Computation of point modules of finitely semi-graded rings

TL;DR

This paper computes the point modules of finitely semi-graded rings, including quantum affine spaces and non-graded quantum algebras, expanding understanding of their module structures.

Contribution

It introduces methods to compute point modules for finitely semi-graded rings, especially for non-graded quantum algebras, which was previously less understood.

Findings

Point modules for quantum affine n-space are parametrized.

Set of point modules for certain non-ℕ-graded quantum algebras is explicitly computed.

Provides new insights into the structure of semi-graded rings.

Abstract

In this paper we compute the set of point modules of finitely semi-graded rings. In particular, from the parametrization of the point modules for the quantum affine n-space, the set of point modules for some important examples of non $\mathbb{N}$-graded quantum algebras is computed.

Equations

$$E=\{\begin{array}{ll}{\mathbb{P}}^2\text{if}{q}_{12}{q}_{23}=q_{13},& \\ \{(x_0:y_0:z_0)\in {\mathbb{P}}^2\mid x_0{y}_0{z}_0=0\}\text{if}{q}_{12}{q}_{23}\ne q_{13},& \end{array}$$

$$\left[\begin{array}{ccc}{y}_0& -q_{12}{x}_0& 0\\ z_0& 0& -q_{13}{x}_0\\ 0& z_0& -q_{23}{y}_0\end{array}\right]\left[\begin{array}{c}{x}_1\\ y_1\\ z_1\end{array}\right]=0.$$

$$\sigma :E\to {\mathbb{P}}^2,\sigma (p_0):=p_1.$$

$$\left[\begin{array}{cccccccc}{x}_2^0& -q_{12}{x}_1^0& 0& 0& 0& \cdots & 0& 0\\ x_3^0& 0& -q_{13}{x}_1^0& 0& 0& \cdots & 0& 0\\ x_4^0& 0& 0& -q_{14}{x}_1^0& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ x_n^0& 0& 0& 0& 0& \cdots & 0& -q_{1n}{x}_1^0\\ 0& x_3^0& -q_{23}{x}_2^0& 0& 0& \cdots & 0& 0\\ 0& x_4^0& 0& q_{24}{x}_2^0& 0& \cdots & 0& 0\\ 0& x_5^0& 0& 0& q_{25}{x}_2^0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ 0& x_n^0& 0& 0& 0& \cdots & 0& -q_{2n}{x}_2^0\\ ⋮& ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ 0& 0& 0& 0& 0& \cdots & x_n^0& -q_{n-1,n}{x}_{n-1}^0\end{array}\right]\left[\begin{array}{c}{x}_1^1\\ x_2^1\\ ⋮\\ \\ ⋮\\ \\ ⋮\\ x_{n-1}^1\\ x_n^1\end{array}\right]=0.$$

$$\sigma :E\to {\mathbb{P}}^{n-1},\sigma (p_0):=p_1.$$

$$E=\bigcap _{f\in \mathcal{F}}V(f)=V(I_F),$$

$$F_n(B):=B_0\oplus \cdots \oplus B_n.$$

$$x_{i}r-c_{i,r}{x}_i\in R.$$

$$x_j{x}_i-c_{i,j}{x}_i{x}_j\in R+R{x}_1+\cdots +R{x}_n.$$

$$x_{i}r=c_{i,r}{x}_i.$$

$$x_j{x}_i=c_{i,j}{x}_i{x}_j.$$

$$F_m:=\{\begin{array}{ll}R,& \text{ if }m=0\\ \{f\in A|deg(f)\le m\},& \text{ if }m\ge 1\end{array}$$

$$A\cong R[z_1;{\theta }_1]\dots [z_n;{\theta }_n].$$

$${\mathcal{B}}_0$$

$$S$$

$$S\to T$$

$$P(B)$$

$$\displaystyle[M]\$$

$$(P(B)/\sim )$$

$$\displaystyle[[M]]\$$

Full text

Key words and phrases. Point modules, point functor, Zariski topology, strongly noetherian algebras, finitely semi-graded rings, skew $PBW$ extensions.

2010 Mathematics Subject Classification. Primary: 16S38. Secondary: 16W50, 16S80, 16S36.

1 Introduction

In algebraic geometry a key role play the point modules and its parametrization. The point modules of algebras, and the spaces parameterizing them, were first studied by Artin, Tate and Van den Bergh ([2]) in order to complete the classification of Artin-Schelter regular algebras of dimension $3$. Many commutative and non-commutative graded algebras have nice parameter spaces of point modules. For example, let $K$ be a field and $A=K[x_{0},\dots,x_{n}]$ be the commutative $K$-algebra of usual polynomials, then there exists a bijective correspondence between the projective space $\mathbb{P}^{n}$ and the set of isomorphism classes of point modules of $A$; this correspondence generalizes to any finitely graded commutative algebra generated in degree $1$. Similarly, the point modules of the quantum plane and the Jordan plane are parametrized by $\mathbb{P}^{1}$. The study of the point modules and its parametrization has been concentrated in finitely graded algebras (see [2], [4], [6], [7], [15], [16]), in the present paper we are interested in the computation of point modules for algebras that are not necessarily $\mathbb{N}$-graded, examples of such algebras are the quantum algebra $\mathcal{U}^{\prime}(so(3,K))$, the dispin algebra \mbox{{\cal U}}(osp(1,2)), the Woronowicz algebra \mbox{{\cal W}}_{\nu}(\mathfrak{sl}(2,K)), among many others. In [11] were introduced the semi-graded rings and for them was proved a classical theorem of non-commutative algebraic geometry, the Serre-Artin-Zhang-Verevkin theorem about non-commutative projective schemes (see also [8]). Thus, the semi-graded rings are nice objects for the investigation of geometric properties of non-commutative algebras that are not $\mathbb{N}$-graded, and include as a particular case the class of finitely graded algebras. In [10] was initiated the study the point modules of finitely semi-graded rings, in the present paper we complete this investigation and we will compute the point modules of finitely semi-graded rings generated in degree one. In particular, from the parametrization of the point modules for the quantum affine $n$-space, the set of point modules for some important examples of non $\mathbb{N}$-graded quantum algebras is computed

