]}_{1}$. 2. (2)

If $n$ is prime, then $\operatorname{GKdim}E=0$. Consequently, $n$ is admissible.

Proof.

(1) We start with the trivial observation that if $u,v\in\overline{B}_{1}\overline{A}$, then $[u,v]\in\overline{B}_{1}\overline{A}$. The strategy is to start with $b_{1}\in\overline{B}_{1}\overline{A}$ and apply a sequence of graded commutations with selected elements of $\overline{B}_{1}\overline{A}$ to obtain $c_{0}^{n-1}\in\overline{B}_{1}\overline{A}$. Note that in $\overline{A}$, we have $[b_{i},b_{j}]=0$ unless $i+j=0\mod n$.

Claim 1: Suppose that $0\leq j\leq n-1$ and $2j+1\neq 0\mod n$ (or $2j+1\neq n$). If $c_{0}^{s}b_{j}\in\overline{B}_{1}\overline{A}$ for some $s$, then $c_{0}^{s+1}b_{j+1}\in\overline{B}_{1}\overline{A}$

Proof of Claim 1: First of all, $b_{j+1}b_{n-j}\in\overline{B}_{1}\overline{A}$ (which is not central). Since $2j+1\neq 0\mod n$, $b_{j}b_{j+1}=-b_{j+1}b_{j}$ in $\overline{A}$ (E6.6.1). We obtain

[TABLE]

The assertion follows.

Claim 2: Suppose otherwise that $2j+1=0\mod n$, so that $2j+1=n$. If $c_{0}^{s}b_{j}\in\overline{B}_{1}\overline{A}$, then $c_{0}^{s+2}b_{j+2}\in\overline{B}_{1}\overline{A}$.

Proof of Claim 2: Under the hypothesis of $j$, we have that $n$ is odd and that $b_{j+1}b_{j+2}b_{n-1}\in\overline{B}_{1}\overline{A}$. Given that $n\neq 3,5$, we have $n\geq 7$, and consequently,

[TABLE]

so that the indices in $b_{j+1}b_{j+2}b_{n-1}$ are strictly increasing. This gives the following commutator computation in $\overline{B}_{1}\overline{A}$,

[TABLE]

We apply the one additional commutator to get

[TABLE]

Therefore, if $c_{0}^{s}b_{j}\in\overline{B}_{1}\overline{A}$ and $2j+1=n$, then $c_{0}^{s+2}b_{j+2}\in\overline{B}_{1}\overline{A}$.

Claim 3: $c_{0}^{n-1}\in\overline{B}_{1}\overline{A}$.

Proof of Claim 3: Starting with $b_{1}$, we can apply Claim 1 $(n-2)$-times to get $c_{0}^{n-2}b_{n-1}\in\overline{B}_{1}\overline{A}$ whenever $n$ is even. Hence $c_{0}^{n-1}=[c_{0}^{n-2}b_{n-1},b_{1}]\in\overline{B}_{1}\overline{A}$ as required.

If $n$ is odd, we apply Claim 1 $(j_{0}-1)$-times to get $c_{0}^{j_{0}-1}b_{j_{0}}\in\overline{B}_{1}\overline{A}$ where $j_{0}=\frac{n-1}{2}$. Next we apply Claim 2 to get $c_{0}^{j_{0}+1}b_{j_{0}+2}\in\overline{B}_{1}\overline{A}$. Then apply Claim 1 again $(j_{0}-2)$-times to get $c_{0}^{n-2}b_{n-1}\in\overline{B}_{1}\overline{A}$. Finally we have

[TABLE]

as desired.

The assertion follows from Claim 3.

(2) Now $n$ is a prime integer $\neq 3,5$. By Proposition 2.3(1), $\{1,\ldots,n-1\}\subseteq\Phi_{n}$, so the hypothesis of part (1) holds when taking $\Phi=\Phi_{n}$. By part (1), $0\in\overline{\Psi}^{[n]}_{1}$. Since $c_{k}=0$ in $\overline{A}$ for all $k\neq 0\mod n$, we have that $c_{k}^{N}=0$ in $\overline{B}_{1}\overline{A}$ for all $k$. This implies that $\operatorname{GKdim}(\overline{A}/\overline{B}_{1}\overline{A})=0$. By Lemma 6.5(2), $\operatorname{GKdim}\overline{E}=0$. The assertion follows from Lemma 6.2. ∎

Proposition 6.6(2) is one of the initial steps in the proof of Theorem 0.2 and Proposition 6.6(1) is a step of reduction. The following technical lemma is needed for the proof of the proposition below.

Lemma 6.7**.**

Suppose $i$ and $j$ satisfy the following conditions:

1. (1)

$i$* is an (odd) integer with $0\leq i\leq n-1$ and $\gcd(i,n)>1$,* 2. (2)

$2\leq j\leq n-1$* such that $\gcd(i,j,n)=1$.*

Then there is an integer $t\geq 0$ such that $\gcd(j+ti,n)=1$.

Proof.

Let $n=p_{1}^{n_{1}}\cdots p_{s}^{n_{s}}\cdots p_{r}^{n_{r}}$ for some $1\leq s\leq r$, where $\{p_{i}\}$ are the prime factors of $n$ and $n_{u}\geq 1$ for all $1\leq u\leq r$. The ordering of the prime and the integer $s$ are chosen so that $j=p_{1}^{j_{1}}\cdots p_{s-1}^{j_{s-1}}j^{\prime}$ where $\gcd(j^{\prime},n)=1$ and $j_{w}\geq 1$ for all $1\leq w\leq s-1$. Since we assume that $\gcd(i,j,n)=1$, we can write $i=p_{s}^{i_{s}}\cdots p_{r}^{i_{r}}i^{\prime}$ where $\gcd(i^{\prime},n)=1$ with $i_{v}\geq 0$ for all $s\leq v\leq r$. Let $t=p_{s}\cdots p_{r}$. Then it is easy to see that each $p_{u}$, for $1\leq u\leq r$, does not divide $j+ti$. Thus $\gcd(j+ti,n)=1$. ∎

Proposition 6.8**.**

Let $n\geq 2$ and denote $\Phi=\Phi_{n}$. Suppose that

1. (a)

every proper factor of $n$ is admissible, and that 2. (b)

for each $0\leq i\leq n-1$, $i\in\overline{\Psi}^{[n]}_{1}$.

Then $n$ is admissible.

Proof.

By Lemmas 6.2 and 6.3, and the ideas in Lemma 3.4, it suffices to show that $i\in\overline{\Psi}^{[n]}_{j}$ for all $0\leq i\leq n-1$ and $1\leq j\leq n-1$ with $j\mid n$. We use induction on $j$ and then on $i$. The minimal possible $j$ is $1$, in which the assertion follows from hypothesis (b). Now assume that $j>1$.

If $\gcd(i,n)=1$, or $i=2^{w}q$ with $q$ odd and $\gcd(q,n)=1$, then the assertion follows from Proposition 2.3(1). This shows that the assertion holds for $i=1$. So we can assume that $i\geq 2$ and proceed with induction on $i$.

Suppose $i$ is even and write $i=2i^{\prime}$. If $n$ is even, then $i^{\prime}\in\Psi^{[n/2]}_{j}$ by hypothesis (a). By Lemma 3.9(3), $i\in\Psi^{[n]}_{2j}$. Consequently, $i\in\overline{\Psi}^{[n]}_{2j}$. By the $\overline{A}$-version of Lemma 5.5(3), $i\in\overline{\Psi}^{[n]}_{j}$. If $n$ is odd, then by the induction hypothesis, $i^{\prime}\in\overline{\Psi}^{[n]}_{j}$. Applying the automorphism $f_{2}$ in Definition 3.3, we obtain that $i\in\overline{\Psi}^{[n]}_{2j}$. By the $\overline{A}$-version of Lemma 5.5(3), we have $i\in\overline{\Psi}^{[n]}_{j}$.

For the rest of proof we assume that $i$ is odd and $\gcd(i,n)>1$. If $\gcd(i,j,n)=:q>1$, write $n=qn^{\prime}$, $i=qi^{\prime}$ and $j=qj^{\prime}$. By hypothesis (a), $i^{\prime}\in\Psi^{[n^{\prime}]}_{j^{\prime}}$. By Lemma 3.9(3), $i\in\Psi^{[n]}_{j}$. Consequently, $i\in\overline{\Psi}^{[n]}_{j}$. The other alternative is $\gcd(i,j,n)=1$. By Lemma 6.7, there is a $t\geq 0$ such that $\gcd(j+ti,n)=1$. By hypothesis (b), $i^{\prime}\in\overline{\Psi}^{[n]}_{1}$ for all $i^{\prime}$. Let $\lambda$ be $j+ti$, which is invertible in ${\mathbb{Z}}_{n}$ by Lemma 6.7, and let $f_{\lambda}$ be the ($\overline{A}$-version of the) algebra automorphism defined as in Definition 3.3. Pick $i^{\prime}=i\lambda^{-1}$ in ${\mathbb{Z}}_{n}$. Then, after applying $f_{\lambda}$, $i^{\prime}\in\overline{\Psi}^{[n]}_{1}$ becomes $i\in\overline{\Psi}^{[n]}_{j+ti}$. By the $\overline{A}$-version of Lemma 5.5(3), $i\in\overline{\Psi}^{[n]}_{j}$ for all $i$. Thus we have finished the inductive step and the whole proof. ∎

7. Proof of Theorem 0.2

The proof of Theorem 0.2 follows the strategy of Proposition 6.6. Let us recall the argument for showing that some power of $c_{0}$ is in $\overline{B}_{1}\overline{A}$. Given an element $c_{0}^{s}b_{j}\in\overline{B}_{1}\overline{A}$, depending on whether $j$ satisfies a certain congruence, we applied commutators to $c_{0}^{s}b_{j}$ to conclude that $c_{0}^{s+1}b_{j+1}$ or $c_{0}^{s+2}b_{j+2}$ is in $\overline{B}_{1}\overline{A}$. Starting with $b_{1}$ and continuing in this way, we eventually reach $c_{0}^{n-2}b_{n-1}\in\overline{B}_{1}\overline{A}$. Applying $[-,b_{1}]$ to this gives $c_{0}^{n-1}\in\overline{B}_{1}\overline{A}$.

