New Artin-Schelter regular and Calabi-Yau algebras via normal extensions

TL;DR

This paper presents a novel method for constructing 4-dimensional Artin-Schelter regular and Calabi-Yau algebras from 3-dimensional ones using normal extensions, expanding the class of known regular algebras.

Contribution

Introduces a new technique to generate 4D Artin-Schelter regular and Calabi-Yau algebras via normal extensions of 3D algebras, including explicit relations and automorphism analysis.

Findings

Constructs large classes of new 4D Artin-Schelter regular algebras.

Produces flat families of 4-Calabi-Yau extensions from 3-Calabi-Yau algebras.

Shows the homological determinant of the Nakayama automorphism is 1 for these algebras.

Abstract

We introduce a new method to construct 4-dimensional Artin-Schelter regular algebras as normal extensions of (not necessarily noetherian) 3-dimensional ones. The method produces large classes of new 4-dimensional Artin-Schelter regular algebras. When applied to a 3-Calabi-Yau algebra our method produces a flat family of central extensions of it that are 4-Calabi-Yau, and all 4-Calabi-Yau central extensions having the same generating set as the original 3-Calabi-Yau algebra arise in this way. Each normal extension has the same generators as the original 3-dimensional algebra, and its relations consist of all but one of the relations for the original algebra and an equal number of new relations determined by "the missing one" and a tuple of scalars satisfying some numerical conditions. We determine the Nakayama automorphisms of the 4-dimensional algebras obtained by our method and as a consequence show that their homological determinant is 1. This supports the conjecture by Mori-Smith that the homological determinant of the Nakayama automorphism is 1 for all Artin-Schelter regular connected graded algebras. Reyes-Rogalski-Zhang proved this is true in the noetherian case.

Tables

Table 1. Table 1. Good ( p 1 , p 2 ) subscript 𝑝 1 subscript 𝑝 2 (p_{1},p_{2}) for the generic cubic AS-regular algebras in [ 1 , Table (3.9)]

Type	$(q_1,q_2)$	$k=1$	$k=2$
$A$	$(1,1)$	$(1,\pm 1)$	$(\pm 1,1)$
$E$	$(1,{\zeta }_3)$	$(1,{\zeta }_3^r),r=0,1,2$	$({\zeta }_3,{\zeta }_3)$
$H$	$({\zeta }_8,-{\zeta }_8)$	none	none
$S_1$	$(\alpha ,{\alpha }^{-1})$	$(\alpha ,{\alpha }^{-1/2})$	$({\alpha }^{1/2},{\alpha }^{-1})$
$S_2$	$(\alpha ,-{\alpha }^{-1})$	$(\alpha ,{\alpha }^{-1/2})$	$({\left(-\alpha \right)}^{1/2},-{\alpha }^{-1})$
$S_2^{\prime }$	$(1,-1)$	$(1,\pm 1)$	none

Table 2. Table 2. Good ( p 1 , p 2 , p 3 ) subscript 𝑝 1 subscript 𝑝 2 subscript 𝑝 3 (p_{1},p_{2},p_{3}) for the generic quadratic AS-regular algebras in [ 1 , Table (3.11)]

Type	$(q_1,q_2,q_3)$	$k=1$	$k=2$	$k=3$
$A$	$(1,1,1)$	$(1,1,1)$, $(1,{\zeta }_3,{\zeta }_3^2)$	$(1,1,1)$, $({\zeta }_3^2,1,{\zeta }_3)$	$(1,1,1)$, $({\zeta }_3,{\zeta }_3^2,1)$
$B$	$(1,1,-1)$	$(1,1,\pm 1)$	$(1,1,\pm 1)$	$(-1,-1,-1)$
$E$	$({\zeta }_9,{\zeta }_9^4,{\zeta }_9^7)$	none	none	none
$H$	$(1,-1,{\zeta }_4)$	$(1,1,\pm 1)$, $(1,-1,\pm {\zeta }_4)$	$(-1,-1,\pm 1)$	$(-{\zeta }_4,-{\zeta }_4,{\zeta }_4)$
$S_1$	$(\alpha ,\beta ,{\left(\alpha \beta \right)}^{-1})$	$(\alpha ,\lambda ,{\lambda }^{-1})$	$(\lambda ,\beta ,{\lambda }^{-1})$	$(\lambda ,{\lambda }^{-1},{\left(\alpha \beta \right)}^{-1})$
$S_1^{\prime }$	$(\alpha ,{\alpha }^{-1},1)$	$(\alpha ,{\alpha }^{-1/3},{\alpha }^{1/3})$	$({\alpha }^{1/3},{\alpha }^{-1},{\alpha }^{-1/3})$	$(\lambda ,{\lambda }^{-1},1)$
$S_2$	$(\alpha ,-\alpha ,{\alpha }^{-2})$	$(\alpha ,\pm \alpha ,{\alpha }^{-1})$	$(\pm \alpha ,-\alpha ,-{\alpha }^{-1})$	$(1,\pm 1,{\alpha }^{-2})$, $(-1,\pm 1,{\alpha }^{-2})$

Equations

$${\partial }_i\mathsf{w}=x_i({\partial }_k\mathsf{w})-p_i({\partial }_k\mathsf{w})x_i=0(i=1,\dots ,n,i\ne k),$$

$${\partial }_i\mathsf{w}=[x_i,{\partial }_k\mathsf{w}]=0(i=1,\dots ,n,i\ne k)$$

$${\mathrm{Ext}}_A^i(\mathbb{k},A)\cong \{\begin{array}{ll}\mathbb{k}(\ell )& \text{if i=d}\\ 0& \text{if i=d}\end{array}$$

$${\mathrm{Ext}}_{A^e}^i(A,A^e)\cong \{\begin{array}{ll}{}_{\nu }{A}_1(\ell )& \text{if i=d}\\ 0& \text{if i=d}\end{array}$$

$$\Omega x=\tau (x)\Omega$$

$$x\cdot m:=\sigma (x)m$$

$$x\cdot m\cdot y:=\alpha (x)m\beta (y).$$

$$0$$

$${\mathrm{Ext}}_D^{p+1}(\mathbb{k},D)\cong {\mathrm{Ext}}_A^p(\mathbb{k},A)$$

$${\mathrm{Ext}}_D^{p+1}(N,M)\cong {\mathrm{Ext}}_A^p(N,{}_{\tau }(M/\Omega M))$$

$$\mathsf{Mod}(D)$$

$${\mathrm{RHom}}_D(N,M)\cong {\mathrm{RHom}}_A(N,{\mathrm{RHom}}_D(A,M))$$

$${\mathrm{RHom}}_D(A,M)\cong {}_{\tau }(M/\Omega M)[-1]$$

$${\mathrm{RHom}}_D(N,M)\cong {\mathrm{RHom}}_A(N,{}_{\tau }(M/\Omega M))[-1],$$

$$\dim (D_k)-\dim (D_{k-m})\le \dim (A_k)$$

$${\mathrm{Ext}}_{D^e}^{p+2}(A,D^e)\cong {}_{\tau }{\mathrm{Ext}}_{A^e}^p(A,A^e){}_{{\tau }^{-1}}$$

$$c\cdot (a\otimes b)\cdot d=\tau (\overline{c})a\otimes b{\tau }^{-1}(\overline{d})$$

$${\mathrm{Ext}}_{D^e}^{p+2}(A,D^e)\cong {\mathrm{Ext}}_{A^e}^p(A,{}_{\tau }{A}^e{}_{{\tau }^{-1}})$$

$${}_{\tau }{A}_1\cong {}_1{A}_{{\tau }^{-1}},{}_1{A}_{{\tau }^{-1}}\cong {}_{\tau }{A}_1$$

$${\mathrm{Ext}}_{D^e}^{d+1}(D,D^e)\cong {}_{\nu }{D}_1\cong {}_1{D}_{{\nu }^{-1}};$$

$${\mathrm{Ext}}_{D^e}^{d+2}(A,D^e)\cong {}_{\tau }{A}_{{\nu }_{{}_D}^{-1}}$$

$${\mathrm{Ext}}_{D^e}^{d+1}(D,D^e)$$

$${\mathrm{Ext}}_{D^e}^{d+1}(FM,N)$$

$${\mathrm{Ext}}_{D^e}^{d+1}(D,D^e)$$

$${}_1{A}_{{\nu }_{{}_A}^{-1}}\cong {\mathrm{Ext}}_{A^e}^d(A,A^e)\cong {}_{{\tau }^{-1}}{\mathrm{Ext}}_{D^e}^{d+2}(A,D^e{)}_{\tau }\cong {}_1{A}_{{\nu }_{{}_D}^{-1}\tau };$$

$$H_{\mathfrak{m}}^d({}_{\sigma }A)=\underrightarrow{\lim }{\mathrm{Ext}}_A^d(A/A_{\ge k},{}_{\sigma }A)\stackrel{\sim }{\to }\underrightarrow{\lim }{\mathrm{Ext}}_A^d({}_{{\sigma }^{-1}}(A/A_{\ge k}),A)\stackrel{\sim }{\to }\underrightarrow{\lim }{\mathrm{Ext}}_A^d((A/A_{\ge k}{)}_{\sigma },A)={}_{\sigma }{H}_{\mathfrak{m}}^d(A).$$

Full text

1. 1 Introduction
2. 2 Results on normal extensions
3. 3 Normal extensions of $3$-dimensional regular algebras
4. 4 Proof of the main theorem
5. 5 Examples

1. Introduction

1.1. Notation

Throughout this paper we fix a field $\Bbbk$. All vector spaces will be $\Bbbk$-vector spaces, all categories will be $\Bbbk$-linear, and all functors will be $\Bbbk$-linear. All algebras will be finitely presented ${\mathbb{N}}$-graded $\Bbbk$-algebras. Such an algebra, $A=A_{0}\oplus A_{1}\oplus\cdots$, is connected if $A_{0}=\Bbbk$. All graded algebras in this paper are connected and generated by their degree-one components. Thus, we are concerned with algebras $TV/(R)$ where $TV$ denotes the tensor algebra on a finite dimensional vector space $V$ placed in degree $1$ and $(R)$ denotes the ideal generated by a finite dimensional graded subspace $R$ of $TV$ whose elements have degrees $\geq 2$.

The algebras we study are not required to be noetherian or to have finite Gelfand-Kirillov dimension.

1.2.

The assumption that a non-commutative algebra has finite global dimension does not imply that it satisfies good analogues of the Cohen-Macaulay and Gorenstein properties. For example, all free algebras on a finite positive number of generators have global dimension 1; for all $d\geq 0$, there are finite dimensional algebras having global dimension $d$; for all $d\geq 1$, there are algebras of global dimension $d$ whose only ideals are $\{0\}$ and the rings themselves; and so on. However, for graded algebras there are two ways to strengthen the finite global dimension hypothesis that have proven sufficiently effective that they lead to a good non-commutative analogue of projective algebraic geometry.

The algebras satisfying these stronger definitions are called Artin-Schelter regular and twisted Calabi-Yau algebras, respectively. The definitions vary a little from one paper to another but in this paper they are defined in such a way that they are, in fact, the same (see 1.8). A particularly important subset of these algebras are the Calabi-Yau algebras: their place among all twisted Calabi-Yau algebras is analogous to that occupied by the symmetric algebras among all finite dimensional Frobenius algebras.

Many Artin-Schelter regular and Calabi-Yau algebras are known. The 3-dimensional Artin-Schelter regular algebras (meaning those of global dimension 3) that have finite Gelfand-Kirillov dimension were classified by Artin-Schelter [1] and Artin-Tate-Van den Bergh [2]. Using those classifications, the 3-dimensional Calabi-Yau algebras of finite Gelfand-Kirillov dimension were classified by Mori-Smith [25] and Mori-Ueyama [26]. A bewildering variety of 4-dimensional Artin-Schelter regular and Calabi-Yau algebras are known but no classification is in sight. The construction of new examples is an important on-going research goal.

1.3. The main result

This paper introduces a new method for producing 4-dimensional Artin-Schelter regular and Calabi-Yau algebras from 3-dimensional ones.

Before stating an approximate version of our main result we need the following terminology. An element $\Omega$ in a ring $D$ is normal if $\Omega D=D\Omega$, and is regular if it is not a zero-divisor. Following the terminology in [18], we call $D$ a normal extension by $\Omega$ of a ring $A$ if there is a regular normal element $\Omega$ in $D$ such that $A\cong D/(\Omega)$. If, in addition, $\Omega$ belongs to the center of $D$ we call $D$ a central extension of $A$.

Now let $A=TV/(R)$ be a $3$-dimensional Artin-Schelter regular algebra. Since $A$ is generated by its degree-one component we can, without loss of generality, assume that $R\subseteq V^{\otimes m}$ for some $m\geq 2$ (see 1.10). We do that. Moreover, Dubois-Violette proved (see Theorem 3.1) that there is a twisted superpotential ${\sf w}\in V^{\otimes m+1}$ such that $A$ is the algebra $A({\sf w}):=TV/({\operatorname{\partial}}_{1}{\sf w},\ldots,{\operatorname{\partial}}_{n}{\sf w})$ where the ${\operatorname{\partial}}_{i}{\sf w}$ are the partial derivatives of ${\sf w}$ with respect to a basis $x_{1},\ldots,x_{n}$ for $V$. Such ${\sf w}$ is uniquely determined up to scalar multiplication and the definition of $A({\sf w})$ does not depend on the choice of basis for $V$. See §§3.1 and 3.2 for details on twisted superpotentials.

With this notation, our main result is as follows.

Theorem 1.1** (Approximate version of Theorem 3.5).**

Let $A=TV/(R)$ be a $3$-dimensional Artin-Schelter regular algebra and fix a basis $x_{1},\ldots,x_{n}$ for $V$. Let ${\sf w}\in V^{\otimes(m+1)}$ be a twisted superpotential such that $A=TV/({\operatorname{\partial}}_{1}{\sf w},\ldots,{\operatorname{\partial}}_{n}{\sf w})$. Fix an index $1\leq k\leq n$, let ${\sf p}=(p_{1},\ldots,p_{n})$ be a “good” tuple of non-zero scalars, let $D({\sf w},{\sf p})$ be the free algebra $TV$ modulo the $2(n-1)$ relations

[TABLE]

and let $\Omega$ denote the image of ${\operatorname{\partial}}_{k}{\sf w}$ in $D({\sf w},{\sf p})$.

1. (1)

$D({\sf w},{\sf p})$* is a 4-dimensional Artin-Schelter regular algebra and is a normal extension of $A=D({\sf w},{\sf p})/(\Omega)$.* 2. (2)

$D({\sf w},{\sf p})$* is noetherian if and only if $A$ is.* 3. (3)

$D({\sf w},{\sf p})$* has finite Gelfand-Kirillov dimension if and only if $A$ does.* 4. (4)

$D({\sf w},{\sf p})$* is a domain if $A$ is.*

In Theorem 1.1, the algebra $A$ has $n$ generators and $n$ defining relations, all of degree $m$, and the algebra $D({\sf w},{\sf p})$ has $n$ generators and $2n-2$ relations, $n-1$ of degree $m$ and $n-1$ of degree $m+1$.

It is well-known that items (2)-(4) follow from (1) so the real content of Theorem 1.1 is (1).

The condition that the tuple ${\sf p}$ is “good” can be stated in several ways, perhaps the simplest of which is Theorem 3.54 (see 3.4).

It is not immediately obvious that there are any “good” ${\sf p}$. However, as we show in §5, there are good ${\sf p}$’s for almost all the generic 3-dimensional Artin-Schelter regular algebras found by Artin and Schelter. Furthermore, if $A$ is a 3-Calabi-Yau algebra, then ${\sf p}=(1,\ldots,1)$ is good and in that case $D({\sf w},{\sf p})$ is 4-dimensional Calabi-Yau algebra and is a central extension of $A$.