The paper is organized as follows: In the first section we recall some basic facts and examples on point modules for classical $\mathbb{N}$-graded algebras, we include a complete proof (usually not available in the literature) of the parametrization of the point modules for the quantum affine $n$-space (Example 1.7); we review the definition and elementary properties of semi-graded rings, in particular, the subclass of finitely semi-graded rings and modules. As was pointed out above, the finitely semi-graded rings generalize the finitely graded algebras, so in order present sufficient examples of finitely semi-graded rings not being necessarily finitely graded algebras, we recall in the last part of the first section the notion of skew $PBW$ extension defined firstly in [9]. Skew $PBW$ extensions represent a way of describing many important non-commutative algebras not necessarily $\mathbb{N}$-graded, remarkable examples are the operator algebras, algebras of diffusion type, quadratic algebras in $3$ variables, the quantum algebra $\mathcal{U}^{\prime}(so(3,K))$, the dispin algebra \mbox{{\cal U}}(osp(1,2)), the Woronowicz algebra \mbox{{\cal W}}_{\nu}(\mathfrak{sl}(2,K)), among many others. The second and the third sections contain the novelty of the paper. The main results are Theorem 2.7, which completes the parametrization started in [10], and Theorem 3.1 where we apply the results of the previous sections to compute the point modules of some skew $PBW$ extensions, and hence, the point modules of some examples of non $\mathbb{N}$-graded quantum algebras. Concrete examples are presented in Examples 3.2 and 3.3.

1.1 Point modules for finitely graded algebras

In this first subsection we recall some key facts and examples on point modules for classical $\mathbb{N}$-graded algebras that we will use later in the paper (see [2], [4] and [15]). Let $K$ be a field, a $K$-algebra $A$ is finitely graded if the following conditions hold: (i) $A$ is $\mathbb{N}$-graded, $A=\bigoplus_{n\geq 0}A_{n}$. (ii) $A$ is connected, i.e., $A_{0}=K$. (iii) $A$ is finitely generated as $K$-algebra. Let $A$ be a finitely graded $K$-algebra that is generated in degree $1$. A point module for $A$ is a graded left module $M$ such that $M$ is cyclic, generated in degree [math], i.e., there exists an element $m_{0}\in M_{0}$ such that $M=Am_{0}$, and $\dim_{K}(M_{n})=1$ for all $n\geq 0$ (since $A$ is $\mathbb{N}$-graded and $m_{0}\in M_{0}$, then $M$ is necessarily $\mathbb{N}$-graded). The collection of isomorphic classes of point modules for $A$ is denoted by $P(A)$.

The following examples show the description (parametrization) of point modules for some well-known finitely graded algebras.

Example 1.1** ([15]).**

Let $K$ be a field and $A=K[x_{0},\dots,x_{n}]$ be the commutative $K$-algebra of usual polynomials. Then,

1. (i)

There exists a bijective correspondence between the projective space $\mathbb{P}^{n}$ and the set of isomorphism classes of point modules of $A$, $\mathbb{P}^{n}\longleftrightarrow P(A)$. 2. (ii)

The correspondence in $(i)$ generalizes to any finitely graded commutative algebra which is generated in degree $1$.

Example 1.2** ([15]).**

(i) The point modules of the quantum plane are parametrized by $\mathbb{P}^{1}$, i.e., there exists a bijective correspondence between the projective space $\mathbb{P}^{1}$ and the collection of isomorphism classes of point modules of the quantum plane $A:=K\{x,y\}/\langle yx-qxy\rangle$:

$\mathbb{P}^{1}\longleftrightarrow P(A)$.

(ii) For the Jordan plane, $A:=K\{x,y\}/\langle yx-xy-x^{2}\rangle$, can be proved a similar equivalence:

$\mathbb{P}^{1}\longleftrightarrow P(A)$.

Example 1.3** ([15]).**

The isomorphism classes of point modules for the free algebra $A:=K\{x_{0},\dots,x_{n}\}$ are in bijective correspondence with $\mathbb{N}$-indexed sequences of points in $\mathbb{P}^{n}$, $\{(\lambda_{0,i}:\cdots:\lambda_{n,i})\in\mathbb{P}^{n}|i\geq 0\}$, in other words, with the points of the infinite product $\mathbb{P}^{n}\times\mathbb{P}^{n}\times\cdots=\prod_{i=0}^{\infty}\mathbb{P}^{n}$(note that an element of $\prod_{i=0}^{\infty}\mathbb{P}^{n}$ is a sequence of the form $((\lambda_{0,i}:\cdots:\lambda_{n,i}))_{i\geq 0})$. Thus, there exists a bijective correspondence between $\prod_{i=0}^{\infty}\mathbb{P}^{n}$ and $P(K\{x_{0},\dots,x_{n}\})$:

$\prod_{i=0}^{\infty}\mathbb{P}^{n}\longleftrightarrow P(A)$.