In the more general situation of Theorem 0.2, we have to show that for each divisor $i$ of $n$, some power of $c_{i}$ is in $\overline{B}_{i_{0}}\overline{A}$ for all $i_{0}$. Given an element $c_{i}^{s}b_{j}\in\overline{B}_{i_{0}}\overline{A}$ we apply certain commutators to $c_{i}^{s}b_{j}$ depending on congruences satisfied by $i,j,i_{0}$ to conclude that other elements of the form $c_{i}^{s^{\prime}}b_{j^{\prime}}$ are in $\overline{B}_{i_{0}}\overline{A}$ (see Lemmas 7.4 and 7.5). These congruences are described in Definition 7.1. Then we show that there are indeed integers satisfying Definition 7.1 (see Lemma 7.2) and that we eventually reach $c_{0}^{s^{\prime}}b_{i-i_{0}}$ (see Lemma 7.6), so that applying $[-,b_{i_{0}}]$ gives what we want. The final induction steps needed for the proof of Theorem 0.2 are given in Proposition 7.7 and Corollary 7.8.

To simplify notation, let

[TABLE]

It is clear that $i_{0}\in\Lambda_{i,i_{0}}$. Write

[TABLE]

and

[TABLE]

for all integers $\xi$. Since $i_{0}$ will be a fixed integer in most of proofs below, we hope that the probability of serious confusion is not high. Let

[TABLE]

where for $n\geq 2$

[TABLE]

It is clear that $0\in\Xi_{i,i_{0}}$.

Let $\mathbb{Z}_{n}^{\times}$ be the invertible elements in $\mathbb{Z}_{n}$ and $S$ be the set of odd integers between $1$ and $n$ which are not in $\phi_{2}(n)$ (E0.6.1). Then define

[TABLE]

It is not hard to show that $\Omega_{2}(n)\subseteq\phi_{2}(n)$.

Definition 7.1**.**

Let $n\geq 2$ be an integer. An integer $i_{0}$ with $1\leq i_{0}\leq n-1$ is called $n$-special if

1. (1)

$i_{0}\in\phi_{2}(n)$. (This is also a consequence of (2) below.)

For all odd integers $i\not\in\phi_{2}(n)$ with $1\leq i\leq n$, the following hold.

1. (2)

$i_{0}-i$ is invertible in ${\mathbb{Z}}_{n}$.

Part (2) is just that $i_{0}\in\Omega_{2}(n)$. Now fix any $i$ as in part (2). For every $1\leq j\leq n-1$, either

1. (3)

$2j+(i_{0}-i)\in\phi_{2}(n)$,

or

1. (4)

(If $2j+(i_{0}-i)\not\in\phi_{2}(n)$, then) there is a $\xi\in\Xi_{i,i_{0}}$ such that

1. (4$(\xi)$i)

$2j+(\xi+2)(i_{0}-i)\in\phi_{2}(n)$, 2. (4$(\xi)$ii)

$j-(\xi+1)(i_{0}-i)\in\phi_{2}(n)$, and 3. (4$(\xi)$iii)

$j+i_{0}+2(\xi+1)(i_{0}-i)\in\phi_{2}(n)$.

For any $\xi\in\Xi_{i,i_{0}}$, conditions (4$(\xi)$i), (4$(\xi)$ii) and (4$(\xi)$iii) all together are denoted by (4$(\xi)$).

Let $Spl(n)$ denote the set of integers $i_{0}$ that are $n$-special.

Lemma 7.2**.**

Let $n=2^{a}p^{b}$ where $p$ is a prime $\geq 3$.

1. (1)

$2\in\Omega_{2}(n)$. 2. (2)

If $p\geq 7$, then $i_{0}=2$ is $n$-special.

Proof.

(1) For every odd integer $i\not\in\phi_{2}(n)$, we have $p\mid i$. Therefore $2$ and $p$ do not divide $i_{0}-i=2-i$, so Definition 7.1(2) holds, and the assertion follows.

(2) Now assume $p\geq 7$. Note that Definition 7.1(1) is obvious. Definition 7.1(2) holds by part (1). For Definition 7.1(3,4), note that $0\in\Xi_{i,i_{0}}$. Note that, for every $i$ given in Definition 7.1(2), $i\not\in\phi_{2}(n)$. Hence $i$ is divisible by $p$. When Definition 7.1(3) fails, namely, $2j+(i_{0}-i)$ (or equivalently, $2j+i_{0}$) is divisible by $p$, then $j+1=\frac{1}{2}(2j+i_{0})$ is divisible by $p$, that is, $j=-1\mod p$. By taking $\xi=0$, we have

[TABLE]

This means that (4$(0)$i), (4$(0)$ii) and (4$(0)$iii) hold. Therefore $i_{0}=2$ is $n$-special. ∎

We have another case when $\Omega_{2}(n)$ is non-empty. The following lemma is not needed for the proof of Theorem 0.2. It will be used in §8 (see Theorem 8.7).

Lemma 7.3**.**

Let $p_{1}$ and $p_{2}$ be two distinct odd primes.

1. (1)

If $n=p_{1}p_{2}$, then $\Omega_{2}(n)\neq\emptyset$. 2. (2)

If $n$ is either $p_{1}p_{2}^{2}n^{\prime}$ or $2p_{1}p_{2}n^{\prime}$ for some $n^{\prime}\geq 1$, then $\Omega_{2}(n)=\emptyset$. As a consequence, $Spl(n)=\emptyset$.

Proof.

(1) Let $p_{1}<p_{2}$. Every integer $m$ can be written uniquely as $m=ap_{1}+bp_{2}$ where $0\leq a<p_{2}$. In particular, $1=a_{1}p_{1}+b_{1}p_{2}$. If $a_{1}$ is odd, we claim that $i_{0}=-1\in\Omega_{2}(n)$. If $a_{1}$ is even, we claim that $i_{0}=1\in\Omega_{2}(n)$. Since the proofs are very similar, we only consider the first case.

Suppose that $i$ is an odd integer $1\leq i\leq n$ that is not in $\phi_{2}(n)$ such that $i-i_{0}=i+1$ is not invertible in ${\mathbb{Z}}_{n}$. Then $i$ and $i+1$ are divisible by different prime factors. We need to consider two cases.

Case 1: $p_{1}\mid i$ and $p_{2}\mid i+1$. Write $i=i^{\prime}p_{1}$ (where $i^{\prime}$ is odd as $i$ is odd) and $i+1=jp_{2}$. Then

[TABLE]

where $i^{\prime}$ is odd, which implies that

[TABLE]

Note that $p_{2}-i^{\prime}$ is even, which contradicts the fact that $a_{1}$ is odd.

Case 2: $p_{2}\mid i$ and $p_{1}\mid i+1$. Write $i=i^{\prime}p_{2}$ (where $i^{\prime}$ is odd as $i$ is odd) and $i+1=jp_{1}$ (where $j$ is even as $i+1$ is even). Then

[TABLE]

which implies that

[TABLE]

Note that $j$ is even, which contradicts the fact that $a_{1}$ is odd.

(2) Since $p_{1}$ and $p_{2}$ are distinct, $1=ap_{1}+bp_{2}$. For every $i_{0}$, one can write it as

[TABLE]

for some $c,d$ with $0\leq c<p_{2}$. If $c$ is odd, take $i=cp_{1}<n$, which is odd and not in $\phi_{2}(n)$. Then $i_{0}-i=dp_{2}$ is not invertible in ${\mathbb{Z}}_{n}$. If $c$ is even, take $i=(c+p_{2})p_{1}<n$, which is odd and not in $\phi_{2}(n)$. Then $i_{0}-i=(d-p_{1})p_{2}$ is not invertible in ${\mathbb{Z}}_{n}$. This means that $i_{0}\not\in\Omega_{2}(n)$ for every $i_{0}$. ∎

The next two lemmas describe a family of partially defined maps \Lambda_{i,i_{0}}\mathrel{-\mkern-6.0mu{\to}\mkern-20.0mu{\color[rgb]{1,1,1}\bullet}\mkern 12.0mu}\Lambda_{i,i_{0}}, where $i_{0}$ and $i$ satisfy hypotheses (1) and (2) of Definition 7.1.

Lemma 7.4**.**

Retain the above notation.

1. (1)

Let $j\in\Lambda_{i,i_{0}}$. If $j$ satisfies hypothesis (3) of Definition 7.1, that is, $2j+(i_{0}-i)\in\phi_{2}(n)$, then $j+(i_{0}-i)\in\Lambda_{i,i_{0}}$. In particular, we get a partially defined map \omega_{0}:\Lambda_{i,i_{0}}\mathrel{-\mkern-6.0mu{\to}\mkern-20.0mu{\color[rgb]{1,1,1}\bullet}\mkern 12.0mu}\Lambda_{i,i_{0}} given by $\omega_{0}(j):=j+(i_{0}-i)$. 2. (2)

If $i_{\xi}\in\Lambda_{i,i_{0}}$ and $2i_{\xi}+(i_{0}-i)=2i_{0}+(2\xi+1)(i_{0}-i)\in\phi_{2}(n)$, then $i_{\xi+1}\in\Lambda_{i,i_{0}}$. 3. (3)

If $2i_{\xi}+(i_{0}-i)=2i_{0}+(2\xi+1)(i_{0}-i)\in\phi_{2}(n)$ for all $0\leq\xi<\frac{1}{2}(\operatorname{mop}(n)-3)$, then $\Xi_{i,i_{0}}=[0,1,\ldots,\frac{1}{2}(\operatorname{mop}(n)-3)]$.