Corollary 1.2** (Corollary 3.6).**

Let $TV/(R)=A({\sf w})$ be a 3-Calabi-Yau algebra and fix an index $1\leq k\leq n$. Then $TV$ modulo the relations

[TABLE]

is a $4$-Calabi-Yau algebra and is a central extension of $A({\sf w})$ by the image of ${\operatorname{\partial}}_{k}{\sf w}$.

Although the terminology “central extension” and “normal extension” follows that in [18] there is no overlap between this paper and that one. In this paper the natural map $D\to A$ from one of the normal or central extensions in Theorem 1.1 is an isomorphism in degree 1, i.e., $A$ and $D$ have the “same” generating sets, whereas the corresponding map $D\to A$ in [18] has a 1-dimensional kernel in degree 1.

1.4. A flat family of 4-Calabi-Yau algebras

Although it is not immediately apparent, if $A({\sf w})$ is fixed, then Corollary 1.2 can be used to produce a flat family of 4-Calabi-Yau algebras over ${\mathbb{P}}^{n-1}$ that map onto $A({\sf w})$. To do this we must state Corollary 1.2 in a basis-free way.

Let $V$ be an $n$-dimensional $\Bbbk$-vector space and let $m\in{\mathbb{Z}}_{\geq 1}$. For each $x\in V$, let $x^{\perp}:=\{\psi\in V^{*}\;|\;\psi(x)=0\}$. Define $\langle-,-\rangle:V^{*}\times V^{\otimes(m+1)}\to V^{\otimes m}$ by $\langle\psi,{\sf w}\rangle:=(\psi\otimes\mathrm{id}^{\otimes m})({\sf w})$.

Theorem 1.3**.**

Let $A({\sf w})$ be a 3-Calabi-Yau algebra where ${\sf w}\in V^{\otimes(m+1)}$ is a superpotential. For each $x\in V$, let $I_{x}$ be the ideal in $TV$ generated by $\{\langle\psi,{\sf w}\rangle\;|\;\psi\in x^{\perp}\}$ and $\{[y,\langle\psi,{\sf w}\rangle]\;|\;y\in V,\,\psi\in V^{*}\}$. Define $D_{x}=TV/I_{x}$. By definition $D_{0}=A({\sf w})$.

1. (1)

$\{D_{x}\;|\;x\in{\mathbb{P}}^{n-1}={\mathbb{P}}(V)\}$* is a flat family of $4$-Calabi-Yau algebras having degree-$m$ central regular elements $\Omega_{x}$ such that $A({\sf w})=D_{x}/(\Omega_{x})$ for all $x\in V$.* 2. (2)

All degree-$m$* central extensions of $A({\sf w})$ are obtained in this way and they are $4$-Calabi-Yau (Theorem 3.8).*

In §5, we give some explicit examples to show that not every normal extension of a 3-Calabi-Yau algebra need be a central extension.

1.5.

Given part 1 of Theorem 1.3 above, it is natural to consider the algebras $D_{x}$ as homogeneous coordinate rings of a well-behaved family of non-commutative schemes. In order to make this precise, we need some preparation.

Let $\mathcal{D}$ be the sheaf of $\mathcal{O}_{{\mathbb{P}}(V)}$-algebras on ${\mathbb{P}}(V)$ defined in 3.6 below. The fiber of $\mathcal{D}$ over a $\Bbbk$-point $x\in{\mathbb{P}}(V)$ is the algebra $D_{x}$ defined in Theorem 1.3. For each open subscheme $U\subseteq{\mathbb{P}}(V)$ we denote by $\mathcal{D}_{U}$ the restriction of $\mathcal{D}$ to $U$.

Definition 1.4**.**

If $D$ is a connected graded $\Bbbk$-algebra we write ${\sf Gr}(D)$ for the category of ${\mathbb{Z}}$-graded left $D$-modules and $\operatorname{\sf QGr}(D)$ for the quotient category ${\sf Gr}(D)/\operatorname{\sf Tors}(D)$ where $\operatorname{\sf Tors}(D)$ is the full subcategory of graded $D$-modules that are the sum of their finite dimensional submodules.

Similarly, ${\sf Gr}(\mathcal{D}_{U})$ is the category of graded quasi-coherent sheaves of $\mathcal{D}_{U}$-modules and $\operatorname{\sf QGr}(\mathcal{D}_{U})$ is the quotient by the subcategory $\operatorname{\sf Tors}(\mathcal{D}_{U})$ consisting of modules whose fibers are in $\operatorname{\sf Tors}(D_{x})$ for all $x\in{\mathbb{P}}(V)$. $\blacklozenge$

In [19, Defn. 3.2], Lowen and Van den Bergh introduced the notion of a flat abelian $R$-linear category over a commutative ring $R$.

Theorem 1.5**.**

For every open affine $\operatorname{Spec}(R)=U\subseteq{\mathbb{P}}(V)$, the $R$-linear abelian category $\operatorname{\sf QGr}(\mathcal{D}_{U})$ is flat over $R$ in the sense of [19]. We can express this by saying that $\operatorname{\sf QGr}(\mathcal{D})$ is flat over ${\mathbb{P}}(V)$.

The intuition here is that the non-commutative scheme represented by the category $\operatorname{\sf QGr}(\mathcal{D})$ aggregates its fibers, the non-commutative schemes represented by $\operatorname{\sf QGr}(D_{x})$, $x\in{\mathbb{P}}(V)$, into a flat family over ${\mathbb{P}}(V)$.

1.6.

The 3-dimensional Artin-Schelter regular algebras of finite Gelfand-Kirillov dimension (or, equivalently, the noetherian ones) have been classified. They fall into two broad classes: those on 3 generators subject to 3 quadratic relations and those on 2 generators subject to 2 cubic relations. Applying Theorem 1.1 to these produces 4-dimensional Artin-Schelter regular algebras on 3 generators subject to two quadratic and two cubic relations, and 4-dimensional Artin-Schelter regular algebras on 2 generators subject to one cubic and one quartic relation, respectively.

Following the terminology in [20], we sometimes say that these two classes of 3-dimensional Artin-Schelter regular algebras have types $(1331)$ and $(1221)$, respectively. The terminology comes from the ranks of the free modules that appear in the minimal projective resolution of the trivial module (see 1.7). Extending this terminology to the 4-dimensional case in the obvious way, Theorem 1.1 produces 4-dimensional Artin-Schelter regular algebras of types $(13431)$ and $(12221)$, respectively.

Once one drops the requirement that the 3-dimensional Artin-Schelter regular algebra have finite Gelfand-Kirillov dimension (or, equivalently, be noetherian), other “types” can appear. For example, in [13], Eisenschlos constructs 3-Calabi-Yau algebras of type $(1,n,n,1)$ for many $n\geq 3$. Theorem 1.1 applies to these algebras and produces 4-Calabi-Yau algebras of type $(1,n,2n-2,n,1)$.

Normal extensions of Artin-Schelter regular algebras of types $(1331)$ and $(1221)$ were also studied by Cassidy [8]. He obtained an analogous result to Theorem 3.5 for those two classes of algebras without any assumption on the matrix $Q$. The proof in [8] was given separately for each type, while our proof is given for all types of 3-dimensional regular algebras, which include those do not have finite Gelfand-Kirillov dimensions, simultaneously.

Most of the algebras produced by Theorem 1.1 are new. A few are not. For example, two of the four families of 4-dimensional regular algebras introduced by Lu-Palmieri-Wu-Zhang [20] can be obtained by Theorem 1.1. Thus the regularity for their algebras follows from Theorem 1.1; in contrast, the proof of their regularity in [20] proceeded by finding an explicit basis of the defining ideal using Bergman’s diamond lemma [5], or by observing that some of those algebras are Ore extensions of skew polynomial rings. The algebras produced by Theorem 1.1 need not have PBW bases.

1.7. Artin-Schelter regular algebras

When $A$ is connected graded, $\Bbbk=A/A_{\geq 1}$ is a graded left $A$-module concentrated in degree 0. We call it the trivial module.

Let $d$ be a non-negative integer. A connected graded algebra $A$ is Gorenstein of dimension $d$ if it has left injective dimension $d$ as a module over itself and

[TABLE]

for some $\ell\in{\mathbb{Z}}$. The number $\ell$ in 1-1 is called the Gorenstein parameter of $A$.

A connected graded algebra $A$ is Artin-Schelter regular (or AS-regular, or simply regular) of dimension $d$ if it

1. (1)

has finite global dimension $d$ and 2. (2)

is Gorenstein of dimension $d$.

The Gorenstein and AS-regularity properties are known to be left-right symmetric. The projective resolution of $\Bbbk$ as a graded right $A$-module can be obtained by applying $\operatorname{Hom}_{A}(-,A)(-\ell)$. This also explains why the types of regular algebras are palindromic sequences; see the proof of [1, Thm. 1.5]. The left and right global dimensions coincide since both are equal to the projective dimension of $A$ as a bimodule ([33, Lem. 4.3]; see also 1.8 below).

Commutative Artin-Schelter regular algebras are polynomial rings; see [22, Prop. 2.2].

The notion of regularity we use does not require or imply that the algebra be noetherian or have finite Gelfand-Kirillov dimension. However, by [30, Prop. 3.1(1)] it does imply that ${}_{A}\Bbbk$ has a finite projective resolution in which every term is a finitely generated free left $A$-module.

1.8. Twisted Calabi-Yau algebras

Let $A$ be a graded algebra. We denote its opposite algebra by $A^{\circ}$ and its enveloping algebra by $A^{e}=A\otimes_{\Bbbk}A^{\circ}$. An $A$-bimodule $M$ is regarded as a left $A^{e}$-module via $(a\otimes b)\cdot x=axb$ for all $a,b\in A$ and $x\in M$. If $\nu$ is an automorphism of $A$ we denote by ${{}_{{}_{\nu}}}A_{{}_{1}}$ the left $A^{e}$-module that is $A$ as a graded vector space with action $(a\otimes b)\cdot c=\nu(a)cb$. We say that $A$ is twisted Calabi-Yau of dimension $d$ (abbreviated as $d$-tCY or simply tCY) if $A$, as left $A^{e}$-module, has a finite-length projective resolution in which each term is finitely generated, and there is an isomorphism

[TABLE]

of graded right $A^{e}$-modules for some $\ell\in{\mathbb{Z}}$ and some graded algebra automorphism $\nu$ of $A$. Such a $\nu$ is uniquely determined and is called the Nakayama automorphism of $A$. We also denote it by $\nu_{\!{}_{A}}$ to clarify the algebra. (Some authors call $\nu^{-1}$ the Nakayama automorphism; see, e.g., [27].) We say that $A$ is $d$-Calabi-Yau (or simply Calabi-Yau, abbreviated as $d$-CY or CY) if it is twisted Calabi-Yau of dimension $d$ and $\nu=\mathrm{id}_{A}$.

Proposition 1.6**.**

[27*, Lem. 1.2]**

A connected graded algebra is an Artin-Schelter regular algebra of dimension $d$ if and only if it is a twisted Calabi-Yau algebra of dimension $d$. If this is the case, the numbers $\ell$ in 1-1 and 1-2 are the same.*

1.9.

The next result, which summarizes some of the results in §2, applies to a larger class of algebras than those covered by Theorem 1.1.

Theorem 1.7**.**

Let $D$ be a connected graded algebra generated as a $\Bbbk$-algebra by $D_{1}$. Let $\Omega\in D$ be a homogeneous normal regular element of degree $\geq 2$ and let $A=D/(\Omega)$. If $A$ is twisted Calabi-Yau then so is $D$ and in that case the following results hold:

1. (1)

on the common space of generators $A_{1}=D_{1}$, $\nu_{\!{}_{A}}=\tau^{-1}\nu_{\!{}_{D}}$ where $\tau$ is the unique automorphism of $D$ such that $\Omega x=\tau(x)\Omega$ for all $x\in D$; 2. (2)

$\Omega$* is an eigenvector for the Nakayama automorphism $\nu_{\!{}_{D}}$;* 3. (3)

if $\sigma$ is a degree-preserving automorphism of $D$ such that $\sigma(\Omega)=\lambda\Omega$, then $\operatorname{hdet}(\sigma)$, the homological determinant of $\sigma$ acting on $D$, is $\lambda\operatorname{hdet}(\sigma|_{A})$.

1.10. $m$-Koszul algebras

Because they are generated by their degree-one components the 3-dimensional Artin-Schelter regular algebras $A({\sf w})$ in this paper are $m$-Koszul by [1, Thm. 1.5]. A connected graded algebra $A$ is $m$-Koszul if it is finitely presented and its relations are homogeneous of degree $m$ and $\operatorname{Ext}^{i}_{A}(\Bbbk,\Bbbk)$ is concentrated in degree $n(i)$ for all $i$, where $n(2j)=jm$ and $n(2j+1)=jm+1$ for integers $j$. The class of $m$-Koszul algebras was identified and first studied by R. Berger [4], the justification being that they are a natural generalization of Koszul algebras (the $2$-Koszul algebras are exactly the Koszul algebras in the “classical” sense) and that there are many “nice” and “natural” $m$-Koszul algebras that are not $2$-Koszul.

Reyes-Rogalski-Zhang [28, Cor. 5.4] proved that the homological determinant of the Nakayama automorphism of a noetherian Artin-Schelter regular connected graded algebras is 1. Mori-Smith [24] proved a similar result in the non-noetherian case when the algebra is $m$-Koszul (and Artin-Schelter regular and connected) and conjectured that the equality holds without the $m$-Koszul assumption. It follows from Theorem 1.7(3) that the conjecture is true for the algebras $D({\sf w},{\sf p})$ (which are never $m$-Koszul and often not noetherian).

1.11. The structure of the paper

In §2, we prove some new results about the homological properties of a connected graded algebra and a normal extension of it. The main theorem is stated in §3 and some of its consequences are stated and proved there. Its proof is postponed to §4. In the last section, §5, we determine the good tuples ${\sf p}$ for all the generic 3-dimensional Artin-Schelter regular algebras of finite Gelfand-Kirillov dimension listed in [1, Tables (3.9) and (3.11)]. This provides a host of new noetherian 4-dimensional Artin-Schelter regular algebras on 2 and 3 generators. Some new normal and central extensions of non-noetherian 3-Calabi-Yau algebras are also presented in §5.

Acknowledgements

We would like to thank Izuru Mori for pointing out errors in an earlier version, Michaela Vancliff for providing information on Cassidy’s work [8], and the anonymous referee for a very careful reading and numerous insightful comments.

A.C. was partially supported by NSF grants DMS-1565226 and DMS-1801011. R.K. was a JSPS Overseas Research Fellow and supported by JSPS KAKENHI Grant Numbers JP17K14164 and JP16H06337.

2. Results on normal extensions

2.1.

In this section we are concerned about the relation between a ring $D$ having a normal regular element $\Omega$ and the quotient $A:=D/(\Omega)$.

To do this we make use of the automorphism $\tau:D\to D$ defined by the requirement that

[TABLE]

for all $x\in D$. Since $\tau$ fixes $\Omega$, it descends to an automorphism of $A$ that we also denote by $\tau$.

Let $\sigma$ be an automorphism of a ring $D$. If $M$ is a left $D$-module we write ${}_{\sigma}M$ for the left $D$-module that is the abelian group $M$ endowed with the action

[TABLE]

for all $x\in D$ and $m\in M$. There is an auto-equivalence $M\rightsquigarrow{}_{\sigma}M$ of the category ${\sf Mod}(D)$ of left $D$-modules. The inverse $\sigma^{-1}:D\to D$ is an isomorphism ${}_{\sigma}D\to D$ of left $D$-modules. Likewise, if $\sigma$ descends to $A$, then $\sigma^{-1}:A\to A$ is an isomorphism ${}_{\sigma}A\to A$ of left $D$-modules.