Using the previous example it is possible to compute the collection of point modules for any finitely presented algebra. For this we need a definition first, see [15]. Suppose that $f\in K\{x_{0},\dots,x_{n}\}$ is a homogeneous element of degree $m$ $($assuming that $\deg(x_{v}):=1$, for all v$$). Consider the polynomial ring $B=K[y_{v,u}]$ in a set of $(n+1)m$ commuting variables $\{y_{v,u}|0\leq v\leq n,1\leq u\leq m\}$. The multilinearization of $f$ is the element of $B$ given by replacing each word $w=x_{v_{m}}\cdots x_{v_{2}}x_{v_{1}}$ occurring in $f$ by $y_{v_{m},m}\cdots y_{v_{2},2}y_{v_{1},1}(=y_{v_{1},1}y_{v_{2},2}\cdots y_{v_{m},m})$.

Proposition 1.4** ([15], Proposition 3.5).**

Let $A=K\{x_{0},\dots,x_{n}\}/\langle f_{1},\dots,f_{r}\rangle$ be a finitely presented $K$-algebra, where the $f_{l}$ are homogeneous of degree $d_{l}\geq 2$, $1\leq l\leq r$. For each $f_{l}$, let $g_{l}$ be the multilinearization of $f_{l}$. Then,

1. (i)

The isomorphism classes of point modules for $A$ are in bijection with the closed subset $X$ of $\prod_{i=0}^{\infty}\mathbb{P}^{n}$ defined by

$X:=\{(p_{0},p_{1},\dots)|g_{l}(p_{i},p_{i+1},\dots,p_{i+d_{l}-1})=0\ \text{for all}\ 1\leq l\leq r,i\geq 0\}$.

Consider for each $m\geq 1$ the closed subset

$X_{m}:=\{(p_{0},p_{1},\dots,p_{m-1})|g_{l}(p_{i},p_{i+1},\dots,p_{i+d_{l}-1})=0\ \text{for all}\ 1\leq l\leq r,0\leq i\leq m-d_{l}\}$**

of $\prod_{i=0}^{m-1}\mathbb{P}^{n}$. The canonical projection onto the first m coordinates defines a function $\phi_{m}:X_{m+1}\to X_{m}$. Then $X$ is equal to the inverse limit $\underleftarrow{\lim}X_{m}$ of the $X_{m}$ with respect to the functions $\phi_{m}$. In particular, if $m_{0}$ is such that $\phi_{m}$ is a bijection for all $m\geq m_{0}$, then the isomorphism classes of point modules of $A$ are in bijective correspondence with the points of $X_{m_{0}}$.

For strongly Noetherian algebras there is $m_{0}$ such that the functions $\phi_{m}:X_{m+1}\to X_{m}$ are bijective for all $m\geq m_{0}$. These algebras were studied by Artin, Small and Zhang in [1], and appears naturally in the study of point modules in non-commutative algebraic geometry (see [2], [3] and [15]). Let $K$ be a field and let $A$ be a left Noetherian $K$-algebra, it is said that $A$ is left strongly Noetherian if for any commutative Noetherian $K$-algebra $C$, $C\otimes_{K}A$ is left Noetherian.

Corollary 1.5** ([4], Corollary E4.12).**

Let $A$ be a finitely graded strongly noetherian algebra, with presentation as in Proposition 1.4. Then there is $m_{0}$ such that $\phi_{m}:X_{m+1}\to X_{m}$ is bijective for all $m\geq m_{0}$, and the point modules of $A$ are in bijective correspondence with the points of $X_{m_{0}}$.

Example 1.6**.**

It is known (see Example 3.8 in [15] and also [7]) that for the multi-parameter quantum affine $3$-space, $A:=K_{\textbf{q}}[x_{1},x_{2},x_{3}]$, defined by

$x_{2}x_{1}=q_{12}x_{1}x_{2}$, $x_{3}x_{1}=q_{13}x_{1}x_{3}$, $x_{3}x_{2}=q_{23}x_{2}x_{3}$,

$m_{0}=2$* and*

$X_{2}=\{(p_{0},\sigma(p_{0}))\mid p_{0}\in E\}$,

where

[TABLE]

with $\sigma:E\to E$ bijective. Thus, for $A$ we have the bijective correspondence $P(A)\longleftrightarrow X_{2}\longleftrightarrow E$. We will prove this including all details omitted in the above cited references. Following Proposition 1.4, in this case we have $3$ variables, $r=3$ and every generator of $A$ has degree $2$. We have to show that the description of $X_{2}$ is as above and for all $m\geq 2$ the function $\phi_{m}:X_{m+1}\to X_{m}$ is bijective.