Proof.

Part (2) is a special case of part (1) by taking $j=i_{\xi}$. Part (3) follows from part (2) and induction. So we only prove part (1) below.

Let $s=i-j$ and $r=j-(i-i_{0})$. Then $j+r=2j-i+i_{0}$ is in $\phi_{2}(n)$ by the hypothesis. By Proposition 2.3(1), $j+r\in\Phi_{n}$ and $b_{j}b_{r}=-b_{r}b_{j}$ in $\overline{A}$. We start with $b_{j}c_{i}^{t}\in\overline{B}_{i_{0}}\overline{A}$ for some $t\geq 0$ (as $j\in\Lambda_{i,i_{0}}$). By the choice of $r,s$, we have $b_{r}b_{s}=b_{j-i+i_{0}}b_{i-j}\in\overline{B}_{i_{0}}\overline{A}$.

Consider the commutator $[b_{j}c_{i}^{t},b_{r}b_{s}]$, we have the following elements in $\overline{B}_{i_{0}}\overline{A}$

[TABLE]

The assertion follows. ∎

Lemma 7.5**.**

*Let $\xi\in\Xi_{i,i_{0}}$ and $j\in\Lambda_{i,i_{0}}$. Suppose that $\xi$ and $j$ satisfy the hypotheses $(4(\xi))$ in Definition 7.1. Then $j+(\xi+2)(i_{0}-i)\in\Lambda_{i,i_{0}}$. In particular, we get a partially defined map \omega_{\xi+1}:\Lambda_{i,i_{0}}\mathrel{-\mkern-6.0mu{\to}\mkern-20.0mu{\color[rgb]{1,1,1}\bullet}\mkern 12.0mu}\Lambda_{i,i_{0}} given by $\omega_{\xi+1}(j):=j+(\xi+2)(i_{0}-i)$. *

Proof.

Note that $\xi\in\Xi_{i,i_{0}}$ means that $i_{\xi}\in\Lambda_{i,i_{0}}$ where $i_{\xi}=i_{0}+\xi(i_{0}-i)$.

Let $a=i-j$, $c=j+(\xi+2)(i_{0}-i)$ and $d=-(\xi+1)(i_{0}-i)$. By hypotheses (4$(\xi)$i)-(4$(\xi)$iii), we have that $c+j$, $d+j$ and $i_{\xi}+c$ are in $\phi_{2}(n)$. This means that $[b_{c},b_{j}]=0=[b_{d},b_{j}]=[b_{c},b_{i_{\xi}}]$ in $\overline{A}$.

Starting with $b_{j}c_{i}^{t}\in\overline{B}_{i_{0}}\overline{A}$ for some $t\geq 0$ (as $j\in\Lambda_{i,i_{0}}$), we have the two sets of equations in $\overline{B}_{i_{0}}\overline{A}$. The first set is

[TABLE]

In the above computation, note that $b_{a}b_{c}b_{d}=b_{i-j}b_{j+(\xi+2)(i_{0}-i)}b_{-(\xi+1)(i_{0}-i)}\in\overline{B}_{i_{0}}$. We also need $b_{j}$ to skew commute with $b_{d}$ and $b_{c}$, and $a+j=i$. Since $\xi\in\Xi_{i,i_{0}}$, there is a $t^{\prime}\geq 0$ such that $b_{i_{\xi}}c_{i}^{t^{\prime}}\in\overline{B}_{i_{0}}\overline{A}$. The second set of equations is

[TABLE]

Therefore $j+(\xi+2)(i_{0}-i)\in\Lambda_{i,i_{0}}$ and the assertion follows. ∎

We usually apply the above lemma with the additional hypothesis $2j+(i_{0}-i)\not\in\phi_{2}(n)$.

If $i_{0}$ is $n$-special and $i_{0}-i\in\mathbb{Z}_{n}^{\times}$, then by Definition 7.1, the unions of the domains of definition of $\omega_{0}$ and $\omega_{\xi(i)+1}$ for $\xi\in\Xi_{i,i_{0}}$ is equal to $\Lambda_{i,i_{0}}$. In other words, for any $j\in\Lambda_{i,i_{0}}$ there is some $\omega_{t}$ which can be applied to $j$.

Lemma 7.6**.**

Suppose that $i_{0}$ is $n$-special. Then $-(i_{0}-i)\in\Lambda_{i,i_{0}}$. Consequently $c_{i}^{N}\in\overline{B}_{i_{0}}\overline{A}$ for some $N\geq 0$.

Proof.

Suppose that $\bar{i}_{\xi}\in\Lambda_{i,i_{0}}$ for some $\xi\in[0,M]$, where $M=\frac{1}{2}(\operatorname{mop}(n)-3)$. Then $2\bar{i}_{\xi}+(i_{0}-i)=(1-2\xi)(i_{0}-i)\in\phi_{2}(n)$ since $2\xi-1$ is invertible if $0\leq\xi\leq M$. By Lemma 7.4(1) we have $\bar{i}_{\xi}$ is in the domain of definition of $\omega_{0}$, so $\omega_{0}(\bar{i}_{\xi})=\bar{i}_{\xi-1}\in\Lambda_{i,i_{0}}$. Repeating the argument gives $\bar{i}_{-1}\in\Lambda_{i,i_{0}}$.

Now, let $r$ be the maximal integer such that $\bar{i}_{\xi}\not\in\Lambda_{i,i_{0}}$ for every $\xi\in\{-1,0,\ldots,r\}$. In other words, $\bar{i}_{r+1}\in\Lambda_{i,i_{0}}$. Since $i_{0}$ is $n$-special, either Lemma 7.4 or 7.5 applies. That is, $\bar{i}_{r+1}$ is in the domain of definition of $\omega_{\zeta}$ for some $\zeta\in[0,M+1]$. Applying any such $\omega_{\zeta}$ gives $\omega_{\zeta}(\bar{i}_{r+1})=\bar{i}_{r-\zeta}\in\Lambda_{i,i_{0}}$. By maximality of $r$, we have $r-\zeta<-1$ so $r<\zeta-1\leq M$. Thus we may apply the argument in the first paragraph to conclude that $\bar{i}_{-1}\in\Lambda_{i,i_{0}}$.

The above shows that we always have $\bar{i}_{-1}\in\Lambda_{i,i_{0}}$ under the hypotheses that $i_{0}$ is $n$-special. Equivalently, $b_{i-i_{0}}c_{i}^{t}\in\overline{B}_{i_{0}}\overline{A}$. Finally, $c_{i}^{t+1}=[b_{i-i_{0}}c_{i}^{t},b_{i_{0}}]\in\overline{B}_{i_{0}}\overline{A}$. ∎

Here is one of the main results of this section, which leads to Theorem 0.2.

Theorem 7.7**.**

Let $n\geq 2$. Suppose that

1. (1)

every proper factor of $n$ is admissible, 2. (2)

there is an $n$-special integer $i_{0}$.

Then, for each $0\leq i\leq n-1$, $i\in\overline{\Psi}^{[n]}_{1}$.

Proof.

By Definition 7.1(1), we can express the $n$-special integer $i_{0}$ as the product $i_{0}=2^{w}g$ where $w\geq 0$ with $g$ odd and $\gcd(n,g)=1$. Since $g$ is invertible in ${\mathbb{Z}}_{n}$, by using the automorphism $f_{g}$ of $A$ defined in Definition 3.3, the assertion is equivalent to $i\in\overline{\Psi}^{[n]}_{g}$ for all $0\leq i\leq n-1$. By the $\overline{A}$-version of Lemma 5.5(3), it suffices to show the following claim.

Claim: for each $0\leq i\leq n-1$, $i\in\overline{\Psi}^{[n]}_{i_{0}}$.

Proof of Claim: We prove the Claim by induction on $i$ starting at $i=1$ and ending at $i=n$ (which is also [math] in ${\mathbb{Z}}_{n}$). We consider several cases.

Case 1: $i=1$.

The assertion follows from Proposition 2.3(1). For the inductive step, we assume that $i\geq 2$ and that $i^{\prime}\in\overline{\Psi}^{[n]}_{i_{0}}$ for all $i^{\prime}<i$.

Case 2: $i$ is even.

If $n$ is also even, it follows from Lemma 5.5(4) that $i\in\Psi^{[n]}_{i_{0}}$. Passing to the quotient ring, we have $i\in\overline{\Psi}^{[n]}_{i_{0}}$ as desired.

If $n$ is not even, then $f_{2}$ in Definition 3.3 is an automorphism. Write $i=2i^{\prime}$ for some $1\leq i^{\prime}<i$. By the induction hypothesis, $i^{\prime}\in\overline{\Psi}^{[n]}_{i_{0}}$. Applying $f_{2}$, we obtain that $i=2i^{\prime}\in\overline{\Psi}^{[n]}_{2i_{0}}$. By the $\overline{A}$-version of Lemma 5.5(3), we have $i\in\overline{\Psi}^{[n]}_{i_{0}}$. This takes care of the case when $i$ is even. For cases 3 and 4 below, we assume that $i$ is odd.

Case 3: $i$ is odd and $i\in\phi_{2}(n)$.

In this case, the assertion follows from Proposition 2.3(1) as $c_{i}=0$ in $\overline{A}$.