We extend the notation in the previous paragraph to $D$-bimodules: if $\alpha$ and $\beta$ are automorphisms of $D$ and $M$ is a $D$-bimodule, we write ${}_{\alpha}M_{\beta}$ for the $D$-bimodule that is $M$ as an abelian group endowed with the action

[TABLE]

If $\alpha^{-1}\beta=\gamma^{-1}\delta$, then $\gamma\alpha^{-1}:{}_{\alpha}D_{\beta}\to{}_{\gamma}D_{\delta}$ is an isomorphism of $D$-bimodules. For example, ${}_{\alpha}D_{1}\cong{}_{1}D_{\alpha^{-1}}$.

2.2.

In the situation of 2.1, the projective resolution

[TABLE]

of $A$ as a left $D$-module is, in fact, an exact sequence

[TABLE]

in the category of $D$-bimodules. The unadorned $D$ and $A$ in this sequence are given their natural $D$-bimodule structures.

If $M$ is a left $D$-module, the result of applying $\operatorname{Hom}_{D}(-,M)$ to 2-3 is the exact sequence

[TABLE]

The map $\Phi:{}_{\tau}M\to\operatorname{Hom}_{D}({}_{1}D_{\tau},M)$ given by $\Phi(m)(d)=dm$ is an isomorphism of left $D$-modules when $\operatorname{Hom}_{D}({}_{1}D_{\tau},M)$ is given the left $D$-module structure coming from the right action of $D$ on ${}_{1}D_{\tau}$. Thus, $\operatorname{Ext}^{1}_{D}(A,M)$ is isomorphic as a left $D$-module to ${}_{\tau}(M/\Omega M)$.

Proposition 2.1**.**

Let $D$ be a connected graded algebra, $\Omega\in D_{m}$ a normal element of degree $m$, and let $A=D/(\Omega)$. Consider the following statements.

1. (1)

$\Omega$* is regular.* 2. (2)

The Hilbert series of $A$ and $D$ are related by $h_{D}(t)=h_{A}(t)\left(1-t^{m}\right)^{-1}$. 3. (3)

If $A$ is AS-regular of dimension $d$, then $D$ is AS-regular of dimension $d+1$.

There are implications $\lx@cref{creftype~refnum}{item.enh.elm}\Leftrightarrow\lx@cref{creftype~refnum}{item.enh.h}\Rightarrow\lx@cref{creftype~refnum}{item.enh.alg}$

Proof.

1 $\Leftrightarrow$ 2 This follows from the fact that the complex $0\longrightarrow D(-m)\stackrel{{\scriptstyle\cdot\Omega}}{{\longrightarrow}}D\longrightarrow A\longrightarrow 0$ is exact if and only if $\Omega$ is regular.

1 $\Rightarrow$ 3 As explained in [2, §2], the global dimensions of each of $A$ and $D$ equal the projective dimension of the trivial module $\Bbbk$ over the respective algebra. Since $\Omega$ is regular and ${\operatorname{pdim}}({}_{A}\Bbbk)<\infty$, [23, Thm. 7.3.5(i)] implies that ${\operatorname{pdim}}({}_{D}\Bbbk)$ equals ${\operatorname{pdim}}({}_{A}\Bbbk)+1=d+1$. Thus, $\mathrm{gldim}(D)=d+1$. The Gorenstein property for $D$ is equivalent to that of $A$ via the isomorphism

[TABLE]

which is a consequence of Lemma 2.2 below applied to $N=\Bbbk$ and $M=D$. ∎

Lemma 2.2**.**

Let $D$ be a connected graded algebra, $\Omega\in D_{m}$ a normal element of degree $m$, and let $A=D/(\Omega)$. Then, for every left $A$-module $N$ and all left $D$-modules $M$ on which $\Omega$ acts regularly, there are isomorphisms

[TABLE]

which is functorial on both $N$ and $M$, for all $p$.

Proof.

The functor $\operatorname{Hom}_{D}(N,-)$ from left $D$-modules to vector spaces factors as

[TABLE]

Since the lower left functor preserves injectivity, [15, Thm. III.7.1] gives an isomorphism

[TABLE]

in the left-bounded derived category of vector spaces, which is functorial on both $M$ and $N$.

Since $\Omega$ acts faithfully on $M$, $\operatorname{Hom}_{D}(A,M)=0$. As observed above, the canonical map $\operatorname{Hom}_{D}({}_{1}D_{\tau},M)\to\operatorname{Ext}^{1}_{D}(A,M)$ induces an isomorphism {}_{\tau}(M/\Omega M)\xrightarrow{\smash{\raisebox{-2.5pt}{\sim}}}\operatorname{Ext}^{1}_{D}(A,M) of left $D$-modules. Since ${\operatorname{pdim}}({}_{D}A)=1$, $\operatorname{Ext}^{p}_{D}(A,M)=0$ for $p\geq 2$. Hence

[TABLE]

in the left-bounded derived category of left $A$-modules. Thus,

[TABLE]

which is nothing but a reformulation of the conclusion of the lemma. ∎

On occasion, one might want to impose growth conditions on connected graded algebras that complement the good homological properties in 1.7 (the more prevalent definition of Artin-Schelter regularity, for instance, requires finite Gelfand-Kirillov dimension; see, e.g., [1, Introduction] or [2, (2.12)]). The following remark supplements Proposition 2.1 in that direction.

Proposition 2.3**.**

Let $D$ be a connected graded algebra and $\Omega\in D_{m}$ a normal element of degree $m\geq 1$, and let $A=D/(\Omega)$. Then

1. (1)

$A$* has finite Gelfand-Kirillov dimension if and only if $D$ does;* 2. (2)

$A$* is left or right noetherian if and only if $D$ is.*

Proof.

1 The finite Gelfand-Kirillov dimension of $D$ certainly entails that of its quotient $A=D/(\Omega)$. Conversely, it follows from the exact sequence $D(-m)\stackrel{{\scriptstyle\cdot\Omega}}{{\longrightarrow}}D\longrightarrow A\longrightarrow 0$ that

[TABLE]

for all $k$. Thus, if $\mathrm{dim}(A_{k})$ has polynomial growth so does $\mathrm{dim}(D_{k})$.

2 Certainly, if $D$ is noetherian so is $A$. The converse follows from [2, Lem. 8.2]. ∎

Later, we determine the relationship between the Nakayama automorphism of $A$ and that of $D$ whenever the equivalent conditions 1 and 2 of Proposition 2.1 are satisfied. As a first step in that direction, we record the following consequence of Lemma 2.2.

Lemma 2.4**.**

Let $D$ be a connected graded algebra, $\Omega\in D_{m}$ a normal regular element of degree $m$, and let $A=D/(\Omega)$. There is an isomorphism

[TABLE]

of $D$-bimodules.

Proof.

We write ${}_{\tau}A^{e}{}_{\tau^{-1}}$ for $A^{e}$ endowed with the $D$-bimodule structure given by

[TABLE]

where $\overline{c}$ and $\overline{d}$ are the images in $A$ of $c,d\in D$ and $b\tau^{-1}(\overline{d})$ is the product $b\times\tau^{-1}(\overline{d})$ in $A$ viewed as an element of $A^{\circ}$. As a left $D^{e}$-module, ${}_{\tau}A^{e}{}_{\tau^{-1}}$ is annihilated by $1\otimes\Omega$ and $\Omega\otimes 1$ so ${}_{\tau}A^{e}{}_{\tau^{-1}}$ is a left $A^{e}$-module (and, equivalently, an $A$-bimodule) where the subscripts $\tau$ and $\tau^{-1}$ should now be viewed as the automorphisms of $A$ induced by the automorphisms $\tau$ and $\tau^{-1}$ of $D$.

Apply Lemma 2.2 twice, first to the quotient $D\otimes A^{\circ}=D^{e}/(1\otimes\Omega)$, then to $A^{e}=(D\otimes A^{\circ})/(\Omega\otimes 1)$. This results in an isomorphism

[TABLE]

that preserves the right $D^{e}$-module structures. The result follows from the isomorphisms

[TABLE]

for the left and respectively right-hand tensorands of $A^{e}$. ∎

2.3.

For the rest of this section we assume that $D$ is a connected graded algebra, $\Omega\in D_{m}$ is a normal regular element of degree $m$, and $A=D/(\Omega)$ is Artin-Schelter regular of dimension $d$. Since condition 3 of Proposition 2.1 holds, $D$ is Artin-Schelter regular of dimension $d+1$.

By Proposition 1.6, $A$ and $D$ are also twisted Calabi-Yau; it is the $A^{e}$- and $D^{e}$-bimodule structures of $A$ and $D$ that will play the central role.

We denote by $\nu=\nu_{\!{}_{D}}$ the Nakayama automorphism of $D$, defined by

[TABLE]

all other $\operatorname{Ext}^{p}(D,D^{e})$ vanish.

2.4.

The next result, together with Lemma 2.4, ensures that $\operatorname{Ext}_{D^{e}}^{d+2}(A,D^{e})$ can function as a bridge between $\operatorname{Ext}_{D^{e}}^{d+1}(D,D^{e})$ and $\operatorname{Ext}_{A^{e}}^{d}(A,A^{e})$, which in turn are used to define the Nakayama automorphisms of $A$ and $D$.

Lemma 2.5**.**

Let $D$ be a connected graded algebra, $\Omega\in D_{m}$ a normal regular element of degree $m$, and let $A=D/(\Omega)$. If $A$ is AS-regular of dimension $d$, then there is an isomorphism

[TABLE]

of $D$-bimodules.

Proof.

The short exact sequence 2-3 of $D$-bimodules yields an exact sequence

[TABLE]

of right $D^{e}$-modules in which $(\cdot\Omega)^{*}$ denotes the morphism induced from the morphism $(\cdot\Omega)$ between the first variables. A morphism induced from that between second variables will be denoted by a lower star. We will show that the diagram

[TABLE]

commutes. The commutativity of the middle square follows from the definition of $\tau$. Note that right multiplication by $1\otimes\Omega$ on $D^{e}=D\otimes_{k}D^{\circ}$ is left multiplication by $\Omega$ on the right tensorand. The commutativity of the right-most square is straightforward.

For the left-most square: The left vertical morphism is equal to $((1\otimes\Omega)\cdot)^{*}$. For each left $D^{e}$-module $M$, set $FM:={}_{1}M_{\tau^{-1}}$ and $\varphi_{M}:=((1\otimes\Omega)\cdot)\colon M\to FM$. This defines a morphism $\varphi\colon 1\to F$ of auto-equivalences on ${\sf Mod}(D^{e})$. Since every left $D^{e}$-homomorphism $f\colon FM\to N$ satisfies $F(f)F(\varphi_{M})=\varphi_{N}f$, we obtain the following commutative diagram by considering an injective resolution of $N$:

[TABLE]

This becomes the left-most square of 2-8 after substituting $M:={}_{1}D_{\tau}$ and $N:=D^{e}$.

In the following diagram, the right-hand square obviously commutes and the left-hand square commutes since the horizontal arrows are isomorphism of $D$-bimodules:

[TABLE]

where the middle vertical arrow is left multiplication by $\Omega$ through the twist by $\nu$, so as a linear map $D\to D$, it is left multiplication by $\nu(\Omega)$. By adjoining this diagram to the right-hand end of 2-8, we obtain the commutative diagram

[TABLE]

which completes the proof of the lemma. ∎

We now determine the relation between the Nakayama automorphisms of $D$ and $A$ when $D$ is generated by $D_{1}$ as a $\Bbbk$-algebra. The algebras $D$ that we will consider in Section 3 are generated by $D_{1}$.

Theorem 2.6**.**

Let $D$ be a connected graded $\Bbbk$-algebra that is generated by $D_{1}$. Let $\Omega\in D$ be a homogeneous normal regular element of degree $\geq 2$ and let $A=D/(\Omega)$. If $A$ is twisted Calabi-Yau so is $D$ and in that case $\nu_{\!{}_{A}}=\tau^{-1}\nu_{\!{}_{D}}$ on the common space of generators $A_{1}=D_{1}$.

Proof.

It follows from 3 of Proposition 2.1 that $D$ is twisted Calabi-Yau of dimension $d+1$. Lemmas 2.4 and 2.5 together with $\operatorname{Ext}^{d}_{A^{e}}(A,A^{e})\cong{}_{\nu_{\!{}_{A}}}A_{1}$ imply that

[TABLE]

the conclusions follow. ∎

We remind the reader that $\tau$ and $\nu_{\!{}_{D}}$ commute since, by [21, Thm. 4.2], Nakayama automorphisms are central under conditions that our algebras certainly satisfy.

Theorem 2.6 is analogous to [27, Lem. 1.5], but we do not use the theory of dualizing complexes, and do not assume here that $\Omega$ is an eigenvector for the Nakayama automorphism $\nu_{\!{}_{D}}$. In fact, the latter condition now follows from the above discussion.

Corollary 2.7**.**

Let $D$ be a connected graded algebra, $\Omega\in D$ a homogeneous normal regular element of degree $\geq 2$. If $D/(\Omega)$ is twisted Calabi-Yau, then $\Omega$ is an eigenvector for the Nakayama automorphism $\nu_{\!{}_{D}}$.

Proof.

By Theorem 2.6, $\nu_{\!{}_{D}}$ descends to an automorphism of $D/(\Omega)$. Hence $\nu_{\!{}_{D}}(\Omega)$ belongs to the ideal $(\Omega)$. Since $D$ is connected and $\nu_{\!{}_{D}}$ preserves degree, $\nu_{\!{}_{D}}(\Omega)\in\Bbbk\Omega$. ∎

2.5. Homological determinant

As before, let $D$ be a connected graded algebra, $\Omega\in D_{m}$ a normal regular element of degree $m$, and assume that $A=D/(\Omega)$ is Artin-Schelter regular of dimension $d$ with Gorenstein parameter $\ell$. We will determine the relation between the homological determinant of an automorphism of $D$ that preserves the ideal $(\Omega)$ with the homological determinant of the induced automorphism of $A$. The main result in this section, Theorem 2.10, will be used later to show that the homological determinant of the Nakayama automorphism of the algebra $D({\sf w},{\sf p})$ is 1.

The homological determinant was introduced by Jørgensen-Zhang [17]. Before defining it we need some notation and other ideas.

Let $M$ be a left $A$-module. The $p^{\rm th}$ local cohomology group of $M$ is $H_{\mathfrak{m}}^{p}(M):=\varinjlim\operatorname{Ext}_{A}^{p}(A/A_{\geq k},M)$. The right action of $A$ on $A/A_{\geq k}$ determines a left $A$-module structure on $H_{\mathfrak{m}}^{p}(M)$, and $H_{\mathfrak{m}}^{p}$ then becomes an endofunctor on the category of graded left $A$-modules.

The Matlis dual of $M$ is the graded right $A$-module $M^{\prime}$ whose degree-$i$ component is $\operatorname{Hom}_{\Bbbk}(M_{-i},\Bbbk)$, the right action of $A$ coming from the left action of $A$ on $M$. The operation $M\rightsquigarrow M^{\prime}$ is a duality between the category of graded left $A$-modules and the category of graded right $A$-modules.

Let $\operatorname{Aut}_{\sf gr}(A)$ denote the group of degree-preserving $\Bbbk$-algebra automorphisms of $A$ and let $\sigma\in\operatorname{Aut}_{\sf gr}(A)$. Then $\sigma:A\to{}_{\sigma}A_{\sigma}$ is an isomorphism of $A$-bimodules.