Recall that

$X_{2}=\{(p_{0},p_{1})\in\mathbb{P}^{2}\times\mathbb{P}^{2}\mid g_{l}(p_{0},p_{1})=0\ \text{for}\ 1\leq l\leq 3\}$,

where $p_{0}:=(x_{0}:y_{0}:z_{0})$, $p_{1}:=(x_{1}:y_{1}:z_{1})$ and $g_{1},g_{2},g_{3}$ are the multilinearization of the defining relations of $A$, i.e.,

$g_{1}(p_{0},p_{1})=y_{0}x_{1}-q_{12}x_{0}y_{1}$, $g_{2}(p_{0},p_{1})=z_{0}x_{1}-q_{13}x_{0}z_{1}$, $g_{3}(p_{0},p_{1})=z_{0}y_{1}-q_{23}y_{0}z_{1}$.

Thus, the relations that define $A$ can be written as

$y_{0}x_{1}-q_{12}x_{0}y_{1}=0$, $z_{0}x_{1}-q_{13}x_{0}z_{1}=0$ and $z_{0}y_{1}-q_{23}y_{0}z_{1}=0$,

and in a matrix form we have

[TABLE]

In order to compute $X_{2}$, let $E$ be the projection of $X_{2}$ onto the first copy of $\mathbb{P}^{2}$. Let $p_{0}\in\mathbb{P}^{2}$, then $p_{0}\in E$ if and only if (1.2) has a non trivial solution $p_{1}$ if and only if the determinant of the matrix $F$ of this system is equal [math], i.e., $x_{0}y_{0}z_{0}(q_{12}q_{23}-q_{13})=0$. In addition, observe that ${\rm rank}(F)=2$. In fact, it is clear that ${\rm rank}(F)\neq 0$ (contrary, $x_{0}=y_{0}=z_{0}=0$, false). Suppose that ${\rm rank}(F)=1$, and assume for example that the $K$-basis of $F$ is the first column, so $0=z_{0}=y_{0}=x_{0}$, a contradiction; in a similar way we get a contradiction assuming that the $K$-basis of $F$ is the second or the third column. Thus, $\dim_{K}(\ker(F))=1$ and given $p_{0}=(x_{0}:y_{0}:z_{0})\in E$ there exists a unique $p_{1}=(x_{1}:y_{1}:z_{1})\in\mathbb{P}^{2}$ that satisfies (1.2), and hence, we define a function

[TABLE]

Therefore, $X_{2}=\{(p_{0},\sigma(p_{0}))\mid p_{0}\in E\}$.

$\sigma$* is injective: For this, note that the defining relations of $A$ can be written also in the following matrix way:*

$\begin{bmatrix}-q_{12}y_{1}&x_{1}&0\\ -q_{13}z_{1}&0&x_{1}\\ 0&-q_{23}z_{1}&y_{1}\end{bmatrix}\begin{bmatrix}x_{0}\\ y_{0}\\ z_{0}\end{bmatrix}=0$;

let $G$ be the matrix of this system; as above, ${\rm rank}(G)=2$ and $\dim_{K}(\ker(G))=1$, thus, if $\sigma(p_{0})=p_{1}=\sigma(p_{0}^{\prime})$, then $p_{0}=p_{0}^{\prime}$.

From the determinant of $F$ arise two cases.

Case 1: $q_{12}q_{23}=q_{13}$. In this case every point $p_{0}=(x_{0}:y_{0}:z_{0})$ of $\mathbb{P}^{2}$ is in $E$, i.e., $E=\mathbb{P}^{2}$. Let $\sigma_{1}:\mathbb{P}^{2}\to\mathbb{P}^{2}$ be the function in (1.3), we have to show that $\sigma_{1}$ is surjective. Since in this case $p_{0}\in E$ if and only if $p_{1}\in E$, we define $\theta_{1}:\mathbb{P}^{2}\to\mathbb{P}^{2}$ by $\theta_{1}(p_{1}):=p_{0}$ with $Gp_{0}=0$. Observe that $\sigma_{1}\theta_{1}=i_{\mathbb{P}^{2}}$: Indeed, $\sigma_{1}(\theta_{1}(p_{1}))=\sigma_{1}(p_{0})=p_{1}^{\prime}$ with $Fp_{1}^{\prime}=0$, but $Gp_{0}=0$ is equivalent to $Fp_{1}=0$, and since $\dim_{K}(\ker(F))=1$, then $p_{1}^{\prime}=p_{1}$.

Case 2: $q_{12}q_{23}\neq q_{13}$. In this case $p_{0}=(x_{0}:y_{0}:z_{0})\in E$ if and only if $x_{0}y_{0}z_{0}=0$, i.e., $E=\{(x_{0}:y_{0}:z_{0})\in\mathbb{P}^{2}\mid x_{0}y_{0}z_{0}=0\}$. We denote by $\sigma_{2}$ the function in (1.3). We know that $\sigma_{2}$ is injective; observe that $\sigma_{2}(E)\subseteq E$: Indeed, let $p_{0}\in E$, then $\sigma_{2}(p_{0})=\sigma(p_{0})=p_{1}$, but since $q_{12}q_{23}\neq q_{13}$ and $\det(G)=0$ (contrary, $x_{0}=y_{0}=z_{0}=0$, false), then $x_{1}y_{1}z_{1}=0$, i.e., $p_{1}\in E$. Only rest to show that $\sigma_{2}:E\to E$ is surjective. As in the case 1, we define $\theta_{2}:E\to E$ by $\theta_{2}(p_{1}):=p_{0}$ with $Gp_{0}=0$, then $\sigma_{2}(\theta_{2}(p_{1}))=\sigma_{2}(p_{0})=p_{1}^{\prime}$, with $Fp_{1}^{\prime}=0$, but $Gp_{0}=0$ is equivalent to $Fp_{1}=0$, and since $\dim_{K}(\ker(F))=1$, then $p_{1}^{\prime}=p_{1}$.