The remaining case to consider is

Case 4: $i$ is odd and $i\not\in\phi_{2}(n)$. By hypothesis, $i_{0}$ is $n$-special. By Lemma 7.6, $i\in\overline{\Psi}^{[n]}_{i_{0}}$

Hence we finished the inductive step and therefore we complete the proof of the Claim. ∎

Corollary 7.8**.**

Let $n\geq 2$. Suppose that

1. (1)

every proper factor of $n$ is admissible, 2. (2)

$Spl(n)\neq\emptyset$, namely, there is an $n$-special integer $i_{0}$.

Then $n$ is admissible.

Proof.

By Theorem 7.7, for each $0\leq i\leq n-1$, we have $i\in\overline{\Psi}^{[n]}_{1}$. The assertion then follows from Proposition 6.8. ∎

Now we are ready to show Theorem 0.2.

Proof of Theorem 0.2.

In this case $n=2^{a}p^{b}$ for some prime $p\geq 7$. If $(a,b)=(0,1)$ or $(1,0)$, the assertion follows from Proposition 6.6(2). This takes care of the initial step for induction.

By Lemma 7.2, $i_{0}=2$ is $n$-special, which is hypothesis (2) in Corollary 7.8. Hypothesis (1) in Corollary 7.8 follows by induction. Hence we can conclude from Corollary 7.8 that $n$ is admissible. By definition, $\operatorname{GKdim}(E)=0$. Hence $\operatorname{{\sf p}}(A,G)=n$ and, by [MU1, Theorem 3.10], $A^{G}$ is a graded isolated singularity. ∎

Proof of Corollary 0.6.

For each $n<77$, $n$ is either divisible by $3$ or $5$, or $n$ is of the form $2^{a}p^{b}$ for some prime $p\geq 7$. Hence the assertion follows by Theorem 0.2 and 0.4. ∎

Proof of Corollary 0.10.

By Theorem 0.2 and [BHZ1, Theorem 5.7(1)], $A^{G}$ is a graded isolated singularity. By [KKZ1, Theorem 1.5], $A^{G}$ is Gorenstein.

(1) This follows from [MU1, Corollary 2.6].

(2) This follows from [MU1, Theorem 3.15].

(3) This follows from [MU1, Theorem 3.14].

(4) This follows from [Ue, Theorem 1.3]. ∎

Below is a slightly more general result than Theorem 0.2.

Theorem 7.9**.**

Let $S$ be a set of integers $n\geq 2$. Suppose that

1. (1)

each $n$ in $S$ is not divisible by $3$ or $5$, 2. (2)

every proper factor of $n\in S$ is still in $S$. 3. (3)

for each $n\in S$, $Spl(n)\neq\emptyset$.

Then every $n\in S$ is admissible.

Proof.

The assertion follows by induction on $n\in S$. Since each $n$ in $S$ is not divisible by $3$ or $5$, the initial step follows from Proposition 6.6(2). Now we assume that the assertion holds for all proper factors of $n$. The induction step follows from hypothesis (3) and Corollary 7.8. ∎

8. Partial results when $n=p_{1}p_{2}$

In this section we give some partial answer to the case when $n=p_{1}p_{2}$ for $p_{i}$ being distinct primes. Some lemmas works for the case when $n=2^{a}p_{1}^{b}p_{2}^{c}$.

Another way of defining $\Omega_{2}(n)$ (E7.0.1) is the following

[TABLE]

As noted in Section 7, $\Omega_{2}(n)\subseteq\phi_{2}(n)$.

In the rest of this section, let $n$ be $2^{a}p_{1}^{b}p_{2}^{c}$ where $p_{1}$ and $p_{2}$ are distinct odd primes $\geq 7$ and $b,c\geq 1$. We start with a linear algebra fact.

Lemma 8.1**.**

Let $i_{0}\in\phi_{2}(n)$ and $i\in[1,\ldots,n]$ such that $p_{1}\mid i$. If $p_{2}$ divides both $c_{11}i_{0}+c_{12}i$ and $c_{21}i_{0}+c_{22}i$, then $p_{2}$ divides

[TABLE]

Proof.

By easy linear algebra, $p_{2}$ divides both $\operatorname{Det}i_{0}$ and $\operatorname{Det}i$. Since $p_{2}$ and $i_{0}$ are coprime, $p_{2}$ divides $\operatorname{Det}$. ∎

Lemma 8.2**.**

Let $i_{0}\in\phi_{2}(n)$ and $i\in[1,\ldots,n]$ be an odd integer not in $\phi_{2}(n)$. There is at most one integer

[TABLE]

such that $2i_{\xi}+(i_{0}-i)\not\in\phi_{2}(n)$.

Proof.

Without loss of generality, we can assume that $p_{1}$ divides $i$. For each $\xi$ in (E8.2.1), $p_{1}$ does not divide $2i_{\xi}+(i_{0}-i)=(2\xi+3)i_{0}-(2\xi+1)i$, as

[TABLE]

If there are $\xi_{1}$ and $\xi_{2}$ in (E8.2.1) such that $2i_{\xi_{1}}+(i_{0}-i)$ and $2i_{\xi_{2}}+(i_{0}-i)$ are not in $\phi_{2}$, then $p_{2}$ must divide both $2i_{\xi_{1}}+(i_{0}-i)$ and $2i_{\xi_{2}}+(i_{0}-i)$ (or equivalently, divide both $(2\xi_{1}+3)i_{0}-(2\xi_{1}+1)i$ and $(2\xi_{2}+3)i_{0}-(2\xi_{2}+1)i$). By Lemma 8.1, $p_{2}$ divides $\operatorname{Det}$, and an easy computation shows that

[TABLE]

By the choices of $\xi_{1},\xi_{2}$ in (E8.2.1), $p_{2}$ does not divide $4(\xi_{2}-\xi_{1})$, a contradiction. The assertion follows. ∎

Lemma 8.3**.**

Retain the hypothesis as in Lemma 8.2. Suppose that $\xi$ in (E8.2.1) is such that $2i_{\xi}+(i_{0}-i)\not\in\phi_{2}(n)$, namely, Definition 7.1(3)* fails for $j=i_{\xi}$. Then Definition 7.1(4$(0)$) holds for $j=i_{\xi}$.*

Proof.

By Lemma 8.2, $\xi$ is unique. It suffices to show that the following elements in (4(0)i), (4(0)ii) and (4(0)iii) of Definition 7.1(4), when $j=i_{\xi}$, are in $\phi_{2}$:

[TABLE]

Since $i\not\in\phi_{2}(n)$, either $p_{1}$ or $p_{2}$ divides $i$. Without loss of generality, we say $p_{1}$ divides $i$. By the proof of Lemma 8.2, we have $p_{2}$ divides $2i_{\xi}+(i_{0}-i)=(2\xi+3)i_{0}-(2\xi+1)i$. Since $\operatorname{mop}(n)\geq 7$, all of $\xi+2,\xi,\xi+4$ are strictly less than $\mod(n)$. Hence $p_{1}$ does not divide elements in (E8.3.1)-(E8.3.3). We claim that $p_{2}$ does not divide elements in (E8.3.1)-(E8.3.3). If this is false, say, the element in (E8.3.3) is divisible by $p_{2}$, then $p_{2}$ divides both $(2\xi+3)i_{0}-(2\xi+1)i$ and $(\xi+4)i_{0}-(\xi+2)i$. By Lemma 8.1, $p_{2}$ divides $\operatorname{Det}$, where

[TABLE]

However, by the choice of $\xi$ in (E8.2.1), $p_{2}$ does not divide $\operatorname{Det}$, a contradiction. The claim is proved.

Finally, by the above, elements in (E8.3.1)-(E8.3.3) are not divisible by either $p_{1}$ or $p_{2}$. Hence these elements are in $\phi_{2}(n)$. The assertion follows. ∎

Lemma 8.4**.**

Retain the hypothesis as in Lemma 8.2. Then either

[TABLE]

or

[TABLE]

The second case can happen only when there is a $\xi$ in (E8.2.1) such that $2i_{\xi}+(i_{0}-i)\not\in\phi_{2}(n)$.

By Lemma 8.2, $\xi$ is Lemma 8.4 above is unique if it exists.

Proof of Lemma 8.4.

If, for each $\xi$ in (E8.2.1), $2i_{\xi}+(i_{0}-i)$ is in $\phi_{2}(n)$, then, by Lemma 7.4(3),

[TABLE]

which is the first case.

To prove the lemma, we may assume that

[TABLE]

and that there is a $\xi$ in (E8.2.1) such that $2i_{\xi}+(i_{0}-i)\not\in\phi_{2}(n)$. By Lemma 8.2, $\xi$ is unique, namely, for all $\xi^{\prime}\neq\xi$ in (E8.2.1), $2i_{\xi^{\prime}}+(i_{0}-i)$ is in $\phi_{2}(n)$. By Lemma 7.4(2) and induction, $0,1,2,\ldots,\xi\in\Xi_{i,i_{0}}$. By Lemma 8.3, Definition 7.1(4$(0)$) holds for $j=i_{\xi}$. By Lemma 7.5 (for a different $\xi=0\in\Xi_{i,i_{0}}$ in the setting of Lemma 7.5),

[TABLE]

is in $\Lambda_{i,i_{0}}$. If $\xi+2\leq\frac{1}{2}(\operatorname{mop}(n)-3)$, then $\xi+2\in\Xi_{i,i_{0}}$ by definition. By Lemma 8.2, $2i_{\xi^{\prime}}+(i_{0}-i)\in\phi_{2}(n)$ for all

[TABLE]

Using Lemma 7.4(2) and induction again, $\xi+3,\ldots,\frac{1}{2}(\operatorname{mop}(n)-3)\in\Xi_{i,i_{0}}$. Therefore

[TABLE]

This finishes the proof. ∎

Corollary 8.5**.**

If $\operatorname{mop}(n)\geq 11$, then $\Xi_{i,i_{0}}$ contains one of the following subsets

[TABLE]

Proof.