Jørgensen-Zhang [17] defined the homological determinant in the following way. First they show that the top local cohomology group $H_{\mathfrak{m}}^{d}(A)$ is isomorphic to $A^{\prime}(\ell)$ and all other $H_{\mathfrak{m}}^{p}(A)$ vanish. Fix an isomorphism $\psi\colon H_{\mathfrak{m}}^{d}(A)\to A^{\prime}(\ell)$. Let H_{\mathfrak{m}}^{d}({}_{\sigma}A)\xrightarrow{\smash{\raisebox{-2.5pt}{\sim}}}{}_{\sigma}H_{\mathfrak{m}}^{d}(A) be the isomorphism

[TABLE]

The homological determinant of $\sigma$ is the unique scalar $\operatorname{hdet}(\sigma)$ that makes the diagram

[TABLE]

commute. When we need to identify the algebra $A$ we write $\operatorname{hdet}_{A}(\sigma)$.

The degree-$(-\ell)$ components of the diagram 2-9 give another commutative diagram:

[TABLE]

where the two vertical morphisms are the same. Note that ${}_{\sigma}\operatorname{Ext}_{A}^{d}(\Bbbk,A)=\operatorname{Ext}_{A}^{d}(\Bbbk,A)$. Because all the objects in 2-10 have dimension 1, the definition of $\operatorname{hdet}(\sigma)$ does not depend on the choice of the isomorphism $\psi$.

Lemma 2.8**.**

Let $\sigma\in\operatorname{Aut}_{\sf gr}(D)$. If $\Omega$ is an eigenvector for $\sigma$, then there is an isomorphism \alpha\colon\operatorname{Ext}_{D}^{d}(\Bbbk,A)\xrightarrow{\smash{\raisebox{-2.5pt}{\sim}}}\operatorname{Ext}_{A}^{d}(\Bbbk,A) that makes the diagram

[TABLE]

commutative.

Proof.

Since $\sigma(\Bbbk\Omega)=\Bbbk\Omega$, $\sigma$ descends to an automorphism of $A$, which is also denoted by $\sigma$. By applying $\operatorname{RHom}_{D}(-,A)$ to the short exact sequence 2-3 of $D$-bimodules, we obtain the triangle in the first row of the diagram

[TABLE]

in the left-bounded derived category of left $D$-modules. Since this diagram commutes, the first morphism in the triangle admits a section passing through $A$ in the bottom. Hence we have a split triangle

[TABLE]

where the first morphism is the composite of canonical morphisms A\xrightarrow{\smash{\raisebox{-2.5pt}{\sim}}}\operatorname{Hom}_{D}(A,A)\to\operatorname{RHom}_{D}(A,A).

Let $A\to I^{{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}}$ be an injective resolution of $A$ as a left $D$-module. Then $\operatorname{Hom}_{D}(A,I^{{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}})$ is a complex of injective left $A$-modules and we have

[TABLE]

This induces

[TABLE]

where the first isomorphism holds since we have 2-12 and $\mathrm{H}^{d}\operatorname{Hom}_{A}(\Bbbk,{}_{\tau}A[-1])\cong\operatorname{Ext}_{A}^{d-1}(\Bbbk,A)=0$ by the Gorenstein condition.

The isomorphism \operatorname{Ext}_{A}^{d}(\Bbbk,A)\xrightarrow{\smash{\raisebox{-2.5pt}{\sim}}}\operatorname{Ext}_{D}^{d}(\Bbbk,A) sends each $d$-extension of $A$ by $\Bbbk$ to itself. To see this, let

[TABLE]

be a $d$-extension of left $A$-modules and let $A\to J^{{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}}$ be an injective resolution of $A$ as a left $A$-module. Lifting up the canonical isomorphism A\xrightarrow{\smash{\raisebox{-2.5pt}{\sim}}}\operatorname{Hom}_{D}(A,A), we obtain a commutative diagram

[TABLE]

The image of $\xi$ by the first morphism in 2-13 is represented by the composite of the right-most vertical arrows. Applying the adjunction $(A\otimes_{A}-)\dashv\operatorname{Hom}_{D}(A,-)$ to the diagram, we obtain

[TABLE]

The right-most vertical morphism represents the image of $\xi$ by the composite 2-13. Hence the image is the element of $\operatorname{Ext}_{D}^{d}(\Bbbk,A)$ represented by $\xi$ regarded as a $d$-extension of left $D$-modules.

The commutativity of the diagram in the statement now follows from the above description using a $d$-extension. ∎

Lemma 2.9**.**

Let $\sigma\in\operatorname{Aut}_{\sf gr}(D)$. If $\Omega$ is an eigenvector for $\sigma$ with eigenvalue $\lambda$, then there is an isomorphism \delta\colon\operatorname{Ext}_{D}^{d}(\Bbbk,A)\xrightarrow{\smash{\raisebox{-2.5pt}{\sim}}}\operatorname{Ext}_{D}^{d+1}(\Bbbk,D) that makes the diagram

[TABLE]

commutative.

Proof.

The short exact sequence 2-2 is extended to the commutative diagram in the derived category

[TABLE]

where each row is a triangle. The third and fourth columns yield the commutative diagram

[TABLE]

Letting $\delta:=\operatorname{Ext}^{d}(\Bbbk,\varepsilon)$ completes the proof. ∎

Theorem 2.10**.**

Let $D$ be a connected graded algebra, $\Omega\in D_{m}$ a normal regular element of degree $m$, and let $A=D/(\Omega)$. Assume that $A$ is twisted Calabi-Yau of dimension $d$. Let $\sigma\in\operatorname{Aut}_{\sf gr}(D)$. If $\Omega\in D$ is an eigenvector for $\sigma$ with eigenvalue $\lambda$, then

[TABLE]

Proof.

The concatenation of the diagram 2-10 for $D$ with 2-11 and 2-14 yields the commutative diagram

[TABLE]

where the right isomorphism is equal to the left isomorphism multiplied by $\lambda^{-1}$. This shows $(\operatorname{hdet}_{A}(\sigma|_{A}))^{-1}=(\operatorname{hdet}_{D}(\sigma))^{-1}\lambda$. ∎

2.6. An equivariant formulation of the homological determinant

In this section we give a slightly different interpretation of the homological determinant. In this section $A$ denotes a $d$-dimensional Artin-Schelter regular $\Bbbk$-algebra with Gorenstein parameter $\ell$ and $\sigma$ is a graded $\Bbbk$-algebra automorphism of $A$.

We assume some familiarity with group actions on categories and equivariantizations. A good reference is [14, $\S$2.7]. Additional details can be found in the appendix of [9]. Since categories in this paper are $\Bbbk$-linear, all functors in the following discussion are required to be $\Bbbk$-linear.

Recall [14, Defns. 2.7.1 and 2.7.2]; because of the wide availability of resources on the topic we merely paraphrase the definitions here.

Definition 2.11**.**

An action of a group $\Gamma$ on a category $\mathcal{C}$ is a collection of auto-equivalences $T_{g}:\mathcal{C}\to\mathcal{C}$, $g\in\Gamma$, and natural isomorphisms $T_{g}\circ T_{h}\cong T_{gh}$ satisfying the obvious compatibility conditions.

Given an action of $\Gamma$ on $\mathcal{C}$, the category $\mathcal{C}^{\Gamma}$ of $\Gamma$-equivariant objects is the category whose objects are collections of isomorphisms

[TABLE]

that are compatible with the endofunctors $T_{g}$ in the sense that the diagrams

[TABLE]

commute for all $g,h\in\Gamma$, where the lower isomorphism is the structural one defining the $\Gamma$-action on $\mathcal{C}$. The morphisms in $\mathcal{C}^{\Gamma}$ are those morphisms $f:x\to y$ in $\mathcal{C}$ for which the diagrams

[TABLE]

commute for all $g\in\Gamma$. $\blacklozenge$

Now consider objects $x,y\in\mathcal{C}^{\Gamma}$ (we are abusing notation by suppressing the isomorphisms $\varphi_{g}:T_{g}x\to x$). The action of $\Gamma$ on $\mathcal{C}$ induces an action $\triangleright$ of $\Gamma$ on $\operatorname{Hom}_{\mathcal{C}}(x,y)$ defined by the commutativity of the diagram

[TABLE]

Remark 2.12**.**

When $\mathcal{C}$ is abelian a $\Gamma$-action passes over to any of the derived categories associated to $\mathcal{C}$ (the bounded derived category $\mathcal{D}^{b}(\mathcal{C})$, the left bounded one $\mathcal{D}^{+}(\mathcal{C})$, etc.). We assume below that $\mathcal{C}$ has enough injectives.

For an equivariant object $y$ in $\mathcal{C}^{\Gamma}$ as above the isomorphisms $\varphi_{g}:T_{g}y\to y$ lift to isomorphisms between the injective resolutions $T_{g}E^{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}$ and $E^{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}$ of $T_{g}y$ and $y$ respectively, and given another equivariant object $x\in\mathcal{C}$ the action of $\Gamma$ on

[TABLE]

can be computed via the action on $\operatorname{Hom}_{\mathcal{C}}(x,E^{d})$ as in 2-15, using the isomorphism $\varphi_{g}:T_{g}x\to x$ as the left-hand vertical arrow and the isomorphism $T_{g}E^{d}\to E^{d}$ as the right-hand vertical map.

All group actions on derived categories considered below arise in this fashion, from an action on the original abelian category, and similarly, we only consider equivariant objects in $\mathcal{D}^{b}(\mathcal{C})$ that are lifts of those in $\mathcal{C}$ in the manner described above. $\blacklozenge$

We now apply the above to the group $\Gamma=\mathbb{Z}$ acting via the twist $M\mapsto{}_{\sigma}M$ for $M$ in the category of $A$-modules, graded modules, complexes thereof, derived categories, etc. Since the group $\Gamma$ is generated by the single element $g$ mapping to $\sigma\in\langle\sigma\rangle$, equivariant structures are determined by single automorphisms $\varphi^{x}=\varphi_{g}:T_{g}x\to x$.

By definition, for (graded) $A$-modules the twist functor

[TABLE]

does not change the underlying vector spaces and morphisms, so the diagram 2-15 above, for equivariant objects $(M,\varphi^{M})$ and $(N,\varphi^{N})$, reads

[TABLE]

We can apply the discussion above to the bounded derived category $\mathcal{D}^{b}=\mathcal{D}^{b}({\sf Mod}(A))$ of left $A$-modules with $\Gamma=\mathbb{Z}$ once more acting by powers by $\sigma$.

Now regard all $A/A_{\geq k}$, $k\leq\infty$ (where $A/A_{\geq\infty}$ means $A$), and their homological shifts $(A/A_{\geq k})[p]$ in $\mathcal{D}^{b}$ as $\sigma$-equivariant $A$-modules via $\varphi^{(A/A_{\geq k})[p]}:=\sigma^{-1}[p]$. This renders the following statement meaningful; it is a repackaging of [17, Defn. 2.3].

Proposition 2.13**.**

The inverse $\operatorname{hdet}^{-1}:=\operatorname{hdet}(\sigma)^{-1}$ of the homological determinant is the scalar by which $g^{-1}\triangleright$ acts on the degree-$(-\ell)$ component of any of the spaces $H_{\mathfrak{m}}^{d}(A)$, $\operatorname{Ext}_{A}^{d}(\Bbbk,A)$, or $\operatorname{Ext}_{A}^{d}(\Bbbk,\Bbbk)$.

Proof.

First, note that the three scalars in the statement are indeed equal:

As far as the last two are concerned, the long exact sequence attached to the short exact sequence

[TABLE]

the vanishing of $\operatorname{Ext}_{A}^{d+1}$ and the fact that $\operatorname{Ext}_{A}^{d}(\Bbbk,A)$ and $\operatorname{Ext}_{A}^{d}(\Bbbk,\Bbbk)$ are both one-dimensional vector spaces jointly ensure that the induced map

[TABLE]

is an isomorphism.

On the other hand, the first two scalars in the statement are equal because the $\sigma$-equivariant maps $A/A_{\geq k+1}\to A/A_{\geq k}$ induce a $\Gamma$-module morphism

[TABLE]

that identifies degree-$(-\ell)$ components.

Thus, it suffices to prove that the scalar $\operatorname{hdet}^{-1}$ defined via 2-10 agrees with the eigenvalue of $g^{-1}$ on $\operatorname{Ext}_{A}^{d}(\Bbbk,A)$. We proceed to do this.

[17, Defn. 2.3] (which 2-10 follows) dictates that the scalar $\operatorname{hdet}^{-1}$ is computed as follows:

Consider the minimal injective resolution

[TABLE]

of $A$ in ${\sf Mod}(A)$, and let $\sigma$ act on it so as to extend the action on $A$. Then, $\operatorname{hdet}^{-1}$ will be the scaling that $\sigma$ induces on the one-dimensional torsion of the $d^{th}$ term $E^{d}$. This is also what we obtain by making $g$ (the generator of $\Gamma=\mathbb{Z}$) act on $\operatorname{Hom}(\Bbbk,E^{d})$ by sending a vector space morphism $f:\Bbbk\to E^{d}$ to $\sigma\circ f\circ\sigma^{-1}$, which is what 2-16 amounts to in the present setting (see Remark 2.12). ∎

3. Normal extensions of $3$-dimensional regular algebras

In this section we describe a technique for producing $4$-dimensional Artin-Schelter regular algebras by modifying $m$-Koszul $3$-dimensional Artin-Schelter regular algebras. Again, our regular algebras need not be noetherian. We first recall some conventions and terminology from various sources on $m$-Koszul and superpotential algebras, such as [11, 12, 7, 24].

3.1. Superpotential algebras

For our purposes, the relevant setup is as follows.

We fix an $n$-dimensional vector space $V$ that will be the common space of degree-one generators for the algebras under discussion. We also fix integers $\ell\geq m\geq 2$, an invertible linear transformation $Q\in{\rm GL}(V)$, and an element ${\sf w}\in V^{\otimes\ell}$ that is invariant under the linear map

[TABLE]

We call such a ${\sf w}$ a $Q$-twisted superpotential or just a twisted superpotential if we don’t want to specify $Q$. If $Q=\mathrm{id}_{V}$ we simply call ${\sf w}$ a superpotential.

Frequently, we assume that $Q$ is diagonalizable, but see (2) in §3.4.

Let $W$ be a subspace of some tensor power, $V^{\otimes p}$ say. We introduce the following notation:

[TABLE]

The space ${\operatorname{\partial}}^{i}W$ appears in [12, §4] where it is denoted $W^{(p-i)}$.

Let ${\sf w}\in V^{\otimes\ell}$ be a $Q$-twisted superpotential. We call the algebra

[TABLE]

a superpotential algebra. In [7], $A({\sf w},i)$ is called a derivation-quotient algebra. We are mostly interested in the case $i=1$ and we then write

[TABLE]

Theorem 3.1** (Dubois-Violette).**

[12, Thm. 11]* [7, Thm. 6.8]

If $A$ is an $m$-Koszul Artin-Schelter regular algebra of dimension $d$ with Gorenstein parameter $\ell$, then there is a twisted superpotential ${\sf w}\in V^{\otimes\ell}$ such that*

[TABLE]

where such ${\sf w}$ is unique up to non-zero scalar multiple.

The global dimension $d$ of $A({{\sf w}},\ell-m)$ can be recovered from the other numerical data as follows:

• •

if $m=2$, then $d=\ell$;

• •

if $m\geq 3$, then $d$ is odd and $\ell=m\frac{d-1}{2}+1$.

We will use the next lemma in Corollary 3.7.

Lemma 3.2**.**

Let $k\in{\mathbb{Z}}$. If ${\sf w}\in V^{\otimes\ell}$ is a $Q$-twisted superpotential, then

[TABLE]

Proof.