Thus, in both cases we have proved that

$X_{2}=\{(p_{0},\sigma(p_{0}))\mid p_{0}\in E\}\longleftrightarrow E$.

To complete the example, we will show that $X\longleftrightarrow X_{2}$. According to Proposition 1.4, we have to prove that for every $m\geq 2$, $\phi_{m}:X_{m+1}\to X_{m}$ is a bijection. For $m=2$, $\phi_{2}(p_{0},p_{1},p_{2})=(p_{0},p_{1})\in X_{2}$, so $p_{1}=\sigma(p_{0})$, note that $p_{2}=\sigma^{2}(p_{0})$ since by the definition of $X$, $g_{l}(p_{1},p_{2})=0$ for $1\leq l\leq 3$, so by the proved above $(\sigma(p_{0}),\sigma^{2}(p_{0}))\in X_{2}$. Thus, $X_{3}=\{(p_{0},\sigma(p_{0}),\sigma^{2}(p_{0}))\mid p_{0}\in E\}$ and $\phi_{2}$ is bijective. By induction, assume that

$X_{m}=\{(p_{0},\sigma(p_{0}),\dots,\sigma^{m-1}(p_{0}))\mid p_{0}\in E\}$* and $X_{m}\longleftrightarrow X_{m-1}$,*

so $X_{m+1}=\{(p_{0},\sigma(p_{0}),\dots,\sigma^{m}(p_{0}))\mid p_{0}\in E\}$ since $(\sigma^{m-1}(p_{0}),\sigma^{m}(p_{0}))\in X_{2}$, and hence $X_{m+1}\longleftrightarrow X_{m}$.

Example 1.7**.**

We will show next that the ideas and results of Example 1.6 can be extended to the multi-parameter quantum affine $n$-space $A:=K_{\textbf{q}}[x_{1},\dots,x_{n}]$, with $n\geq 3$ (see also [7]). We will see that $m_{0}=2$ and $X_{2}$ is as in (1.4) below.

Recall that in $A$ we have the defining relations

$x_{j}x_{i}=q_{ij}x_{i}x_{j}$, $1\leq i<j\leq n$.

We have $n$ variables, $r=\frac{n(n-1)}{2}$ generators, every generator of $A$ has degree $2$ and

$X_{2}=\{(p_{0},p_{1})\in\mathbb{P}^{n-1}\times\mathbb{P}^{n-1}\mid g_{l}(p_{0},p_{1})=0\ \text{for}\ 1\leq l\leq r\}$,

where $p_{0}:=(x_{1}^{0}:\cdots:x_{n}^{0})$, $p_{1}:=(x_{1}^{1}:\cdots:x_{n}^{1})$ and the $g_{l}$ are the multilinearization of the defining relations of $A$. Thus, the relations that define $A$ can be written as

$x_{j}^{0}x_{i}^{1}-q_{ij}x_{i}^{0}x_{j}^{1}=0$, $1\leq i<j\leq n$,

and in a matrix form we have

[TABLE]

The size of the matrix $F$ of this system is $\frac{n(n-1)}{2}\times n$. In order to compute $X_{2}$, let $E$ be the projection of $X_{2}$ onto the first copy of $\mathbb{P}^{n-1}$. Let $p_{0}\in\mathbb{P}^{n-1}$, then $p_{0}\in E$ if and only if the previous system has a non trivial solution $p_{1}$. This last condition is equivalent to ${\rm rank}(F)=n-1$: Indeed, it is clear that ${\rm rank}(F)\leq n$, but ${\rm rank}(F)\neq n$ (contrary, ${\rm dim}_{K}(\ker(F))=0$ and hence $x_{1}^{1}=\cdots x_{n}^{1}=0$, false); thus, ${\rm rank}(F)\leq n-1$, but there exists $i\in\{1,\dots,n\}$ such that $x_{i}^{0}\neq 0$ and the following $n-1$ rows of $F$ are linearly independent (actually, this is true for every $x_{i}^{0}\neq 0$):

$\begin{bmatrix}\cdots x_{i}^{0}&\cdots&-q_{li}x_{i}^{0}&\cdots\end{bmatrix}$* and $\begin{bmatrix}\cdots&\cdots&x_{k}^{0}&\cdots&-q_{ik}x_{i}^{0}&\cdots\end{bmatrix}$*

for all $1\leq l<i<k\leq n$, where $x_{i}^{0}$ is in the $l$-position, $-q_{li}x_{i}^{0}$ is in the $i$-position, $x_{k}^{0}$ is in the $i$-position and $-q_{ik}x_{i}^{0}$ is in the $k$-position. Hence, ${\rm rank}(F)=n-1$. Conversely, if ${\rm rank}(F)=n-1$, then the system has non trivial solution.