The assertion follows from Lemma 8.4 and the fact $\frac{1}{2}(\operatorname{mop}(n)-3)\geq 4$. ∎

Proposition 8.6**.**

If $\operatorname{mop}(n)\geq 17$, then Definition 7.1(4)* holds automatically. As a consequence, $\Omega_{2}(n)=Spl(n)$.*

Proof.

We start with the assumption that $2j+(i_{0}-i)\not\in\phi_{2}(n)$. Without loss of generality, we can assume that $p_{1}$ divides $2j+(i_{0}-i)$. For simplicity, let $j_{0}=i_{0}-i$. So $j_{0}$ is not divisible by either $p_{1}$ or $p_{2}$. The assumption is that $2j+j_{0}$ is divisible by $p_{1}$. By Lemma 8.1 (replacing $(i,i_{0})$ by $(j,j_{0})$), elements of the forms in Definition 7.1(4$(\xi)$i) and (4$(\xi)$ii) for $\xi=0,1,2,3,4$ are not divisible by $p_{1}$ since the corresponding $\operatorname{Det}$ is not divisible by $p_{1}\geq 17$. Further, any two distinct elements of the form in Definition 7.1(4$(\xi)$i) and (4$(\xi)$ii) can not be divided by $p_{2}$ either (using the fact $p_{2}\geq 17$). This implies that there is only one $\xi$, say $\xi_{0}\in[0,1,2,3,4]$, such that either (4$(\xi_{0})$i) or (4$(\xi_{0})$ii) fails. Removing $\xi_{0}$ from the list $\Xi_{i,i_{0}}$, we still have three integers $\{\xi_{1},\xi_{2},\xi_{3}\}\subseteq[0,1,2,3,4]\cap\Xi_{i,i_{0}}$ such that Definition 7.1(4$(\xi_{s})$i) and (4$(\xi_{s})$ii) hold for all $s=1,2,3$. It remains to show that Definition 7.1(4$(\xi_{s})$iii) holds for one of $s$. Suppose on the contrary that Definition 7.1(4$(\xi_{s})$iii) fails for all three $s$. Then there are two $s$ such that $j+i_{0}+2(\xi_{s}+1)(i_{0}-i)$ is divisible by the same prime factor, say $p_{2}$. Applying Lemma 8.1 to these two element with $(j^{\prime},j_{0})=(j+i_{0},i_{0}-i)$, we obtain that $p_{2}$ divides $|\operatorname{Det}|=2|\xi_{s_{1}}-\xi_{s_{2}}|<\operatorname{mop}(n)$. This is impossible. Therefore Definition 7.1(4$(\xi_{s})$iii) holds for one of $s$. Thus we show that Definition 7.1(4) holds automatically.

The consequence is clear. ∎

Theorem 8.7**.**

Suppose $n=p_{1}p_{2}$ where $p_{s}$ are prime $\geq 17$. Then $n$ is admissible. As a consequence $A^{G}$ has a graded isolated singularity.

Proof.

Since every proper factor of $n$ is admissible by Theorem 0.2, hypothesis of Corollary 7.8(1) holds. By Lemma 7.3(1), $\Omega_{2}(n)\neq\emptyset$. By Proposition 8.6, $Spl(n)\neq\emptyset$. Hence hypothesis of Corollary 7.8(2) holds. The assertion now follows from Corollary 7.8. ∎

9. Proof of Proposition 0.8

We start with $n=6$ and $10$.

Lemma 9.1**.**

Retain the notation as in Theorem 0.4. If $n=6$, then $\operatorname{{\sf p}}(A,C_{n})=5$.

Proof.

First let $\Phi:=\Phi_{6}=\{1,2,4,5\}$. By Lemma 6.2, $\operatorname{GKdim}E=\operatorname{GKdim}\overline{E}$. It suffices to show that $\operatorname{GKdim}\overline{E}=1$. By Theorem 0.4, it is enough to show that $\operatorname{GKdim}\overline{E}\leq 1$. By Lemma 6.5(2),

[TABLE]

where $j$ ranges over $\{1,2,3\}$ (all positive integers less than $6$ that divide $6$).

Case 1: $j=3$. Since $c_{3}\in\overline{B}_{3}\overline{A}$, $c_{3}=0$ in $\overline{A}/\overline{B}_{3}\overline{A}$. By the definition of $\overline{A}$, $c_{1}=c_{2}=c_{4}=c_{5}=0$. Therefore $c_{i}=0$ for $i=1,2,3,4,5$ in $\overline{A}/\overline{B}_{3}\overline{A}$. Therefore $\overline{A}/\overline{B}_{3}\overline{A}$ is a finitely generated module over $\Bbbk[c_{0}]$, which implies that $\operatorname{GKdim}(\overline{A}/\overline{B}_{3}\overline{A})\leq 1$.

Case 2: $j=2$. We need to show that $\operatorname{GKdim}(\overline{A}/\overline{B}_{2}\overline{A})\leq 1$. By (E1.0.3), it is enough to show the claim that

[TABLE]

Now we change $\Phi$ from $\{1,2,4,5\}$ to $\{1,2,3,4,5\}$. Re-cycle all notation such as $\overline{A}$, $\overline{B}_{i}$, etc, for the new $\Phi$, claim (E9.1.1) becomes

[TABLE]

with $\Phi=\{1,2,3,4,5\}$. For the rest of the proof in Case 2, let $\Phi=\{1,2,3,4,5\}$. Note that we have the following elements in $\overline{B}_{2}\overline{A}$:

[TABLE]

Taking commutators in $\overline{B}_{2}\overline{A}$, we have the following computations in $\overline{B}_{2}\overline{A}$:

[TABLE]

Therefore $c_{0}^{4}=0$ in $\overline{A}/\overline{B}_{2}\overline{A}$, and consequently, $0\in\overline{\Psi}_{2}$. By definition, $c_{i}=0$ in $\overline{A}/\overline{B}_{2}\overline{A}$ for all $i=1,2,3,4,5$. Therefore (E9.1.2) holds.

Case 3: $j=1$. We need to show that $\operatorname{GKdim}(\overline{A}/\overline{B}_{1}\overline{A})\leq 1$ for $\Phi=\{1,2,4,5\}$. Similar to the proof of Case 2, it is sufficient to show

[TABLE]

with new $\Phi=\{1,2,3,4,5\}$. But this is Proposition 6.6(1).

Combining these three cases, we finish the proof. ∎

Lemma 9.2**.**

Retain the notation as in Theorem 0.4. If $n=10$, then $\operatorname{{\sf p}}(A,C_{n})=9$.

Proof.

This proof is very similar to the proof of Lemma 9.1

First we let $\Phi:=\Phi_{10}=\{1,2,3,4,6,7,8,9\}$. By Lemma 6.2, $\operatorname{GKdim}E=\operatorname{GKdim}\overline{E}$. It suffices to show that $\operatorname{GKdim}\overline{E}=1$. By Lemma 6.5(2),

[TABLE]

where $j$ ranges over $\{1,2,5\}$ (all positive integers less than $6$ that divide $10$).

Case 1: $j=5$. The proof of Case 1 in Lemma 9.1 can be easily modified by replacing $j=3$ to $j=5$.

Case 2: $j=2$. We need to show that $\operatorname{GKdim}(\overline{A}/\overline{B}_{2}\overline{A})\leq 1$. By (E1.0.3), it is enough to show the claim that

[TABLE]

Now we change $\Phi$ from $\{1,2,3,4,6,7,8,9\}$ to $\{1,2,3,4,5,6,7,8,9\}$. Recycle all notation such as $\overline{A}$, $\overline{B}_{i}$, etc, for the new $\Phi$, claim (E9.2.1) becomes

[TABLE]

with new $\Phi=\{1,2,3,4,5,6,7,8,9\}$. For the rest of the proof in Case 2, we use this new $\Phi$. Note that we have the following elements in $\overline{B}_{2}\overline{A}$:

[TABLE]

Taking commutators in $\overline{B}_{2}\overline{A}$, we have the following computations in $\overline{B}_{2}\overline{A}$:

[TABLE]

Therefore $c_{0}^{5}=0$ in $\overline{A}/\overline{B}_{2}\overline{A}$, and consequently, $0\in\overline{\Psi}_{2}$. By definition, $c_{i}=0$ in $\overline{A}/\overline{B}_{2}\overline{A}$ for all $i=1,2,3,4,5,6,7,8,9$. Therefore (E9.2.2) holds.

Case 3: $j=1$. The proof of Case 3 in Lemma 9.1 works.

Combining these three cases with Theorem 0.4, we finish the proof. ∎

Next we consider $n=9$.

Lemma 9.3**.**

Retain the notation as in Theorem 0.4. If $n=9$, then $\operatorname{{\sf p}}(A,C_{n})=8$.

Proof.

First we let $\Phi:=\Phi_{9}=\{1,2,4,5,7,8\}$. By Lemma 6.2, $\operatorname{GKdim}E=\operatorname{GKdim}\overline{E}$. It suffices to show that $\operatorname{GKdim}\overline{E}=1$. By Lemma 6.5(2),

[TABLE]

where $j$ ranges over $\{1,3\}$. So we need to consider two cases.