The $\ell^{\rm th}$ power of the linear operator on $V^{\otimes\ell}$ in the definition of a $Q$-twisted superpotential is $(Q^{-1})^{\otimes\ell}$. Since ${\sf w}$ is invariant under the operator, the statement follows. ∎

Remark 3.3**.**

Our $Q$-twisted superpotential is a $(Q^{-1})$-superpotential in the sense of [24, Defn. 2.5] and [26, Defn. 2.1]. In [7, §2.2], it was called a twisted superpotential of degree $\ell$ with respect to $(-1)^{\ell-1}Q^{-1}$. Our superpotentials can be related to potentials of a quiver in [10, §3] (also to superpotentials in [6, §1]) by considering a quiver $\Gamma$ consisting of one vertex and $\dim V$ loops. The space of potentials of $\Gamma$ concentrated in length $\ell$ modulo cyclic equivalence in the sense of [10, §3] has a basis consisting of cycles modulo cyclic permutation. So the space is isomorphic to the space of superpotentials in $V^{\otimes\ell}$ in our sense. $\blacklozenge$

3.2. Notation

It will be useful in what follows to use a fixed basis $x_{1},\ldots,x_{n}$ for $V$. The matrix representing the linear transformation $Q$ is also denoted by $Q$. That is, $Q(x_{j})=\sum_{i}Q_{ij}x_{i}$.

For each $i=1,\ldots,n$, let $\partial_{i}:TV\to TV$ be the linear map that acts on words ${\sf u}$ in the letters $x_{j}$ as follows:

[TABLE]

We call ${\operatorname{\partial}}_{i}{\sf u}$ the partial derivative of ${\sf u}$ with respect to $x_{i}$. It is easy to see that $A({\sf w},\ell-m)$ is equal to the tensor algebra $TV$ modulo the ideal generated by

[TABLE]

From now on $A$ will denote a 3-dimensional $m$-Koszul Artin-Schelter regular algebra so, by Theorem 3.1, $\ell=m+1$ and there is a twisted superpotential ${\sf w}\in V^{\otimes(m+1)}$ such that

[TABLE]

The minimal projective resolution of ${}_{A}\Bbbk$ is of the form $0\to A(-\ell)\to A\otimes R\to A\otimes V\to A\to\Bbbk\to 0$ where $R$ is the linear span of a “minimal” set of relations. The Gorenstein condition implies that $\dim(V)=\dim(R)$, so $\{{\operatorname{\partial}}_{1}{\sf w},\ldots,{\operatorname{\partial}}_{n}{\sf w}\}$ is linearly independent in $TV$. This follows also from the “pre-regularity” of ${\sf w}$ in [12, Thm. 11].

If $p\in\Bbbk$ and $x$ and $y$ are elements in a $\Bbbk$-algebra, we use the notation

[TABLE]

3.3. The main theorem

Let ${\sf w}\in V^{\otimes(m+1)}$ be a $Q$-twisted superpotential and let ${\sf p}=(p_{1},\ldots,p_{n})$ be a tuple of non-zero scalars. The algebra $D({\sf w},{\sf p})$ is the tensor algebra $TV$ modulo the relations

[TABLE]

Thus $D({\sf w},{\sf p})$ has $n-1$ relations in degree $m$ and $n-1$ relations of degree $m+1$. The scalar $p_{1}$ is not used in the relation and will later be fixed to be a particular entry in $Q$.

We will frequently write $A$ for $A({{\sf w}})$ and $D({\sf p})$, or just $D$, for $D({\sf w},{\sf p})$.

We will write $\Omega$ for ${\operatorname{\partial}}_{1}{\sf w}$ viewed as an element in $D({\sf w},{\sf p})$. Clearly, $A({\sf w})=D({\sf w},{\sf p})/(\Omega)$.

The following notation will be used in what follows. First, we write

[TABLE]

Since ${\sf w}\in V^{\otimes(m+1)}$, there is a unique $n\times n$ matrix ${\sf M}$ with entries in $V^{\otimes(m-1)}$ such that

[TABLE]

It follows from the definition of the partial derivatives ${\operatorname{\partial}}_{j}{\sf w}$ that

[TABLE]

Because ${\sf w}$ is a $Q$-twisted superpotential,

[TABLE]

where $Q$ is regarded as a matrix as explained in 3.2, and the products are matrix products whose entries are in $TV$. The equation 3-2 is written in [1, (2.1)] and the equivalence to the definition of a superpotential follows from [1, (2.12)]. Note that they used the dual space $V^{*}$ in the place of our $V$ and hence the matrices representing linear transformations on $V$ are transposed from ours.

Lemma 3.4**.**

If $Q$ is diagonal with entries $q_{1},\ldots,q_{n}$ with respect to the basis $x_{1},\ldots,x_{n}$ for $V$, then $\Omega$ is a normal element and $A({{\sf w}})=D({\sf w},{\sf p})/(\Omega)$.

Proof.

Since $[x_{j},\partial_{1}{{\sf w}}]_{p_{j}}=0$ in $D({\sf w},{\sf p})$, to prove $\Omega$ is normal it only remains to show that $x_{1}\Omega$ is a non-zero scalar multiple of $\Omega x_{1}$. It follows from the remarks prior to this lemma that there are equalities

[TABLE]

in $TV$. But $f_{i}=0$ in $D({\sf w},{\sf p})$ for all $i\geq 2$, so $x_{1}\Omega=q_{1}\Omega x_{1}$. Thus, $\Omega$ is normal as claimed. ∎

Now we state our main theorem.

Theorem 3.5**.**

Let $A({{\sf w}})$ be an $m$-Koszul 3-dimensional Artin-Schelter regular algebra generated by $x_{1},\ldots,x_{n}$. Let ${\sf p}=(p_{1},\ldots,p_{n})$ be a tuple of non-zero scalars and define $D({\sf w},{\sf p})$ as above. If $Q$ is a diagonal matrix $\operatorname{diag}(q_{1},\ldots,q_{n})$ with respect to $x_{1},\ldots,x_{n}$ and $p_{1}=q_{1}$, then the following conditions are equivalent:

1. (1)

$D({\sf w},{\sf p})$* is a 4-dimensional Artin-Schelter regular algebra;* 2. (2)

$[\Omega,{\sf M}_{ij}]_{q_{1}^{-1}p_{i}p_{j}}=0$* in $D({\sf w},{\sf p})$ for all $1\leq i,j\leq n$;* 3. (3)

for all $1\leq i,j\leq d$ and all words $x_{l_{1}}\ldots x_{l_{m-1}}$ appearing in ${\sf M}_{ij}$, $q_{1}=p_{i}p_{j}p_{l_{1}}\ldots p_{l_{m-1}}$; 4. (4)

the automorphism $\varphi\colon V\to V$ defined by $\varphi(x_{i})=p_{i}x_{i}$ has the property

[TABLE]

Furthermore, if one of these conditions holds, then $\Omega$ is a regular element of $D({\sf w},{\sf p})$.

We will prove the theorem in Section 4.

3.4. Remarks

(1) We say the tuple ${\sf p}$ is good if it satisfies one of the conditions 2-4 in Theorem 3.5.

(2) Theorem 3.5 can be generalized and extended in a number of ways that we will take for granted below. For example, instead of replacing the relation ${\operatorname{\partial}}_{1}{\sf w}=0$ we could replace the relation $\partial_{k}{{\sf w}}=0$ for any $k\in\{1,\ldots,n\}$; the proof proceeds just as for $k=1$ after making the obvious changes to the indexing. Furthermore, it is not necessary that $Q$ be diagonal; as we note below in Remark 4.3, it is enough to assume that the entry $q_{k}$ relevant to the relation $\partial_{k}{{\sf w}}$ being omitted splits off as a block diagonal entry.

(3) If one of the conditions in Theorem 3.5 holds, then $D({\sf w},{\sf p})$ is a domain if $A({\sf w})$ is. The argument is well-known. Let $R$ be an ${\mathbb{N}}$-graded ring and $\Omega$ a homogeneous regular normal element of positive degree. Suppose $R/(\Omega)$ is a domain. If $R$ is not a domain there are non-zero homogeneous elements $a$ and $b$ in $R$ such that $ab=0$; among all such pairs $(a,b)$ choose one such that $\deg(a)+\deg(b)$ is as small as possible; since $R/(\Omega)$ is a domain either $a$ or $b$ is a multiple of $\Omega$; without loss of generality, assume $a=\Omega a^{\prime}$; hence $\Omega a^{\prime}b=0$; since $\Omega$ is regular, $a^{\prime}b=0$; but $\deg(a^{\prime})+\deg(b)<\deg(a)+\deg(b)$ which contradicts the choice of $a$ and $b$ so we conclude that $R$ is, in fact, a domain.

3.5. Consequences of the main theorem

Theorem 3.5 is particularly useful when $Q=\mathrm{id}_{V}$, i.e., when ${\sf w}$ is a superpotential. In our setting, it follows from [24, Cor. 4.5] that ${\sf w}$ is a superpotential if and only if $A({\sf w})$ is a $3$-Calabi-Yau algebra. In that case, ${\sf p}=(1,\ldots,1)$ is good because it obviously satisfies condition (3), and equally obviously satisfies condition (4), in Theorem 3.5. We record this observation since we will use it later.

Corollary 3.6**.**

Let $A=A({{\sf w}})$ be a $3$-Calabi-Yau algebra generated by $x_{1},\ldots,x_{n}$. Then the algebra $D$ generated by $x_{1},\ldots,x_{n}$ subject to relations

[TABLE]

is a $4$-dimensional Artin-Schelter regular algebra and is a central extension of $A$ by $\Omega=\partial_{1}{{\sf w}}$.

There is a more general version of Corollary 3.6.

Corollary 3.7**.**

Let $k\in\mathbb{Z}$. Let $A({{\sf w}})$ be $3$-dimensional Artin-Schelter regular (= twisted Calabi-Yau) algebra generated by $x_{1},\ldots,x_{n}$. If $Q=\operatorname{diag}(1,q_{2},\ldots,q_{n})$ with respect to the basis $x_{1},\ldots,x_{n}$, then the algebra $D$ generated by $x_{1},\ldots,x_{n}$ subject to the relations

[TABLE]

is a $4$-dimensional Artin-Schelter regular algebra and is a normal extension of $A({\sf w})$ by $\Omega=\partial_{1}{{\sf w}}$.

Proof.

It suffices to show that ${\sf p}=(1,q_{2}^{k},\ldots,q_{n}^{k})$ is good. By Lemma 3.2, for every monomial $x_{l_{1}}\ldots x_{l_{m+1}}$ appearing in ${\sf w}$ the product $\prod_{i=1}^{m+1}q_{l_{i}}^{k}$ equals $1$. Hence ${\sf p}$ satisfies the criterion for goodness in Theorem 3.54. ∎

In Corollary 3.15 we will show that the algebra $D$ in Corollary 3.6 is $4$-Calabi-Yau. The next result shows that every $4$-Calabi-Yau central extension of $A({\sf w})$ can be obtained in this way.

Theorem 3.8**.**

Let $A({\sf w})$ be a $m$-Koszul $3$-Calabi-Yau algebra. Then all central extensions of $A({\sf w})$ by a degree-$m$ element can be obtained by the construction in Corollary 3.6. Consequently, they are $4$-Calabi-Yau.

Proof.

Let $D^{\prime}$ be a central extension of $A({\sf w})$. Explicitly, let $\Omega\in D^{\prime}$ be a degree-$m$ central regular element such that $D^{\prime}/(\Omega)\cong A({\sf w})$. Note that $D^{\prime}_{1}=A_{1}=V$. Since the relations for $A$ are concentrated in degree $m$ and span an $n$-dimensional subspace of $TV$ (see 3.2), there exist degree-$m$ elements $f_{1},\ldots,f_{n}$ of $TV$ such that $A=TV/(f_{1},\ldots,f_{n})$, and $f_{2}=\cdots=f_{n}=0$ in $D^{\prime}$, and $\Omega$ is the image of $f_{1}$ in $D^{\prime}$.

Let $y_{1},\ldots,y_{n}$ be a basis of $V$ and write ${\sf w}=y_{1}h_{1}+\cdots+y_{n}h_{n}$. Since $A({\sf w})$ is defined by the relations $h_{1},\ldots,h_{n}$, there is a matrix $P\in{\sf GL}(n)$ such that $(h_{1},\ldots,h_{n})^{t}=P(f_{1},\ldots,f_{n})^{t}$. Define the basis $x_{1},\ldots,x_{n}$ of $V$ by $(x_{1},\ldots,x_{n})=(y_{1},\ldots,y_{n})P$. Then

[TABLE]

Hence $f_{i}={\operatorname{\partial}}_{i}{\sf w}$ where the partial derivatives are taken with respect to the basis $x_{1},\ldots,x_{n}$. Since $\Omega\in D^{\prime}$ is central, the defining relations for the algebra $D$ in Corollary 3.6 are satisfied by $D^{\prime}$. Hence there is a surjective homomorphism $D\to D^{\prime}$. But $D$ and $D^{\prime}$ have the same Hilbert series because $A$ is a quotient of each of them by a central regular element of degree $m$. Thus $D^{\prime}=D$. ∎

3.6. A flat family

The other consequence of Theorem 3.5 we wish to highlight is Theorem 1.3 in the introduction. That result says that if $A({\sf w})$ is a 3-Calabi-Yau algebra on $n$ generators, then the recipe in Corollary 3.6 produces a flat family over ${\mathbb{P}}^{n-1}$ of 4-Calabi-Yau algebras $D_{x}$, $x\in V-\{0\}$, that are central extensions of $A({\sf w})$.

In the statement of Theorem 1.3, we defined $D_{x}:=TV/I_{x}$ where $I_{x}$ is the ideal generated by

[TABLE]

and

[TABLE]

If $\lambda\in\Bbbk-\{0\}$, then $D_{\lambda x}=D_{x}$ so we will often write $x$ for the point in ${\mathbb{P}}(V)$ that corresponds to the line $\Bbbk x$ in ${\mathbb{P}}(V)$. Thus, we have an algebra $D_{x}$ for each $\Bbbk$-point $x\in{\mathbb{P}}(V)$.

Before showing that the $D_{x}$’s provide a flat family we check they have the other properties we want.

Lemma 3.9**.**

Let $A({{\sf w}})$ be a $3$-Calabi-Yau algebra generated by $x_{1},\ldots,x_{n}$.

1. (1)

The algebra $D_{x_{1}}$ is the same as the algebra $D$ constructed in Corollary 3.6. 2. (2)

The algebras $D_{x}$ are 4-Calabi-Yau algebras having central regular elements $\Omega_{x}$ such that $D_{x}/(\Omega_{x})=A({\sf w})$.

Proof.

1 Let $D$ be the algebra constructed in Corollary 3.6.

The hypothesis that ${\sf w}$ is a superpotential says there are elements ${\sf w}_{1},\ldots,{\sf w}_{n}\in V^{\otimes m}$ such that

[TABLE]

By definition, $D=TV/I$ where $I$ is the ideal generated by $\{{\sf w}_{2},\ldots,{\sf w}_{n}\}$ and $\{[x_{i},{\sf w}_{1}]\;|\;i=2,\ldots,n\}$. This is clearly the same as the ideal generated by $\{{\sf w}_{2},\ldots,{\sf w}_{n}\}$ and $\{[x_{i},{\sf w}_{j}]\;|\;1\leq i,j\leq n\}$. We note that

[TABLE]

and

[TABLE]

It is now apparent that $I=I_{x_{1}}$, i.e., $D=D_{x_{1}}$.