Thus, $p_{0}\in E$ if and only if ${\rm rank}(F)=n-1$ if and only if ${\rm dim}_{K}(\ker(F))=1$, and hence, given $p_{0}\in E$ there exists a unique $p_{1}\in\mathbb{P}^{n-1}$ that satisfies the above matrix system, so we define a function

[TABLE]

Therefore, $X_{2}=\{(p_{0},\sigma(p_{0}))\mid p_{0}\in E\}$. Rewriting the defining relations of $A$ as we did in Example 1.6, we conclude that $\sigma$ is injective. Moreover, $Im(\sigma)=E$: In fact, recall that ${\rm rank}(F)=n-1$ if and only if every minor of $F$ of size $n$ is equal [math]; let $\mathcal{F}$ be the set of minors of size $n$ of the matrix $F$, then $E$ is the projective variety

[TABLE]

where $I_{F}$ is the ideal of $K[x_{1}^{0},\dots,x_{n}^{0}]=K[x_{1},\dots,x_{n}]$ generated by all $f\in\mathcal{F}$. Hence, from the rewriting of relations we conclude also that $p_{1}=\sigma(p_{0})\in E$ and, as in Example 1.6, we define a function $\theta:E\to E$ such that $\sigma\theta=i_{E}$.

Finally, $X_{2}=\{(p_{0},\sigma(p_{0}))\mid p_{0}\in E\}\longleftrightarrow E$ and the proof of $X\longleftrightarrow X_{2}$ is exactly as in the final part of Example 1.6.

Remark 1.8**.**

(i) For $n=3$, $\mathcal{F}=\{x_{1}^{0}x_{2}^{0}x_{3}^{0}(q_{12}q_{23}-q_{13})\}$ and hence (1.4) extends (1.1). For $n=2$, $A=K_{q}[x_{1},x_{2}]$ is the quantum plane and from Example 1.2 we know that $P(A)\longleftrightarrow\mathbb{P}^{1}$, but observe that (1.4) also cover this case since $\mathcal{F}=\emptyset$ and hence $E=V(0)=\mathbb{P}^{1}$.

(ii) For the affine algebra $A=K[x_{1},\dots,x_{n}]$ all constants $q_{ij}$ are trivial, i.e, $q_{ij}=1$ and the generators of $I_{F}$ in (1.4) are null, so $I_{F}=0$ and $E=\mathbb{P}^{n-1}$. This result agrees with Example 1.1.

(iii) It is important to recall that there exist quotients of finitely graded algebras $A$ for which $P(A)=\emptyset$. In fact, some quotient algebras of the multi-parameter quantum affine $n$-space have empty set of point modules, see [5] and [18].

1.2 Finitely semi-graded rings

In this preliminary subsection we recall the definition of semi-graded rings and modules introduced firstly in [11]. Let $B$ be a ring. We say that $B$ is semi-graded $(SG)$ if there exists a collection $\{B_{n}\}_{n\geq 0}$ of subgroups $B_{n}$ of the additive group $B^{+}$ such that the following conditions hold:

1. (i)

$B=\bigoplus_{n\geq 0}B_{n}$. 2. (ii)

For every $m,n\geq 0$, $B_{m}B_{n}\subseteq B_{0}\oplus\cdots\oplus B_{m+n}$. 3. (iii)

$1\in B_{0}$.

The collection $\{B_{n}\}_{n\geq 0}$ is called a semi-graduation of $B$ and we say that the elements of $B_{n}$ are homogeneous of degree $n$. Let $B$ and $C$ be semi-graded rings and let $f:B\to C$ be a ring homomorphism, we say that $f$ is homogeneous if $f(B_{n})\subseteq C_{n}$ for every $n\geq 0$. Let $B$ be a $SG$ ring and let $M$ be a $B$-module. We say that $M$ is a $\mathbb{Z}$-semi-graded, or simply semi-graded, if there exists a collection $\{M_{n}\}_{n\in\mathbb{Z}}$ of subgroups $M_{n}$ of the additive group $M^{+}$ such that the following conditions hold:

1. (a)

$M=\bigoplus_{n\in\mathbb{Z}}M_{n}$. 2. (b)

For every $m\geq 0$ and $n\in\mathbb{Z}$, $B_{m}M_{n}\subseteq\bigoplus_{k\leq m+n}M_{k}$.

The collection $\{M_{n}\}_{n\in\mathbb{Z}}$ is called a semi-graduation of $M$ and we say that the elements of $M_{n}$ are homogeneous of degree $n$. We say that $M$ is positively semi-graded, also called $\mathbb{N}$-semi-graded, if $M_{n}=0$ for every $n<0$. Let $f:M\to N$ be an homomorphism of $B$-modules, where $M$ and $N$ are semi-graded $B$-modules; we say that $f$ is homogeneous if $f(M_{n})\subseteq N_{n}$ for every $n\in\mathbb{Z}$.

An important class of semi-graded rings that includes finitely graded algebras is the following (see [11]). Let $B$ be a ring. We say that $B$ is finitely semi-graded $(FSG)$ if $B$ satisfies the following conditions:

1. (1)

$B$ is $SG$. 2. (2)

There exists finitely many elements $x_{1},\dots,x_{n}\in B$ such that the subring generated by $B_{0}$ and $x_{1},\dots,x_{n}$ coincides with $B$. 3. (3)

For every $n\geq 0$, $B_{n}$ is a free $B_{0}$-module of finite dimension.