Case 1: $j=3$. Since $b_{3},c_{3}\in\overline{B}_{3}\overline{A}$, $c_{6},c_{3}\in\overline{B}_{3}\overline{A}$. This shows that $c_{i}=0$ in $\overline{A}/\overline{B}_{3}\overline{A}$ for all $i=1,2,3,4,5,6,7,8$. So $\operatorname{GKdim}(\overline{A}/\overline{B}_{3}\overline{A})\leq 1$.

Case 2: $j=1$. We need to show that $\operatorname{GKdim}(\overline{A}/\overline{B}_{1}\overline{A})\leq 1$. Note that we have the following elements in $\overline{B}_{1}\overline{A}$:

[TABLE]

Taking commutators in $\overline{B}_{1}\overline{A}$, we have the following computations inside $\overline{B}_{1}\overline{A}$:

[TABLE]

Similarly we have the following elements in $\overline{B}_{1}\overline{A}$:

[TABLE]

Taking commutators in $\overline{B}_{1}\overline{A}$, we have the following computations inside $\overline{B}_{1}\overline{A}$:

[TABLE]

It is easy to see that the quotient algebra

[TABLE]

has GKdimension 1. Since $\overline{A}/\overline{B}_{1}\overline{A}$ is a quotient of $D$ by the above computation. Therefore $\operatorname{GKdim}\overline{A}/\overline{B}_{1}\overline{A}\leq 1$ as desired.

Combining these two cases with Theorem 0.4, we finish the proof. ∎

Now we are ready to prove Proposition 0.8.

Proof of Proposition 0.8.

When $n=6,10,9$, the $p$ is $5,9,8$ by Lemmas 9.1, 9.2 and 9.3 respectively. For $n=3,5$, the assertion follows by Propositions 4.1 and 4.2 respectively. For $n=2,4,7,8,11,13,14$, the assertion follows from Theorem 0.2. The statement for $n=12$ follows by combining Theorems 0.4 and 0.7. ∎

10. More examples of graded isolated singularities

To save space, we will omit some non-essential details in Sections 10 and 11.

In this section, we give more examples of graded isolated singularities. Some nice results of He-Y.H. Zhang [HZ] and Gaddis-Kirkman-Moore-Won [GKMW] will be reviewed and used in this section. First we recall some definitions from [HZ].

Let $R$ be a noetherian algebra and $G$ be a finite group acting on $R$. We say that two sequences $(a_{1},\ldots,a_{w})$ and $(b_{1},\ldots,b_{w})$ of elements of $R$ are pertinent under the $G$-action, if

[TABLE]

for all $1\neq g\in G$. In this case we write $(a_{1},\ldots,a_{w})\sim(b_{1},\ldots,b_{w})$. The radical of the $G$-action on $R$ is defined to be

[TABLE]

By [HZ, Section 1], ${\mathfrak{r}}(R,G)$ is a 2-sided ideal of $R$.

Let $e_{0}$ be the element $1\#(\frac{1}{|G|}\sum_{g\in G}g)$ in $R\#G$. By the proof of [HZ, Proposition 2.4], ${\mathfrak{r}}(R,G)=R\cap(e_{0})$. Therefore we have [HZ, (3.1.1)],

[TABLE]

If $R$ is noetherian and Artin-Schelter regular, then $R^{G}$ is a graded isolated singularity if and only if $R/{\mathfrak{r}}(R,G)$ is finite dimensional over the base field $\Bbbk$.

As said in introduction, almost all graded isolated singularities studied in this paper are non-conventional in the following sense.

Definition 10.1**.**

Let $R$ be a noetherian Artin-Schelter regular algebra with graded maximal ideal $\mathfrak{m}:=A_{\geq 1}$. Let $G$ be a finite subgroup of $\operatorname{Aut}_{gr}(R)$ such that $R^{G}$ is a graded isolated singularity. We say the graded isolated singularity $R^{G}$ is non-conventional if there is an element $1\neq\sigma\in G$ such that at least one of the eigenvalues of $\sigma$ restricted to the $\Bbbk$-vector space $\mathfrak{m}/\mathfrak{m}^{2}$ is 1. Otherwise, we say $R^{G}$ is conventional.

If $R$ is the commutative polynomial ring $\Bbbk[V]$, then every graded isolated singularity $R^{G}$ is conventional, see [MU1, Corollary 3.11]. A similar statement holds for skew polynomial rings. Let $\{p_{ij}\mid 1\leq i<j\leq n-1\}$ be a set of nonzero scalars in $\Bbbk^{\times}$. The skew polynomial ring $\Bbbk_{p_{ij}}[x_{0},x_{1},\ldots,x_{n-1}]$ is generated by $\{x_{0},\ldots,x_{n-1}\}$, with $\deg x_{i}>0$ for each $i$, and subject to the relations $x_{j}x_{i}=p_{ij}x_{i}x_{j}$ for all $i<j$. Let $V=\bigoplus_{i=0}^{n-1}\Bbbk x_{i}$.

Lemma 10.2**.**

Let $R$ be a skew polynomial ring $\Bbbk_{p_{ij}}[x_{0},x_{1},\ldots,x_{n-1}]$ and let $G$ be a finite group acting on $R$ linearly and diagonally, namely, each $x_{i}$ is an eigenvector of $G$. Then $R^{G}$ is a graded isolated singularity if and only if the $G$-action on $V\setminus\{0\}$ is free.

Proof.

Let $d=|G|$.

$\Longleftarrow$: Assume that the $G$-action on $V\setminus\{0\}$ is free. In this setting, for each $i$, the $G$-action on $\Bbbk x_{i}\setminus\{0\}$ is also free. This implies that there is an $\sigma\in G$ and a $\xi\in\Bbbk$ being a primitive $d$th root of unity such that $\sigma(x_{i})=\xi x_{i}$. As a consequence, $G$ is generated by $\sigma$ and $\sigma^{w}(x_{i})=\xi^{w}x_{i}$ for all $w\in{\mathbb{Z}}_{d}$. By [HZ, Lemma 3.4], $x_{i}^{d}\in{\mathfrak{r}}(R,G)$. Therefore $R/{\mathfrak{r}}(R,G)$ is finite dimensional. As a consequence, $R^{G}$ is a graded isolated singularity.

$\Longrightarrow$: We prove the statement by contradiction and assume that the $G$-action on $V\setminus\{0\}$ is not free. Pick an element $1\neq\sigma\in G$ so that $\sigma$ has a fixed point in $V\setminus\{0\}$. This implies that $\sigma$ fixes one $x_{i}$. Replacing $G$ by the subgroup $\langle\sigma\rangle$, we can assume that $G=\langle\sigma\rangle$ following [GKMW, Theorem 3.4]. Since $\sigma$ fixes $x_{i}$, one can show that $x_{i}^{N}$ is not in ${\mathfrak{r}}(R,G)$ for all $N\geq 0$ (which also follows from Lemma 10.4(6) in an appropriate setting). Therefore $R/{\mathfrak{r}}(R,G)$ is not finite dimensional, whence $R^{G}$ is not a graded isolated singularity. ∎

As a consequence of [GKMW, Theorem 3.4], if $R^{G}$ is a graded isolated singularity, then so is $R^{H}$ for all subgroups $1\subsetneq H\subseteq G$. The graded isolated singularities in the above lemma are all conventional. One nice example of non-conventional graded isolated singularities is given by Gaddis-Kirkman-Moore-Won [GKMW].

Example 10.3**.**

[GKMW, Theorem 5.2]

Let $R$ be a generic 3-dimensional Sklyanin algebra $S(a,b,c)$ generated by $\{x,y,z\}$ with standard relations, see [GKMW, Introduction]. Let $G$ be the cyclic group of order 3 acting on $R$ by permuting the standard generators $\{x,y,z\}$. Then $R^{G}$ is a graded isolated singularity by [GKMW, Theorem 5.2]. Since $G$ has a fixed point $x+y+z$ in $R_{1}\setminus\{0\}$, we obtain that $R^{G}$ is non-conventional.

We will use a few more lemmas. In Lemma 10.4 below we do not assume that the $G$-actions is inner-faithful.

Lemma 10.4**.**

Let $R$ and $S$ be two connected graded algebra with $G$-action where $G$ is a finite group. Let $e_{0}=1\#(\frac{1}{|G|}\sum_{g\in G}g)$. Suppose that $f:R\to S$ be a graded algebra homomorphism that is compatible with $G$-action.

1. (1)

There is an induced algebra homomorphism $f\#G:R\#G\to S\#G$ such that $f\#G(r\#g)=f(r)\#g$ for all $r\in R$ and $g\in G$. 2. (2)

$f\#G$* maps $e_{0}\in R\#G$ to $e_{0}\in S\#G$. As a consequence, there is an induced algebra homomorphism $\overline{f\#G}:R\#G/(e_{0})\to S\#G/(e_{0})$.* 3. (3)

If $x\in R$ such that $x:=x\#1\in(e_{0})$ in $R\#G$, then $f(x):=f(x)\#1\in(e_{0})$ in $S\#G$. 4. (4)

If $f$ is surjective, so is $f\#G$. If, further, $S\#G/(e_{0})$ is infinite dimensional, so is $R\#G/(e_{0})$. 5. (5)

$f$* maps ${\mathfrak{r}}(R,G)$ to ${\mathfrak{r}}(S,G)$. As a consequence, $f$ induces an algebra homomorphism from $R/{\mathfrak{r}}(R,G)$ to $S/{\mathfrak{r}}(S,G)$.* 6. (6)

Suppose $f$ is surjective. If $S/{\mathfrak{r}}(S,G)$ is infinite dimensional, then so is $R/{\mathfrak{r}}(R,G)$.

The proof of Lemma 10.4 is easy and omitted.

Lemma 10.5**.**

Let $A=\Bbbk_{-1}[\mathbf{x}]$ with $n\geq 2$.