2 Since $\dim_{\Bbbk}(\langle V^{*},{\sf w}\rangle)=n$, $\langle x^{\perp},{\sf w}\rangle$ is a codimension-one subspace of $\langle V^{*},{\sf w}\rangle$ or, equivalently, the image of $\langle V^{*},{\sf w}\rangle$ in $TV/I_{x}$ has dimension 1; thus, if $\Omega_{x}$ is a non-zero element in it, then $D_{x}/(\Omega_{x})=A({\sf w})$. In $D_{x}$, $[y,\langle\psi,{\sf w}\rangle]=0$ for all $y\in V$ and all $\psi\in V^{*}$. This says that $\Omega_{x}$ belongs to the center of $D_{x}$. ∎

Remark 3.10**.**

It is reasonable to ask whether as $x$ varies over $V-\{0\}$ every codimension-1 subspace of the degree-$m$ relations for $A({\sf w})$ arises as the degree-$m$ relations for some $D_{x}$. The answer is “yes” and the proof is as follows. Let $R^{\prime}={\rm span}\{{\sf w}_{2}^{\prime},\ldots,{\sf w}_{n}^{\prime}\}$ be a codimension-1 subspace of $\langle V^{*},{\sf w}\rangle$. Let ${\sf w}_{1}^{\prime}\in\langle V^{*},{\sf w}\rangle$ be an element that is not in $R^{\prime}$. Since the linear map $\langle-,{\sf w}\rangle:V^{*}\to V^{\otimes m}$ has rank $n$, there is a basis $\{x,x_{2}^{\prime},\ldots,x_{n}^{\prime}\}$ for $V$ such that ${\sf w}=x{\sf w}_{1}^{\prime}+x_{2}^{\prime}{\sf w}_{2}^{\prime}+\cdots+x_{n}^{\prime}{\sf w}_{n}^{\prime}$. It is now clear that $R^{\prime}=\langle x^{\perp},{\sf w}\rangle$. $\blacklozenge$

We now know that the $D_{x}$’s have the same Hilbert series for all $x\in V-\{0\}$ but in order to complete the proof of Theorem 1.3 we need to construct a single sheaf of algebras $\mathcal{D}$ on ${\mathbb{P}}(V)$ whose fibers at the $\Bbbk$-points $x\in{\mathbb{P}}(V)$ are the algebras $D_{x}$. The precise statement which we wish to prove is the following.

Theorem 3.11**.**

Let $A({{\sf w}})$ be a $3$-Calabi-Yau algebra generated by $x_{1},\ldots,x_{n}$. Then there are flat sheaves $\mathcal{D}$ and $\mathcal{A}({\sf w})$ of graded $\mathcal{O}_{{\mathbb{P}}(V)}$-algebras and a morphism $\mathcal{D}\to\mathcal{A}({\sf w})$ such that for every $\Bbbk$-point $x\in{\mathbb{P}}(V)$, the fibers of $\mathcal{D}$ and $\mathcal{A}({\sf w})$ at $x$ are the algebras $D_{x}$ (defined in Theorem 1.3) and $A({\sf w})$, respectively, and the morphism $\mathcal{D}\to\mathcal{A}({\sf w})$ induces the projection

[TABLE]

Proof.

We continue to work with a fixed basis $\{x_{1},\ldots,x_{n}\}$ for $V$ and write ${\sf w}=x_{1}{\sf w}_{1}+\cdots+x_{n}{\sf w}_{n}$.

Let $\{c_{1},\ldots,c_{n}\}$ be the basis for $V^{*}$ that is dual to $\{x_{1},\ldots,x_{n}\}$. Let $R=\Bbbk[c_{1},\ldots,c_{n}]$ be the polynomial ring on indeterminates $c_{1},\ldots,c_{n}$ placed in degree-one. Thus, ${\mathbb{P}}(V)=\operatorname{Proj}(R)$. Let $E$ be the $R$-algebra generated by $x_{1},\ldots,x_{n}$ modulo the relations

[TABLE]

or, more succinctly, modulo the relations

[TABLE]

Viewing $E$ as a graded $R$-module with respect to $c_{1},\ldots,c_{n}$, we define $\mathcal{D}$ to be the quasi-coherent sheaf of algebras on ${\mathbb{P}}(V)$ whose sections over a basic open set $U_{f}=\operatorname{Spec}(R[f^{-1}]_{0})$ is $E[f^{-1}]_{0}$. Since $E$ has an algebra structure and also has a grading with $x_{1},\ldots,x_{n}$ placed in degree-one, $\mathcal{D}$ is a sheaf of graded $\mathcal{O}_{{\mathbb{P}}(V)}$-algebras on ${\mathbb{P}}(V)$.111The definition of $\mathcal{D}$ does not depend on the basis $x_{1},\ldots,x_{n}$ for $V$.

The fiber $\mathcal{D}_{\mathfrak{p}}$ at a point $\mathfrak{p}\in{\mathbb{P}}(V)$ (e.g., a $\Bbbk$-point, a closed point, or the generic point), which is by definition the pullback of $\mathcal{D}$ along $\operatorname{Spec}(\Bbbk(\mathfrak{p}))\to{\mathbb{P}}(V)$, is obtained from $E$ by replacing the base ring $R$ by $\Bbbk(\mathfrak{p})$ and $c_{i}$ in the relations by $\gamma_{i}:=c_{i}/c\in\Bbbk(\mathfrak{p})$, where $c\in R_{1}$ is a fixed element not belonging to $\mathfrak{p}$. So the fiber $\mathcal{D}_{\mathfrak{p}}$ is the $\Bbbk(\mathfrak{p})$-algebra generated by $x_{1},\ldots,x_{n}$ modulo the relations

[TABLE]

We will show that this algebra is $D_{x}$ with $\Bbbk$ replaced by $\Bbbk(\mathfrak{p})$ and $x$ replaced by $y:=\gamma_{1}x_{1}+\cdots+\gamma_{n}x_{n}\in V\otimes_{\Bbbk}\Bbbk(\mathfrak{p})$. Since $y^{\perp}={\rm span}_{\Bbbk(\mathfrak{p})}\{\gamma_{i}c_{j}-\gamma_{j}c_{i}\;|\;1\leq i,j\leq n\}\subseteq V^{*}\otimes_{\Bbbk}\Bbbk(\mathfrak{p})$, $\langle y^{\perp},{\sf w}\rangle$ is the span of the elements

[TABLE]

Hence $\mathcal{D}_{\mathfrak{p}}$ is the $\Bbbk(\mathfrak{p})$-algebra generated by $x_{1},\ldots,x_{n}$ modulo the relations

[TABLE]

i.e., $\mathcal{D}_{\mathfrak{p}}$ is $D_{y}$ defined over $\Bbbk(\mathfrak{p})$. In particular, if $\mathfrak{p}$ is a $\Bbbk$-point $x\in{\mathbb{P}}(V)$, then the element $y\in V\otimes_{\Bbbk}\Bbbk(\mathfrak{p})=V$ is equal to $x$ when viewed as a point of ${\mathbb{P}}(V)$. So the fiber $\mathcal{D}_{x}$ at $x$ is the algebra $D_{x}$.

Similarly, we define the sheaf of graded $\mathcal{O}_{{\mathbb{P}}(V)}$-algebras $\mathcal{A}({\sf w})$ to be the sheaf associated to the $R$-algebra $F$ generated by $x_{1},\ldots,x_{n}$ modulo the relations $\langle V^{*},{\sf w}\rangle$. As $\mathfrak{p}$ ranges over the points of ${\mathbb{P}}(V)$, we obtain 3-Calabi-Yau algebras $A({\sf w})\otimes_{\Bbbk}\Bbbk(\mathfrak{p})$ over the fields $\Bbbk(\mathfrak{p})$ all having the same Hilbert series as $A({\sf w})$. The canonical surjective algebra homomorphism $E\to F$ induces an epimorphism $\mathcal{D}\to\mathcal{A}({\sf w})$ that specializes to the projection $D_{y}\to A({\sf w})\otimes_{\Bbbk}\Bbbk(\mathfrak{p})$.

Applying Corollary 3.6 with $\Bbbk$ replaced by the residue field $\Bbbk(\mathfrak{p})$ at a point $\mathfrak{p}$ in ${\mathbb{P}}(V)$, we conclude that the dimension of the homogeneous component $\mathcal{D}_{n}$ at the point $\mathfrak{p}$ is the same for all $\mathfrak{p}$. Hence by [16, Lem. II.8.9], each $\mathcal{D}_{n}$ is locally free and $\mathcal{D}$ is therefore flat. So is $\mathcal{A}({\sf w})$ for the same reason. ∎

Alternative proof.

Consider the free sheaf $\mathcal{V}=V\otimes_{\Bbbk}\mathcal{O}_{{\mathbb{P}}(V)}$ over ${\mathbb{P}}(V)$. We denote its dual by $\mathcal{V}^{*}$; note that we have evaluation morphisms

[TABLE]

pairing $\mathcal{V}^{*}$ against the leftmost tensorand.

We will construct $\mathcal{D}$ as $T\mathcal{V}/(\mathcal{R}_{m}\oplus\mathcal{R}_{m+1})$, where $T\mathcal{V}$ is the tensor algebra on the sheaf $\mathcal{V}$ and $\mathcal{R}_{m}$ and $\mathcal{R}_{m+1}$ are subsheaves of $\mathcal{V}^{\otimes m}$ and $\mathcal{V}^{\otimes(m+1)}$, corresponding to 3-3 and 3-4.

Since over every open affine subset $\operatorname{Spec}(R)\subseteq{\mathbb{P}}(V)$ the sheaf $\mathcal{V}$ corresponds to the free $R$-module on the vector space $V$, ${\sf w}$ induces a section of $\mathcal{V}^{\otimes(m+1)}$ (i.e., a morphism $\mathcal{O}\to\mathcal{V}^{\otimes(m+1)}$) denoted in the sequel by the same symbol.

First, let $\mathcal{R}_{m+1}\subseteq\mathcal{V}^{\otimes(m+1)}$ be the image of the morphism

[TABLE]

where the last morphism is the commutator of the leftmost $\mathcal{V}$ and the tensor product $\mathcal{V}^{\otimes m}$. $\mathcal{R}_{m+1}$ specializes to 3-4 at $\Bbbk$-points of $\mathbb{P}(V)$.

Similarly, we define $\mathcal{R}_{m}\subseteq\mathcal{V}^{\otimes m}$ specializing to 3-3 as follows. Since the space of global sections of $\mathcal{O}(1)$ is canonically isomorphic to $V^{*}$, we have a canonical epimorphism $V^{*}\otimes_{\Bbbk}\mathcal{O}\to\mathcal{O}(1)$. The dual of this morphism is a monomorphism $\mathcal{O}(-1)\to V\otimes_{\Bbbk}\mathcal{O}=\mathcal{V}$, whose image specializes to the $1$-dimensional subspace $\Bbbk x\subseteq V$ at each $\Bbbk$-point $x\in{\mathbb{P}}(V)$. Regard $\mathcal{O}(-1)$ as a subsheaf of $\mathcal{V}$ and consider its annihilator $\mathcal{O}(-1)^{\perp}\subseteq\mathcal{V}^{*}$. We define $\mathcal{R}_{m}$ as the image of the sheaf morphism

[TABLE]

As explained above, we can now define $\mathcal{D}$ as $T\mathcal{V}/(\mathcal{R}_{m}\oplus\mathcal{R}_{m+1})$.

The sheaf $\mathcal{A}({\sf w})$ is defined as $T\mathcal{V}/(\mathcal{R})$, where $\mathcal{R}$ is the image of

[TABLE]

These basis-free definitions of $\mathcal{D}$ and $\mathcal{A}({\sf w})$ agree with those in the previous proof, and we can complete the proof using the same argument. ∎

Proof of Theorem 1.5.

This follows from Theorem 1.31 and the general remark that flatness of families of connected graded algebras transports over to their $\operatorname{\sf QGr}$ categories, recorded in Lemma 3.12 below. ∎

Lemma 3.12**.**

Let $R$ be a commutative ring and $D$ a non-negatively graded $R$-algebra with $D_{0}=R$. If the homogeneous components $D_{n}$ are flat over $R$ then so is the abelian category $\operatorname{\sf QGr}(D)$ where $\operatorname{\sf QGr}(D)$ is the category defined in Definition 1.4.

Proof.

The $R$-linear category ${\sf Gr}(D)$ is equivalent to the category of modules over the small $R$-linear category $\mathfrak{d}$ with objects $i\in\mathbb{Z}$ and $\operatorname{Hom}_{\mathfrak{d}}(i,j)=D_{j-i}$. The hypothesis implies that $\mathfrak{d}$ is $R$-flat as a linear category so ${\sf Mod}(\mathfrak{d})$, and hence ${\sf Gr}(D)$, is flat over $R$ by [19, Prop. 3.7]. Since the quotient functor ${\sf Gr}(D)\to\operatorname{\sf QGr}(D)$ is exact and essentially surjective, flatness of $\operatorname{\sf QGr}(D)$ follows from that of ${\sf Gr}(D)$ by [19, Lem. 8.13]. ∎

3.7. Automorphisms

Let $A=A({\sf w})$ and $D=D({\sf w},{\sf p})$ be algebras satisfying the assumptions of Theorem 3.5. We now consider the relation between $\operatorname{Aut}_{\sf gr}(A)$ and $\operatorname{Aut}_{\sf gr}(D)$.

Proposition 3.13 identifies a subgroup of $\operatorname{Aut}_{\sf gr}(A)$ that consists of automorphisms that lift to $D$.

This result does not depend on Theorem 3.5 and its proof will be used in the proof of Theorem 3.5.

Proposition 3.13**.**

Let $A=A({\sf w})$ and $D=D({\sf w},{\sf p})$ be algebras satisfying the hypotheses of Theorem 3.5. Let $\sigma\in\operatorname{Aut}_{\sf gr}(A)$. If every $x_{i}$ is an eigenvector for $\sigma$, then $\sigma$ lifts to an automorphism of $D$.

Proof.

By definition, the relations $f_{i}=\partial_{i}{{\sf w}}$ are the respective images of $x_{i}^{*}\otimes{{\sf w}}$ through the map

[TABLE]

that evaluates the two leftmost tensorands against each other, where $\{x_{i}^{*}\}\subseteq V^{*}$ is the basis dual to $\{x_{i}\}$.

By [24, Thm. 1.1], the automorphisms of $A$ are (identified in the obvious manner with) precisely those linear automorphisms of $V=A_{1}$ which scale the superpotential ${\sf w}\in V^{\otimes{(m+1)}}$. In other words, the element ${\sf w}$ is an eigenvector for the action of $\Gamma=\langle\sigma\rangle$ on $V^{\otimes(m+1)}$ (with the tensor product being taken in the monoidal category of $\Gamma$-modules).

Now suppose that the generators $x_{i}$ are eigenvectors for $\sigma\in GL(V)$. Then each $\Bbbk f_{i}$ is the image of the $\Gamma$-submodule $\Bbbk x_{i}^{*}\otimes\Bbbk{{\sf w}}$ of the left-hand side of the $\Gamma$-module morphism 3-5, so they are $\Gamma$-submodules of $V^{\otimes m}$. Thus, each $f_{i}$ is an eigenvector for $\sigma^{\otimes m}$. Since $x_{1}$ is, by hypothesis, also an eigenvector for $\sigma$, it acts as a scalar on each of the defining relations 3-1 for $D$. ∎

In what follows we assume that the four conditions in Theorem 3.5 are satisfied.

We have the following consequence of Theorem 2.6 on how the respective Nakayama automorphisms interact.