Moreover, if $M$ is a $B$-module, we say that $M$ is finitely semi-graded if $M$ is semi-graded, finitely generated, and for every $n\in\mathbb{Z}$, $M_{n}$ is a free $B_{0}$-module of finite dimension.

Remark 1.9**.**

(i) It is clear that any $\mathbb{N}$-graded ring is $SG$.

(ii) Any finitely graded algebra is a $FSG$ ring.

From the definitions above we get the following elementary facts.

Proposition 1.10** ([11]).**

Let $B=\bigoplus_{n\geq 0}B_{n}$ be a $SG$ ring. Then,

1. (i)

$B_{0}$* is a subring of $B$. Moreover, for any $n\geq 0$, $B_{0}\oplus\cdots\oplus B_{n}$ is a $B_{0}-B_{0}$-bimodule, as well as $B$.* 2. (ii)

$B$* has a standard $\mathbb{N}$-filtration given by*

[TABLE] 3. (iii)

The associated graded ring $Gr(B)$ satisfies

$Gr(B)_{n}\cong B_{n}$, for every $n\geq 0$ $($isomorphism of abelian groups$)$. 4. (iv)

Let $M=\bigoplus_{n\in\mathbb{Z}}M_{n}$ be a semi-graded $B$-module and $N$ a submodule of $M$. The following conditions are equivalent:

1. (a)

$N$* is semi-graded.* 2. (b)

For every $z\in N$, the homogeneous components of $z$ are in $N$. 3. (c)

$M/N$* is semi-graded with semi-graduation given by*

$(M/N)_{n}:=(M_{n}+N)/N$, $n\in\mathbb{Z}$.

Remark 1.11**.**

(i) If $B$ is a $FSG$ ring, then for every $n\geq 0$, $Gr(B)_{n}\cong B_{n}$ as $B_{0}$-modules.

(ii) Observe if $B$ is $FSG$ ring, then $B_{0}B_{p}=B_{p}$ for every $p\geq 0$, and if $M$ is finitely semi-graded, then $B_{0}M_{n}=M_{n}$ for all $n\in\mathbb{Z}$.

1.3 Skew $PBW$ extensions

In order to present enough examples of $FSG$ rings not being necessarily finitely graded algebras, we recall in this subsection the notion of skew $PBW$ extension defined firstly in [9].

Definition 1.12** ([9]).**

Let $R$ and $A$ be rings. We say that $A$ is a skew $PBW$ extension of $R$ $($also called a $\sigma-PBW$ extension of R$$), if the following conditions hold:

1. (i)

$R\subseteq A$. 2. (ii)

There exist finitely many elements $x_{1},\dots,x_{n}\in A$ such $A$ is a left $R$-free module with basis

${\rm Mon}(A):=\{x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}\mid\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}\}$, with $\mathbb{N}:=\{0,1,2,\dots\}$.

The set $Mon(A)$ is called the set of standard monomials of $A$. 3. (iii)

For every $1\leq i\leq n$ and $r\in R-\{0\}$, there exists $c_{i,r}\in R-\{0\}$ such that

[TABLE] 4. (iv)

For every $1\leq i,j\leq n$, there exists $c_{i,j}\in R-\{0\}$ such that

[TABLE]

Under these conditions we will write $A:=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$.

Associated to a skew $PBW$ extension $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$, there are $n$ injective endomorphisms $\sigma_{1},\dots,\sigma_{n}$ of $R$ and $\sigma_{i}$-derivations $\delta_{i}$, as the following proposition shows.

Proposition 1.13** ([9]).**

Let $A$ be a skew $PBW$ extension of $R$. Then, for every $1\leq i\leq n$, there exists an injective ring endomorphism $\sigma_{i}:R\rightarrow R$ and a $\sigma_{i}$-derivation $\delta_{i}:R\rightarrow R$ such that

$x_{i}r=\sigma_{i}(r)x_{i}+\delta_{i}(r)$,

for each $r\in R$.

Some particular cases of skew $PBW$ extensions are the following.

Definition 1.14** ([9]).**

Let $A$ be a skew $PBW$ extension.

1. (a)

$A$* is quasi-commutative if the conditions (iii) and (iv) in Definition 1.12 are replaced by*

1. (iii’)

For every $1\leq i\leq n$ and $r\in R-\{0\}$ there exists $c_{i,r}\in R-\{0\}$ such that

[TABLE] 2. (iv’)

For every $1\leq i,j\leq n$ there exists $c_{i,j}\in R-\{0\}$ such that

[TABLE] 2. (b)

$A$* is bijective if $\sigma_{i}$ is bijective for every $1\leq i\leq n$ and $c_{i,j}$ is invertible for any $1\leq i<j\leq n$.*

Definition 1.15**.**

Let $A$ be a skew $PBW$ extension of $R$ with endomorphisms $\sigma_{i}$, $1\leq i\leq n$, as in Proposition 1.13.