1. (1)

Let $p$ be a prime number such that $p\neq 3,5$ and $p\leq n$. Then there is a group $G\subseteq\operatorname{Aut}_{gr}(A)$ of order $p$ such that $A^{G}$ is a non-conventional graded isolated singularity. 2. (2)

Let $p=3^{a}5^{b}$ for some $a,b\geq 0$. If $G$ is a subgroup of $\operatorname{Aut}_{gr}(A)$ of order $p$ such that $A^{G}$ is a graded isolated singularity, then it is conventional.

Proof.

We omit the proof of part (2). For part (1), we show give a proof when $p=2$.

We construct the group $G=\langle\sigma\rangle$ as follows. If $n$ is even, let $\sigma\in\operatorname{Aut}_{gr}(A)$ be defined by

[TABLE]

for all $i\in{\mathbb{Z}}_{n}$. If $n$ is odd, let $\sigma\in\operatorname{Aut}_{gr}(A)$ be defined by

[TABLE]

for all $i\in{\mathbb{Z}}_{n}$ not equal to $\frac{n-1}{2}$. By [HZ, Example 1.6(ii)] and [HZ, Lemma 3.4], $x_{i}^{2}\in{\mathfrak{r}}(A,G)$ for all $i$. (Some details are omitted.) Therefore $A/{\mathfrak{r}}(A,G)$ is finite dimensional and $A^{G}$ is a graded isolated singularity. Since $G$ preserves $x_{0}+x_{n-1}$, it is non-conventional. ∎

The next lemma is due to Jason Bell. We thank him for sharing his result with us. We say an algebra $B$ is PI if it satisfies a polynomial identity.

Lemma 10.6** (Jason Bell).**

Let $B$ be a noetherian connected graded PI algebra generated in degree 1. If every linear combination of homogenous elements of odd degrees is nilpotent, then $B$ is finite dimensional.

Proof.

Suppose on the contrary that $B$ is infinite dimensional. Let $W$ be the set of graded ideals $I$ of $B$ such that $B/I$ is infinite dimensional. Since $B$ is noetherian, there is a maximal element $J$ in $W$. Replacing $B$ by $B/J$, we may assume that every nonzero ideal of $B$ has finite codimension. Since $B$ is graded, every minimal prime of $B$ is graded. As a consequence, the nilradical $N$ of $B$ is graded. Since $B$ is noetherian, $B$ is infinite dimensional if and only if $B/N$ is infinite dimensional. This implies that $N=0$. As a consequence, a product of minimal prime ideals is zero. This in turn implies that one of minimal prime is zero, or $B$ is prime.

Since $B$ is PI, there is a nonzero central element in $B$. We can further assume that this element, say $z$, is homogeneous and a nonzerodivisor (or regular element). By the last paragraph, $B/(z)$ is finite dimensional. Then $\operatorname{GKdim}B=1$ by (E1.0.3).

By Small-Warfield’s theorem [SW], the center $Z(B)$ of $B$ is a finitely generated graded algebra of GKdimension one and $B$ is a finite module over $Z(B)$. Note that every nonzero element in $Z(B)$ is regular. Hence $Z(B)$ is contained in the second Veronese subring of $B$ since all odd degree elements are nilpotent.

Let $Q:=Q_{gr}(B)$ be the graded quotient ring of $B$. By a graded version of Posner’s theorem, this is just the result of inverting the homogeneous nonzero central elements, all of which have even degree. The important point here is that every element of odd degree in $Q$ can be written in the form $az^{-1}$ with $a,z$ homogeneous and $a\in B$ of odd degree and $z\in Z(B)$ of even degree. Let $T$ be the (ungraded) total quotient ring of $B$ (or of $Q$). Then $T$ can be embedded into a matrix algebra over a field $F$. With this embedding, we fix a trace map $tr$ (the usual matrix trace). (With a bit more care one can even show that $T\cong M_{n}(F)$ where $F$ is the fraction field of $Z(B)$.) In particular, $tr(1)\neq 0$.

As a general fact, since $B$ is generated in degree 1, $Q$ is strongly ${\mathbb{Z}}$-graded in the sense of [NvO, A.I.3]. Let $Q_{odd}:=\bigoplus_{i\;{\text{is odd}}}Q_{i}$ and $Q_{even}:=\bigoplus_{i\;{\text{is even}}}Q_{i}$. Then $Q=Q_{odd}\oplus Q_{even}$ is a strongly ${\mathbb{Z}}_{2}$-graded algebra, namely, $Q_{odd}^{2}=Q_{even}$. By the last paragraph, every element $u$ in $Q_{odd}$ is of the form $az^{-1}$ where $a\in B$ is a linear combination of homogeneous elements of odd degrees and where $z\in Z(B)$ is of even degree. Therefore $u$ is nilpotent by hypothesis. Let $u,v$ be any two elements in $Q_{odd}$. Then $u,v,u+v$ are all in $Q_{odd}$; and consequently, all nilpotent. By [MOR, Lemma 1], $tr(uv)=0$. Since $Q_{odd}^{2}=Q_{even}$, $tr(Q_{even})=0$. This contradicts $tr(1)\neq 0$. ∎

Now we consider twisted tensor products. Let $\{B(i)\}_{i=1}^{w}$ be a family of connected graded algebras. Then the tensor product

[TABLE]

is a connected graded and ${\mathbb{Z}}^{\oplus n}$-graded algebra. Let $u_{i}$ denote the $i$th unit element $(0,\ldots,0,1,0,\ldots,0)\in{\mathbb{Z}}^{\oplus n}$ where $1$ is in the $i$th position. Let $\{p_{ij}\in\Bbbk^{\times}\mid 1\leq i<j\leq n\}$ be a set of nonzero scalar. Define $f_{u_{i}}$ to be the ${\mathbb{Z}}^{\oplus n}$-graded algebra automorphism of $\bigotimes^{n}B(i)$ determined by

[TABLE]

for all $i\geq j$ and $x_{j}\in B(j)$ and

[TABLE]

for all $i<j$ and homogeneous elements $x_{j}\in B(j)$. Then

[TABLE]

is an twisting system of $\bigotimes^{n}B(i)$ in the sense of [Zh1, Definition 2.1]. By [Zh1, Proposition and Definition 2.3], one can define a twisted algebra of $\bigotimes^{n}B(i)$ associated to the twisting system $F$. This twisted algebra is denoted by $\bigotimes^{n}_{\{p_{ij}\}}B(i)$. If $B(i)=\Bbbk[x]$ for all $i$, then $\bigotimes^{n}_{\{p_{ij}\}}B(i)$ is canonically isomorphic to skew polynomial ring $\Bbbk_{p_{ij}}[x_{1},\ldots,x_{n}]$, see [Zh1, p.310]. Note that if $a=1^{\otimes(i-1)}\otimes x_{i}\otimes 1^{\otimes(n-i)}$ and $b=1^{\otimes(j-1)}\otimes x_{j}\otimes 1^{\otimes(n-j)}$ for two homogeneous elements $x_{i}\in B(i)$ and $x_{j}\in B(j)$ for $i<j$. Then one can check that

[TABLE]

Suppose each $B(i)$ is a noetherian PI Artin-Schelter regular algebra (and it is possible that the “PI” hypothesis can be weakened). One can easily check that $\bigotimes^{n}_{\{p_{ij}\}}B(i)$ is noetherian and Artin-Schelter regular. Further, $\bigotimes^{n}_{\{p_{ij}\}}B(i)$ has enough normal elements in the sense of [Zh2, p.392]. By [Zh2, Theorem 1], it is Auslander regular and Cohen-Macaulay.

Suppose $G$ is a finite group and $\phi_{i}:G\to\operatorname{Aut}_{gr}(B(i))$ is an injective map for each $i$. Then there is a unique extension of the $G$-action on $\bigotimes^{n}_{\{p_{ij}\}}B(i)$.

Proposition 10.7**.**

Retain the above notation. Suppose $G$ is a finite group and $\phi_{i}:G\to\operatorname{Aut}_{gr}(B(i))$ is an injective map for each $i$. Let $B=\bigotimes^{n}_{\{p_{ij}\}}B(i)$.

1. (1)

$B^{G}$* is a graded isolated singularity if and only if each $B(i)^{G}$ is a graded isolated singularity.* 2. (2)

Assume $B^{G}$ is a graded isolated singularity. Then $B^{G}$ is conventional if and only if each $B(i)^{G}$ is conventional.

Proof.

The proof follows from Lemma 10.4(5,6). Details are omitted. ∎

Proposition 10.7 provides a lot examples of graded isolated singularities.

Next let $B(i)=B$, for $i=1,\ldots,n$, be a noetherian PI Artin-Schelter regular algebra generated in degree 1. Let $p_{ij}=-1$ for all $i<j$. We consider $(-1)$-twisted tensor product $\bigotimes^{n}_{\{-1\}}B$ and the permutation automorphism $\sigma\in\operatorname{Aut}_{gr}(\bigotimes^{n}_{\{-1\}}B)$ determined by

[TABLE]

for all $x_{j},x_{n}\in B$.

Proposition 10.8**.**

Retain the above notation. Assume that $n\geq 2$ is admissible in the sense of Definition 5.2(2). Let $B$ be any noetherian PI Artin-Schelter regular algebra generated in degree 1. Let $G$ be the group $\langle\sigma\rangle$ where $\sigma$ is defined in (E10.7.1). Then $(\bigotimes^{n}_{\{-1\}}B)^{G}$ is a non-conventional graded isolated singularity.

Proof.

Let $S=\bigotimes^{n}_{\{-1\}}B$. It suffices to show that $S/{\mathfrak{r}}(S,G)$ is finite dimensional.