Proposition 3.14**.**

Let $A({\sf w})$ and $D({\sf w},{\sf p})$ be Artin-Schelter regular algebras satisfying the hypotheses, and hence the conclusions, of Theorem 3.5. If $Q={\rm diag}(q_{1},\ldots,q_{n})$, then the Nakayama automorphism of $D({\sf w},{\sf p})$ scales $x_{i}$ by $p_{i}^{-1}q_{i}^{-1}$, i.e., $\nu_{\!{}_{D}}(x_{i})=(p_{i}q_{i})^{-1}x_{i}$ for all $i$.

Proof.

This follows from Theorem 2.6, together with the fact that $\tau$, “conjugation” by $\Omega$, acts by

[TABLE]

and [24, Thm. 1.8] which, in our setting, says that $\nu_{\!{}_{A}}$ scales $x_{i}$ by $q_{i}^{-1}$. ∎

Applying this result to the algebras in Corollary 3.7 we obtain the following result.

Corollary 3.15**.**

Suppose $q_{1}=1$. If ${\sf p}=(1,q_{2}^{-1},\ldots,q_{n}^{-1})$, then $D({\sf w},{\sf p})$ is 4-Calabi-Yau.

Proof.

Proposition 3.14 implies that $\nu_{\!{}_{D}}=\mathrm{id}_{D}$ in this case. ∎

By [28, Cor. 5.4], the homological determinant of the Nakayama automorphism is 1 for all noetherian Artin-Schelter regular algebras. By [24, Thm. 1.6], the same result holds in the non-noetherian case provided the algebra is $m$-Koszul and Artin-Schelter regular. It is conjectured at [24, Conj. 4.12] that this holds for all Artin-Schelter regular algebras. Theorem 3.17 below shows that the normal extensions $D({\sf w},{\sf p})$ produced by Theorem 3.5 have this property.

Before stating that result, we give a relatively self-contained proof of the following auxiliary result. It is essentially [24, Thm. 1.2], with the minor caveat that the homological determinant in [24] is inverse to that of [17]. Nevertheless, [24, Thm. 1.2] appeals to [32], which agrees with [17]. For this reason, [24, Thm. 1.2] is correct for the version of the homological determinant used here and in [17, 32], but not precisely as stated, for the version of $\operatorname{hdet}$ defined in [24, $\S$2.2].

Throughout the statement and the proof of Lemma 3.16, and in later sections, we identify each graded algebra automorphism of $A=TV/(R)$ (when the relations $R$ have degree $\geq 2$) with the automorphism of $TV$ that agrees on the degree-1 component $V$.

Lemma 3.16**.**

Let $A$ be an $m$-Koszul Artin-Schelter regular algebra and let ${\sf w}\in V^{\otimes\ell}$ be a $Q$-twisted superpotential such that $A\cong A({\sf w},\ell-m)$ (see Corollary 3.7). If $\sigma\in\operatorname{Aut}_{\sf gr}(A)$, then ${\sf w}$ is an eigenvector for $\sigma^{\otimes\ell}$ with eigenvalue $\operatorname{hdet}(\sigma)$.

Proof.

By Proposition 2.13, $\operatorname{hdet}(\sigma)^{-1}$ is the scalar by which $\sigma\triangleright$ acts on the degree-$(-\ell)$ component of $\operatorname{Ext}^{d}_{A}(\Bbbk,\Bbbk)$. Resolving ${}_{A}\Bbbk$ by projective modules $P_{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}$, this amounts to the action obtained by pre-composing morphisms in $\operatorname{Hom}_{A}(P_{d},\Bbbk)$ with the action of $(\varphi^{P_{d}})^{-1}$.

Now, according to [12, equation (4.2)], the minimal resolution $P_{{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}}$ consists of free left $A$-modules generated by the partial derivatives of ${{\sf w}}$, and the left-most projective $P_{d}$ is precisely the free $A$-module with basis ${{\sf w}}$. It follows from that equation that $(\varphi^{P_{d}})^{-1}$ is the application of $(\sigma^{-1})^{\otimes(\ell+1)}$ to

[TABLE]

Proposition 2.13 now implies that $\operatorname{hdet}(\sigma)^{-1}$ is the eigenvalue associated to the eigenvector ${{\sf w}}$ of $(\sigma^{-1})^{\otimes\ell}$, hence the conclusion. ∎

Theorem 3.17**.**

Let $A=A({\sf w})$ and $D=D({\sf w},{\sf p})$ be Artin-Schelter regular connected graded algebras satisfying all conditions in Theorem 3.5. Then $\operatorname{hdet}(\nu_{\!{}_{D}})=1$.

Proof.

Theorem 2.6 shows that the restriction $\nu_{\!{}_{D}}|_{A}$ is $\tau|_{A}\circ\nu_{\!{}_{A}}$, where $\tau$ is defined on $D$ by

[TABLE]

and $\tau|_{A}$ is its descent to $A$ modulo $\Omega$. Theorem 2.10 shows that

[TABLE]

where $\lambda$ is defined by $\nu_{\!{}_{D}}(\Omega)=\lambda\Omega$. Hence, by Theorem 2.6 and the multiplicativity of $\operatorname{hdet}$ ([17, Prop. 2.5]),

[TABLE]

But $\operatorname{hdet}(\nu_{\!{}_{A}})=1$ by [24, Thm. 1.6], so the rightmost factor in 3-7 is 1.

On the other hand, since $\tau$ fixes $\Omega$, the equation $\nu_{\!{}_{D}}|_{A}=\tau\circ\nu_{\!{}_{A}}$ shows that $\lambda$ is defined by

[TABLE]

where we have abused notation by using $\nu_{A}$ to denote the automorphism of the free algebra $TV$ that agrees with $\nu_{A}$ on $A_{1}=V$ (see the discussion preceding Lemma 3.16).

By [24, Thm. 1.6] and Lemma 3.16, $\nu_{\!{}_{A}}$ fixes ${\sf w}$, and [24, Thm. 1.8] shows that the same automorphism scales $x_{1}$ by $q_{1}^{-1}$. It now follows from the fact that $x_{1}\Omega$ is a sum of monomials appearing in ${\sf w}$ that $\nu_{\!{}_{A}}$ scales $\Omega$ by $q_{1}$. In other words, 3-8 implies $\lambda=q_{1}$.

Finally, [24, Thm. 1.2] shows that $\operatorname{hdet}(\tau|_{A})$ is equal to the eigenvalue of the $\tau|_{A}$-eigenvector ${\sf w}$ which condition 3 in Theorem 3.5 shows is equal to $q_{1}^{-1}$.

To summarize:

• •

the leftmost factor in the right-hand side of 3-7 is $q_{1}$;

• •

the middle factor is $q_{1}^{-1}$;

• •

the rightmost factor is $1$.

The desired conclusion that $\operatorname{hdet}(\nu_{\!{}_{D}})=1$ now follows from 3-7. ∎

3.8. Zhang twists

We now consider how our construction of normal extensions behaves with respect to Zhang twists. Zhang twisting produces new graded algebras by using families of graded $k$-linear automorphisms called twisting systems [34]. The idea extends the idea in [3, §8]. Here we only consider the Zhang twist associated to a graded algebra automorphism of $A$.

Definition 3.18**.**

Let $A$ be a graded algebra and $\sigma$ an automorphism on $A$. The Zhang twist ${}^{\sigma}A$ of $A$ by $\sigma$ is defined by ${}^{\sigma}A:=A$ as an graded vector space with a new multiplication $a*b:=\sigma^{\deg b}(a)b$ where $a,b\in A$ are homogeneous elements.222Our definition of Zhang twist agrees with that of [27] but differs from [24]. $\blacklozenge$

For each $f\in V^{\otimes r}$, set ${}^{\sigma}f:=(\mathrm{id}_{V}\otimes\sigma\otimes\cdots\otimes\sigma^{r-1})(f)$. Then the relations in ${}^{\sigma}A$ are given by ${}^{\sigma}f$ where $f$ runs over all relations in $A$.

Proposition 3.19**.**

Let $A({{\sf w}})$ and $D({\sf w},{\sf p})$ be Artin-Schelter regular algebras as in Theorem 3.5. Let $\sigma$ be an automorphism of $A({{\sf w}})$ such that $\sigma(x_{i})=s_{i}x_{i}$ for some $s_{i}\in\Bbbk^{\times}$. Then $\sigma$ lifts to an automorphism of $D({\sf w},{\sf p})$, and

[TABLE]

where ${\sf p}^{\prime}=(p^{\prime}_{1},\ldots,p^{\prime}_{n})$ is defined by $p^{\prime}_{i}=p_{i}s_{1}s_{i}^{m}(\operatorname{hdet}_{A}(\sigma))^{-1}$.

Proof.

As observed in Proposition 3.13 and its proof, $\sigma$ lifts to $D$, and $\sigma^{\otimes m}$ scales each $f_{i}$ in $V^{\otimes m}$. The algebra ${}^{\sigma}D({\sf w},{\sf p})$ is defined by the relations ${}^{\sigma}f_{i}={}^{\sigma}[x_{i},\partial_{1}{\sf w}]_{p_{i}}=0$ ($i=2,\ldots,n$). By [24, Thm. 5.5], ${}^{\sigma}A({\sf w})=A({}^{\sigma}{\sf w})$. Note that $\partial_{i}{}^{\sigma}{\sf w}=s_{i}^{-1}(\operatorname{hdet}_{A}(\sigma))({}^{\sigma}f_{i})$. By Lemma 3.16,

[TABLE]

and hence $\sigma^{\otimes m}(f_{i})=s_{i}^{-1}(\operatorname{hdet}_{A}(\sigma))f_{i}$. Therefore

[TABLE]

The defining ideal of ${}^{\sigma}D({\sf w},{\sf p})$ is generated by $\partial_{i}{}^{\sigma}{\sf w}$ and $[x_{i},\partial_{1}{}^{\sigma}{\sf w}]_{p^{\prime}_{i}}$ ($i=2,\ldots,n$). ∎

4. Proof of the main theorem

In this section we prove Theorem 3.5. Let $A=A({\sf w})$ and $D=D({\sf w},{\sf p})$ be the algebras as in the theorem.

4.1.

In this subsection we show that 1 $\Rightarrow$ 4 $\Leftrightarrow$ 3 $\Rightarrow$ 2.

The proof that 1 $\Rightarrow$ 4 also shows that $\Omega$ is a regular element in $D({\sf w},{\sf p})$.

1 $\Rightarrow$ 4 We first show that the relations 3-1 form a minimal set of relations for $D({\sf w},{\sf p})$. Suppose not. Since ${\operatorname{\partial}}_{1}{\sf w},\ldots,{\operatorname{\partial}}_{n}{\sf w}$ are linearly independent (see 3.2), there are scalars $a_{i},b_{ij},c_{ij}\in\Bbbk$, not all zero, such that

[TABLE]

in $TV$. In other words,

[TABLE]

Let $R$ be the subspace of $TV$ spanned by ${\operatorname{\partial}}_{1}{\sf w},\ldots,{\operatorname{\partial}}_{n}{\sf w}$. The left-hand side of 4-1 belongs to $V\otimes R$ and the right-hand side of 4-1 belongs to $R\otimes V$. Since $(V\otimes R)\cap(R\otimes V)=\Bbbk{\sf w}$ by [24, Prop. 2.12], the left-hand side of 4-1 is a scalar multiple of ${\sf w}=x_{1}({\operatorname{\partial}}_{1}{\sf w})+\cdots+x_{n}({\operatorname{\partial}}_{n}{\sf w})$. This implies that all $a_{i}=0$ and all $b_{ij}=0$. Similarly, the right-hand side of 4-1 is a scalar multiple of ${\sf w}=q_{1}({\operatorname{\partial}}_{1}{\sf w})x_{1}+\cdots+q_{n}({\operatorname{\partial}}_{n}{\sf w})x_{n}$ which implies that all $a_{i}=0$ and all $c_{ij}=0$. This is a contradiction. Hence 3-1 is a minimal set of relations.

Thus the minimal free resolution of ${}_{D}\Bbbk$ is of the form

[TABLE]

That of ${}_{A}\Bbbk$ is of the form

[TABLE]

These resolutions determine the Hilbert series of $D$ and $A$ and it is then easy to see that $h_{D}(t)=h_{A}(t)(1-t^{m})^{-1}$. Therefore by Proposition 2.1, $\Omega$ is a regular element of $D({\sf w},{\sf p})$.

Since $\Omega$ is regular, we can define an automorphism $\tau\in\operatorname{Aut}_{\sf gr}(D({\sf w},{\sf p}))$ by the formula in 2-1. The argument in the proof of Lemma 3.4 shows that $\tau(x_{i})=p_{i}^{-1}x_{i}$ for each $i=1,\ldots,n$. Of course, $\tau$ can also be viewed as an element in ${\sf GL}(V)$ and, as observed in the proof of Proposition 3.13, ${\sf w}\in V^{\otimes(m+1)}$ and $f_{1}\in V^{\otimes m}$ are eigenvectors for $\tau^{\otimes(m+1)}$ and $\tau^{\otimes m}$, respectively. Since $\tau(\Omega)=\Omega$, $\tau^{\otimes m}(f_{1})=f_{1}$. Since ${\sf w}=x_{1}f_{1}+\cdots+x_{n}f_{n}$, the $\tau^{\otimes(m+1)}$-eigenvalue of ${\sf w}$ is the same as the $\tau$-eigenvalue of $x_{1}$. Therefore $\tau^{\otimes(m+1)}({\sf w})=p_{1}^{-1}{\sf w}=q_{1}^{-1}{\sf w}$. Since the automorphism $\varphi$ defined in Theorem 3.54 is the inverse of $\tau$, 4 holds.

3 $\Leftrightarrow$ 4 Since ${{\sf w}}=\sum_{i,j}x_{i}{\sf M}_{ij}x_{j}$, $\varphi^{\otimes(m+1)}$ scales ${\sf w}$ by $q_{1}$ if and only if it scales $x_{i}x_{l_{1}}\cdots x_{l_{m-1}}x_{j}$ by $q_{1}$ for all $i,j$ and all words $x_{l_{1}}\cdots x_{l_{m-1}}$ appearing in ${\sf M}_{ij}$. But $\varphi^{\otimes(m+1)}(x_{i}x_{l_{1}}\cdots x_{l_{m-1}}x_{j})=p_{i}p_{j}p_{l_{1}}\cdots p_{l_{m-1}}x_{i}x_{l_{1}}\cdots x_{l_{m-1}}x_{j}$ by definition of $\varphi$.

3 $\Rightarrow$ 2 Fix $1\leq i,j\leq d$ and write ${\sf M}_{ij}=\sum_{l}a_{l}x_{l_{1}}\ldots x_{l_{m-1}}$ where every $a_{l}\neq 0$ and $l=(l_{1},\ldots,l_{m-1})$. As described in the proof of Lemma 3.4, $[x_{j},\Omega]_{p_{j}}=0$ for each $j$. Hence

[TABLE]

The right-most side is zero by 3.

4.2.

To complete the proof of Theorem 3.5 we show that 2 $\Rightarrow$ 1. The proof of this occupies the rest of §4.

First we identify a candidate for the minimal projective resolution of the trivial module ${}_{D}\Bbbk$. The minimal resolution of $\Bbbk$ as a left $A$-module is

[TABLE]

By definition, the relations defining $D$ are $f_{i}$ and $g_{i}:=[x_{i},\partial_{1}{{\sf w}}]_{p_{i}}$ for $2\leq i\leq n$. We aggregate them into two column vectors of size $2(n-1)\times 1$, namely

[TABLE]

The subscripts $l$ and $r$ denote “left” and “right” respectively. There are unique matrices ${\sf M}_{l}$ and ${\sf M}_{r}$ such that

[TABLE]

Let $P_{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}$ denote the sequence

[TABLE]

of left $D$-modules and $D$-module homomorphisms. Eventually, we will show that

[TABLE]

is a minimal resolution of the trivial $D$-module ${}_{D}\Bbbk$.

As usual, when working with resolutions, we place $\Bbbk$ in homological degree $-1$ in 4-3, 4-5, and 4-6.

We denote the homology of the complex 4-6 by $H_{i}$, $0\leq i\leq 4$. Each $H_{i}$ is a graded vector space; its homogeneous components will be denoted by $H_{i\ell}$. The dimension of $H_{i\ell}$ will be denoted by $h_{i\ell}$ or $(h_{i})_{\ell}$.

4.3. Notation

Let $X$ be a matrix. For subsets $S$ and $T$ of the set of rows and vectors respectively, we denote by $X_{S,T}$ (or $X_{ST}$ if there is no danger of ambiguity) the matrix consisting of only those entries of $X$ whose row and column indices belong to $S$ and $T$. Moreover, if the sets $S$ and $T$ are one-sided intervals, we simply substitute the corresponding inequality for the set.

For example, $X_{\geq 2,\leq 6}$ is the sub-matrix of $X$ consisting of entries $x_{ij}$ for $i\geq 2$ and $j\leq 6$.

We apply the same convention to row or column vectors, except that we adorn these with a single symbol as in ${{\sf v}}_{\geq 5}$ denoting the vector consisting of the entries of ${{\sf v}}$ with index $\geq 5$.

We use the symbol   for no condition at all, e.g., $X_{{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}}=X$, ${{\sf v}}_{{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}}={{\sf v}}$, and $X_{\geq 8,{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}}$ is the matrix obtained from $X$ by deleting the first 7 rows.

The hat indicates the negation of the condition in question (or the complement of the set). For instance, $X_{{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}\widehat{j}}$ is the matrix obtained from $X$ by discarding its $j^{th}$ column.

We define

[TABLE]

4.4.

The next lemma follows immediately from the definitions of ${\sf M}_{l}$ and ${\sf M}_{r}$ in 4-4. The horizontal and vertical bars in its statement split the respective matrices into blocks of equal size.

Lemma 4.1**.**

The $2(n-1)\times n$ matrix ${\sf M}_{r}$ is

[TABLE]

The $n\times 2(n-1)$ matrix ${\sf M}_{l}$ is

[TABLE]

Lemma 4.2**.**

If condition 2 in Theorem 3.5 holds, then the diagram $0\to P_{{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}}\to\Bbbk\to 0$ in 4-5 is a complex.

Proof.

The composition $P_{1}\to P_{0}\to\Bbbk$ is zero because, the image of the map $P_{1}\to P_{0}=D$ is the maximal ideal $(x_{1},\ldots,x_{n})$. The definitions of ${\sf M}_{l}$ and ${\sf M}_{r}$, respectively, imply that the compositions $P_{4}\to P_{3}\to P_{2}$ and $P_{2}\to P_{1}\to P_{0}$ are zero. It remains to show that the composition

[TABLE]

is zero, i.e., that the entries of the $n\times n$ matrix {\sf M}_{l}{\sf M}_{r}\in M_{n}\big{(}V^{\otimes(2m-1)}\big{)} are zero in $D$.

To this end, we consider the product of the $i^{th}$ row of ${\sf M}_{l}$ by the $j^{th}$ column of ${\sf M}_{r}$. The $i^{th}$ row of ${\sf M}_{l}$ is

[TABLE]

and the $j^{th}$ column of ${\sf M}_{r}$ is

[TABLE]

Since ${{\sf x}}^{t}{\sf M}=(Q{{\sf f}})^{t}$, the product of the left-hand side of 4-7 and the top half of 4-8 is

[TABLE]

Since ${\sf M}{{\sf x}}={{\sf f}}$, the product of the right-hand half of 4-7 by the bottom half of 4-8 is

[TABLE]

Upon adding 4-9 and 4-10 the terms containing $x_{1}$ cancel out, and the remaining sums, in the four cases

1. (1)

$i=1=j$; 2. (2)

$i=1\neq j$; 3. (3)

$i\neq 1=j$; 4. (4)

$i\neq 1\neq j$

precisely coincide with the respective expressions in the statement of Theorem 3.5. The conclusion then follows from our assumption that these vanish in $D$. ∎

Remark 4.3**.**

A simultaneous change of basis for the spaces spanned by $x_{i}$ and $f_{i}$ for $i\geq 2$ induces, as explained in [1, $\S$2], a conjugation of the diagonal matrix $Q$ by a block diagonal scalar matrix of the form $T=\operatorname{diag}(1,T^{\prime})$, where $T^{\prime}$ is an $(n-1)\times(n-1)$ matrix. The resulting matrix $TQT^{-1}$ is then of the form $\operatorname{diag}(q_{1},Q^{\prime})$ for some $Q^{\prime}\in M_{n-1}(\Bbbk)$.

Under this basis change the above proof of Lemma 4.2 consists of an identity involving the entries of the lower right-hand corner $Q^{\prime}$ of $Q$, and hence functions equally well for any diagonalizable $Q^{\prime}$. But then, by Zariski continuity, it is valid for any $Q^{\prime}$ whatsoever. This justifies the remark in §3.4 that the proof of Theorem 3.5 extends to the case $Q=\operatorname{diag}(1,Q^{\prime})$. $\blacklozenge$

4.5. Inductive argument

Define $Z:=\{a\in D\;|\;a\Omega=0\}$ and $z_{i}:=\dim_{\Bbbk}(Z_{i})$.

Lemma 4.4**.**

Suppose condition 2 in Theorem 3.5 holds. If $(h_{3})_{k-m}=z_{k-2m-1}=z_{k-2m}=0$, then $(h_{3})_{k}=0$.

Proof.

Let ${\bf\varphi}^{t}=(\varphi_{1},\ldots,\varphi_{n})$ be an element in $D_{k-2m}^{d}$ representing a class in $(H_{3})_{k}$. Thus, ${\bf\varphi}^{t}{\sf M}_{l}=0$ in $D$.

Let $\eta:=\varphi^{t}{\sf M}$. Because $f_{1}=\Omega=0$ in $A$, the expression for ${\sf M}_{l}$ in Lemma 4.1 implies that

[TABLE]

in $A$. Hence $\eta_{1}({\sf x}_{\geq 2})^{t}=0$ and $\eta_{\geq 2}=0$ in $A$. It also follows that

[TABLE]

in $A$. We have obtained $\eta_{1}{\sf x}^{t}=0$. The exact sequence 4-3 ensures $\eta_{1}=0$, and hence $\varphi^{t}{\sf M}=0$ in $A$.

Again by the exactness of 4-3, ${\bf\varphi}^{t}$ is in the image of $\cdot{{\sf x}}^{t}$ modulo $\Omega$, i.e.,

[TABLE]

for some $u\in D_{k-2m-1}$ and a column vector ${{\sf y}}$ with entries in $D_{k-3m}$. The equations 4-4 show that the first summand of 4-11 is annihilated by right multiplication by ${\sf M}_{l}$, and hence ${{\sf y}}^{t}\Omega{\sf M}_{l}=0$ in $D$.

Since $[\Omega,{\sf M}_{ij}]_{q_{1}^{-1}p_{i}p_{j}}=0$, Lemma 4.1 implies that $\Omega{\sf M}_{l}$ equals ${\sf M}_{l}\Omega$ up to multiplication on the left by an invertible $n\times n$ diagonal matrix and on the right by an invertible diagonal matrix of size $2(n-1)$, namely

[TABLE]

It follows that ${{\sf y}}^{t}\operatorname{diag}(q_{1},p_{2},\ldots,p_{n}){\sf M}_{l}$ belongs to $Z_{k-2m}^{d-1}\oplus Z_{k-2m-1}^{d-1}$ which is assumed to be zero, and hence ${{\sf y}}^{t}\operatorname{diag}(q_{1},p_{2},\ldots,p_{n})$ represents a class in $(H_{3})_{k-m}$ which is again assumed zero. In conclusion we have ${{\sf y}}^{t}\operatorname{diag}(q_{1},p_{2},\ldots,p_{n})=v{{\sf x}}^{t}$ for some $v\in D(-2m-1)$ and hence, by 4-11,

[TABLE]

In other words $\varphi^{t}$ is a boundary, so the homology $(H_{3})_{k}$ vanishes, as desired. ∎

Now define

[TABLE]

It follows from the exact sequence $0\to Z(-m)\to D(-m)\stackrel{{\scriptstyle\cdot\Omega}}{{\longrightarrow}}D\to A\to 0$ that $e_{k}\geq 0$ (as noted in 2-6) and

[TABLE]

for all $k$. Moreover, subtracting

[TABLE]

from the same expression with $d^{\prime}$ in place of $d$ (in which case the right-hand side $=0$) we get

[TABLE]

This is now sufficient preparation for

Lemma 4.5**.**

Suppose condition 2 in Theorem 3.5 holds. Let $k$ be an integer. If $e_{i}=0$ for all $i\leq k-1$ and $z_{j}=0$ for all $j\leq k-2m$, then $e_{k}=z_{k+1-2m}=0$.

Proof.

By Lemma 4.4, the vanishing of the $z_{j}$’s implies that $(h_{3})_{k}=0$ (by an induction argument since $(h_{3})_{\ell}=0$ when $\ell<0$). Equation 4-13 then gives

[TABLE]

so $e_{k}=0$ and $(h_{2})_{k}+(h_{4})_{k}=0$.

Finally, $z_{k+1-2m}=0$ because the $e_{i}$’s vanish and $e_{k+1-m}=e_{k+1-2m}+z_{k+1-2m}$ (by 4-12). ∎

To complete the proof of Theorem 3.5 we must show that $D$ is Artin-Schelter regular of dimension 4. By Proposition 2.1, $D$ is Artin-Schelter regular of dimension 4 if $h_{D}(t)=h_{A}(t)\left(1-t^{m}\right)^{-1}$, i.e., if $e_{k}=0$ for all $k$. Certainly, $e_{k}=z_{k}=0$ if $k<0$; Lemma 4.5 then provides the induction step thereby showing that the vanishing of $e_{k}$ and $z_{k}$ perpetuates for all $k\geq 0$.

The proof of Theorem 3.5 is now complete.

5. Examples

In this section we apply Theorem 3.5 to some important classes of 3-dimensional Artin-Schelter regular algebras thereby producing some new 4-dimensional Artin-Schelter regular algebras.

5.1. The Calabi-Yau case

As observed in Corollary 3.6, if $A({\sf w})$ is 3-Calabi-Yau, then ${\sf p}=(1,\ldots,1)$ is good so the procedure in Theorem 3.5 produces at least one 4-Calabi-Yau algebra that is a normal extension of $A({\sf w})$. It is, in fact, a central extension. The 3-Calabi-Yau algebras of finite Gelfand-Kirillov dimension have been classified by Mori-Smith [25] and Mori-Ueyama [26]. They are noetherian domains.

Proposition 5.1**.**

For all the 3-Calabi-Yau algebras $A$ classified in [25, Table 2] and [26, Table 2] and all numberings of the generators, the procedure described in Corollary 3.6 produces a noetherian $4$-dimensional Artin-Schelter regular central extension of $A$.

Non-noetherian Calabi-Yau algebras are less well understood, but they do exist and a number of them fit into our framework. For example, [13] constructs, usually non-noetherian, $3$-Calabi-Yau algebras from superpotentials ${{\sf w}}$ defined in terms of combinatorial objects known as Steiner systems. The idea for using Steiner (triple) systems to construct 3-Calabi-Yau algebras is due to Mariano Suárez-Alvarez [31]. The smallest Steiner triple system is provided by the points and lines in the Fano plane, ${\mathbb{P}}^{2}_{{\mathbb{F}}_{2}}$, and the 3-Calabi-Yau algebra in that case is the one studied by Smith in [29].

Proposition 5.2**.**

The $3$-Calabi-Yau algebras described in the theorem in the introduction of [13] satisfy the hypotheses of Corollary 3.6 and therefore admit $4$-dimensional Artin-Schelter regular central extensions.

5.2. Generic cases with finite Gelfand-Kirillov dimension

The generic $3$-dimensional Artin-Schelter regular algebras of finite Gelfand-Kirillov dimension are listed in [1, Tables (3.9) and (3.11)]. In the terminology of 1.6, they have type $(1221)$ or $(1331)$.

As remarked after Theorem 3.5, there is a version of Corollary 3.6 in which the relations for $D({\sf w},{\sf p})$ are defined by using $f_{k}$ in place of $f_{1}$. If we do that we set $p_{k}:=q_{k}$ instead of $p_{1}:=q_{1}$.

Let $\zeta_{n}$ be a primitive $n^{th}$ root of unity.

Proposition 5.3**.**

For each 3-dimensional regular algebra $A$ in Tables (3.9) and (3.11) of [1] and each index $k=1,\ldots,n$ of the relation ${\operatorname{\partial}}_{k}{\sf w}$ for $A({\sf w})=A$ that we omit when defining $D({\sf w},{\sf p})$, Tables 1 and 2 list all tuples ${{\sf p}}=(p_{1},\ldots,p_{n})$ that satisfy condition 4 in Theorem 3.5. For each such ${{\sf p}}$, $D({\sf w},{{\sf p}})$ is a 4-dimensional Artin-Schelter regular normal extension of $A$. (In Tables 1 and 2, $\lambda$ is an arbitrary non-zero scalar.)

5.3.

In [20], four classes of 4-dimensional Artin-Schelter regular algebras of type $(12221)$ were discovered. They are labeled by $A(p)$, $B(p)$, $C(p)$, and $D(v,p)$, with parameters $0\neq p\in\Bbbk$ and $v\in\Bbbk$. If the field $\Bbbk$ is algebraically closed, then these algebras are all the noetherian regular algebras of type $(12221)$ that satisfy the $(m_{2},m_{3})$-generic condition in [20]; roughly speaking, these algebras are generic in terms of the $A_{\infty}$-structure on their Yoneda Ext-algebras. In [20], the regularity of the algebras $A(p)$, $B(p)$, $C(p)$ was proved in a computational way using Bergman’s diamond lemma [5], and that of $D(v,p)$ was deduced from the fact that it is an Ore extension of a skew polynomial ring. Among these algebras, all $A(p)$ and $D(v,p)$ appear as $D({\sf w},{\sf p})$ in Theorem 3.5 so their regularity may also be proved as a consequence Proposition 5.3.

Proposition 5.4**.**

1. (1)

Consider the 3-dimensional cubic regular algebra of type $S_{2}$ in Table (3.9) of **[1]**. Then the normal extension at $k=2$ with $p_{1}=(-\alpha)^{1/2}$ is the algebra $A(-p_{1})$ defined in **[20, Thm. A]**. The normal extension at $k=1$ with $p_{2}=\alpha^{-1/2}$ is isomorphic to $A(-p_{2})$ after interchanging the variables $x$ and $y$. 2. (2)

Consider the 3-dimensional cubic regular algebra of type $S_{1}$ in Table (3.9) of **[1]**. Then the normal extension at $k=2$ with $p_{1}=\alpha^{1/2}$ is the algebra $D(a,-p_{1})$ defined in **[20, Thm. A]**. The normal extension at $k=1$ with $p_{2}=\alpha^{-1/2}$ is isomorphic to $D(-ap_{2}^{2},-p_{2})$ after interchanging the variables $x$ and $y$.

Proof.

Let $A$ be the algebra of type $S_{2}$, $k=2$, and $p:=p_{1}=(-\alpha)^{1/2}$. “The” relations for $A$ are

[TABLE]

Thus, “the” relations for $D({\sf w},{\sf p})$ are $f_{1}$ and

[TABLE]

This description agrees with the defining relation of $A(-p)$ up to scalar multiplication. Similar routine calculations settle the other cases. ∎