Let $x\in B$ be a linear combination of homogeneous elements of odd degrees. Let $x_{i}=1^{\otimes i}\otimes x\otimes 1^{\otimes(n-i-1)}\in S$, for $i=0,\ldots,n-1$. Then the subalgebra generated by $\{x_{0},x_{1},\ldots,x_{n}\}$ is the $(-1)$-skew polynomial ring $R:=\Bbbk_{-1}[{\mathbf{x}}]$. So the inclusion $f:R\to S$ is compatible with the $G$-action. (Note that $f$ is not a graded algebra homomorphism.) Since $n$ is admissible, the quotient $R/{\mathfrak{r}}(R,G)$ is finite dimensional. Hence, for each $x_{i}$, we have $x_{i}^{N}\in{\mathfrak{r}}(R,G)$ for some $N\geq 0$. By Lemma 10.4(5), $x_{i}^{N}\in{\mathfrak{r}}(S,G)$. This is true for all $x$ that is a linear combination of homogeneous elements of odd degrees in $B$. By Lemma 10.6, the image of the map

[TABLE]

is finite dimensional. Say this image is $\overline{B}$. By symmetry, $S/{\mathfrak{r}}(S,G)$ is a quotient ring of $\bigotimes^{n}_{\{-1\}}\overline{B}$, which is finite dimensional. Therefore $S/{\mathfrak{r}}(S,G)$ is finite dimensional as desired. ∎

Proposition 10.8 also provides a lot examples of graded isolated singularities by varying $B$.

11. Some questions and comments

It is quite reasonable to adapt Ueyama’s definition of a graded isolated singularity [Ue, Definition 2.2], at least in the connected graded case. By Remark 0.3(2), the straightforward generalization of the freeness criterion for commutative quotient isolated singularities [MSt, Lemma 2.1] fails badly in the noncommutative case. However the freeness of the $G$-action on $V\setminus\{0\}$ is one of the easiest and most effective criterions for isolated singularities. Therefore we ask

Question 11.1**.**

What is the analogue of the freeness criterion of isolated singularities in the (connected graded) noncommutative setting?

Let $R$ be a noetherian Artin-Schelter regular algebra and let $G$ be a finite subgroup of $\operatorname{Aut}_{gr}(R)$. By a result of Mori-Ueyama [MU1, Theorem 3.10] together with [HZ], the following are equivalent:

1. (1)

$R^{G}$ is a graded isolated singularity, 2. (2)

$R/{\mathfrak{r}}(R,G)$ is finite dimensional, 3. (3)

$R\#G/(e_{0})$, where $e_{0}=1\#(\sum_{g\in G}g)$, is finite dimensional, 4. (4)

$\operatorname{{\sf p}}(R,G)=\operatorname{GKdim}R$.

Mori-Ueyama’s criterion of graded isolated singularities is quite convenient. On the other hand, it could be very difficult to verify (2), or (3), or to calculate the exact value of $\operatorname{{\sf p}}(R,G)$.

One of the key steps in the proof of Theorem 0.2 is to show that the set $Spl(n)$ is non-empty. But we can not prove that $Spl(n)\neq\emptyset$ is necessary. In particular, we do not have answers to the following questions.

Question 11.2**.**

Let $n=p_{1}p_{2}$ for two distinct odd primes $p_{1},p_{2}$.

1. (1)

If $7\leq\operatorname{mop}(n)\leq 17$, is then $n$ admissible? 2. (2)

Is $Spl(77)\neq\emptyset$? 3. (3)

If $Spl(77)=\emptyset$, is $77$ admissible?

Hypersurface isolated singularities have been studied extensively, and form a rich topic in algebraic geometry [Mi]. The noncommutative version of a hypersurface was defined in [KKZ2, Definition 1.3(c)].

In the commutative theory, every hypersurface isolated singularity produces a finite dimensional Milnor algebra (as well as the Tjurina algebra). It would be interesting to develop a similar theory for the noncommutative hypersurface isolated singularities. At this point, it is not clear to us what is the best way of defining the noncommutative Jacobian ideal, since there are no canonically defined partial derivatives in the noncommutative case. Here we will like to propose a definition of the Milnor algebra when the hypersurface singularity is defined by “double twisted superpotentials”.

Let $V$ be a finite dimensional vector space $\bigoplus_{s=1}^{v}\Bbbk x_{i}$, or $\{x_{s}\}_{s=1}^{v}$ be a basis of $V$. Let $F$ be the free algebra $\Bbbk\langle x_{1},\ldots,x_{v}\rangle=\Bbbk\langle V\rangle$. Let $\sigma$ denote an element in ${\text{GL}}(V)$. We define two $\Bbbk$-linear maps from $F$ to $F$. The first one is $\phi$, which is determined by

[TABLE]

for all $x_{i_{s}}$ in the basis of $V$. The second one $\sigma\otimes 1$, where $\sigma\in{\text{GL}}(V)$, is determined by

[TABLE]

Following [DV, Definition 1], [BSW, p.1502], [Ka, Definitions 2.1.3 and 2.1.4], [MSm, Definition 2.5] (and taking the quiver with one vertex and $v$ arrows), a twisted superpotential in the free algebra $F$ is an element $w$ in $F$ such that

[TABLE]

for some $\sigma\in{\text{GL}}(V)$. (All papers [DV, BSW, Ka, MSm] use slightly different notation, but one can easily figure out the discrepancies). For every $x_{i}$, we define a partial derivation $\partial_{i}$ as follows

[TABLE]

(This definition of a partial derivative is slightly different from the ordinary partial derivative in calculus. Another possibility is the cyclic, or circular, derivative.) For every $w$, let $\partial(w)$ be the $\Bbbk$-linear span of $\{\partial_{i}(w)\}_{i=1}^{v}$. For an integer $N$, one can define $\partial^{N}(w)$ inductively by $\partial^{N}(w)=\partial(\partial^{N-1}(w))$. Given a twisted superpotential $w$ and an integer $N$, one can define superpotential algebra ${\mathcal{D}}(w,N)$ [Ka, Definition 2.1.6] (which is the same as the derivation-quotient algebra in the sense of [DV, BSW, MSm]) to be

[TABLE]

Dubois-Violette proved a very nice result [DV, Theorem 11]: a Koszul (or higher Koszul) algebra is twisted Calabi-Yau if and only if it is isomorphic to a superpotential algebra for a unique-up-to-scalar-multiples twisted superpotential $w$.

Definition 11.3**.**

Retain the above notation.

1. (1)

A pair of elements $(w_{1},w_{2})$ in $F$ are called double twisted superpotentials if

1. (a)

$w_{1}$ is a twisted superpotential (with an automorphism $\sigma_{1}\in{\text{GL}}(V)$) such that the superpotential algebra $D:={\mathcal{D}}(w_{1},N)$ is a noetherian Artin-Schelter regular algebra. 2. (b)

$w_{2}$ is a twisted superpotential (with an automorphism $\sigma_{2}\in{\text{GL}}(V)$) such that $w_{2}$ is a normal regular element in $D$.

Let $(w_{1},w_{2})$ be double twisted superpotentials in parts (2,3,4). 2. (2)

The algebra $D/(w_{2})$ is called the hypersurface singularity associated to $(w_{1},w_{2})$, and is denoted by $T(w_{1},w_{2})$. 3. (3)

The Milnor algebra associated to $(w_{1},w_{2})$ is defined to be

[TABLE] 4. (4)

The Milnor number associated to $(w_{1},w_{2})$ is defined to be

[TABLE]

With these definitions, we can ask the following:

Question 11.4**.**

Is $T(w_{1},w_{2})$ being a graded isolated singularity equivalent to $m(w_{1},w_{2})$ being finite?

The following example of a hypersurface isolated singularity is non-conventional such that Question 11.4 has an affirmative answer.

Example 11.5**.**

Let $A=\Bbbk_{-1}[x_{0},x_{1}]$ and $G$ be the group of automorphism of $A$ generated by $f$, where $f$ is determined by

[TABLE]

By [KKZ1, Example 3.1], $A^{G}$ is a hypersurface singularity, which can be written as

[TABLE]

where $D$ is an Artin-Schelter regular algebra of global dimension three and $w_{2}$ is a normal element of degree 6 in $D$. In details, $x=x_{0}+x_{1}$ and $y=x_{0}^{3}+x_{1}^{3}$,

[TABLE]

and

[TABLE]

By Theorem 0.2, $A^{G}$ has a non-conventional graded isolated singularity.

Note that $D$ is $(-1)$-twisted Calabi-Yau [RRZ, Example 1.6]. There is a twisted superpotential

[TABLE]

with automorphism $\sigma$ determined by

[TABLE]

and $D$ is the superpotential algebra associated to $w_{1}$. It is easy to check that

1. (1)

$w_{2}$ is a regular normal element in $D$, 2. (2)

$w_{2}$ is a superpotential.

The Milnor ring of the hypersurface singularity $A^{G}$ is

[TABLE]

which is isomorphic to $\Bbbk[x]/(x^{5})$ by an easy calculation. As a consequence, the Milnor number of $A^{G}$ is 5.

Note that the McKay quiver corresponding to $(A,G)$ is of type $\widetilde{L}_{1}$, see [CKWZ1, Proposition 7.1 and pp. 249-250]. This is slightly different from the classical $\widetilde{A}$, $\widetilde{D}$, $\widetilde{E}$ types.

Remark 11.6**.**

Some other noncommutative hypersurface graded isolated singularities are given in [CKWZ2, Theorem 5.2] and [CKWZ2, Table 3 in p.537]. These are related to noncommutative McKay correspondence in dimension two. It would be interesting to answer Question 11.4 for these hypersurface singularities.