1. (i)

For $\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}$, $\sigma^{\alpha}:=\sigma_{1}^{\alpha_{1}}\cdots\sigma_{n}^{\alpha_{n}}$, $|\alpha|:=\alpha_{1}+\cdots+\alpha_{n}$. If $\beta=(\beta_{1},\dots,\beta_{n})\in\mathbb{N}^{n}$, then $\alpha+\beta:=(\alpha_{1}+\beta_{1},\dots,\alpha_{n}+\beta_{n})$. 2. (ii)

For $X=x^{\alpha}\in Mon(A)$, $\exp(X):=\alpha$ and $\deg(X):=|\alpha|$. 3. (iii)

Let $0\neq f\in A$, $t(f)$ is the finite set of terms that conform $f$, i.e., if $f=c_{1}X_{1}+\cdots+c_{t}X_{t}$, with $X_{i}\in Mon(A)$ and $c_{i}\in R-\{0\}$, then $t(f):=\{c_{1}X_{1},\dots,c_{t}X_{t}\}$. 4. (iv)

Let $f$ be as in (iii), then $\deg(f):=\max\{\deg(X_{i})\}_{i=1}^{t}.$

The next theorems establish some results for skew $PBW$ extensions that we will use later, for their proofs see [12], [11] and [13].

Theorem 1.16** ([12]).**

Let $A$ be an arbitrary skew $PBW$ extension of the ring $R$. Then, $A$ is a filtered ring with filtration given by

[TABLE]

and the corresponding graded ring $Gr(A)$ is a quasi-commutative skew $PBW$ extension of $R$. Moreover, if $A$ is bijective, then $Gr(A)$ is quasi-commutative bijective skew $PBW$ extension of $R$.

Theorem 1.17** ([12]).**

Let $A$ be a quasi-commutative skew $PBW$ extension of a ring $R$. Then,

$A$* is isomorphic to an iterated skew polynomial ring of endomorphism type, i.e.,*

[TABLE] 2. 2.

If $A$ is bijective, then each endomorphism $\theta_{i}$ is bijective, $1\leq i\leq n$.

Theorem 1.18** (Hilbert Basis Theorem, [12]).**

Let $A$ be a bijective skew $PBW$ extension of $R$. If $R$ is a left (right) Noetherian ring then $A$ is also a left (right) Noetherian ring.

Theorem 1.19** ([13]).**

Let $K$ be a field and let $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a bijective skew $PBW$ extension of a left strongly Noetherian $K$-algebra $R$. Then $A$ is left strongly Noetherian.

Theorem 1.20** ([11]).**

Any skew $PBW$ extension $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ is a $FSG$ ring with semi-graduation $A=\bigoplus_{k\geq 0}A_{k}$, where

$A_{k}:=_{R}\langle x^{\alpha}\in Mon(A)|\deg(x^{\alpha})=k\rangle$.

Example 1.21**.**

Many important algebras and rings coming from mathematical physics are particular examples of skew $PBW$ extensions, see [12] and [14]: Habitual ring of polynomials in several variables, Weyl algebras, enveloping algebras of finite dimensional Lie algebras, algebra of $q$-differential operators, many important types of Ore algebras, algebras of diffusion type, additive and multiplicative analogues of the Weyl algebra, dispin algebra $\mathcal{U}(osp(1,2))$, quantum algebra $\mathcal{U}^{\prime}(so(3,K))$, Woronowicz algebra $\mathcal{W}_{\nu}(\mathfrak{sl}(2,K))$, Manin algebra $\mathcal{O}_{q}(M_{2}(K))$, coordinate algebra of the quantum group $SL_{q}(2)$, $q$-Heisenberg algebra H${}_{n}(q)$, Hayashi algebra $W_{q}(J)$, differential operators on a quantum space $D_{\textbf{q}}(S_{\textbf{q}})$, Witten’s deformation of $\mathcal{U}(\mathfrak{sl}(2,K))$, multiparameter Weyl algebra $A_{n}^{Q,\Gamma}(K)$, quantum symplectic space $\mathcal{O}_{q}(\mathfrak{sp}(K^{2n}))$, some quadratic algebras in 3 variables, some 3-dimensional skew polynomial algebras, among many others. For a precise definition of any of these rings and algebras see [12] and [14].

2 Parametrization of point modules of $FSG$ rings

In this section we complete the parametrization of point modules of $FSG$ rings that was initiated in [10]. The main result of the present section is Theorem 2.7 below. For completeness, we include the basic facts of parametrization studied in [10], omitting the proofs.

Definition 2.1**.**

Let $B=\bigoplus_{n\geq 0}B_{n}$ be a $FSG$ ring that is generated in degree $1$.

1. (i)

A point module for $B$ is a finitely $\mathbb{N}$-semi-graded $B$-module $M=\bigoplus_{n\in\mathbb{N}}M_{n}$ such that $M$ is cyclic, generated in degree [math], i.e., there exists an element $m_{0}\in M_{0}$ such that $M=Bm_{0}$, and $\dim_{B_{0}}(M_{n})=1$ for all $n\geq 0$. 2. (ii)

Two point modules $M$ and $M^{\prime}$ for $B$ are isomorphic if there exists a homogeneous $B$-isomorphism between them. 3. (iii)

$P(B)$* is the collection of isomorphism classes of point modules for $B$.*

The following result is the first step in the construction of the geometric structure for $P(B)$.

Theorem 2.2**.**

Let $B=\bigoplus_{n\geq 0}B_{n}$ be a $FSG$ ring generated in degree $1$. Then, $P(B)$ has a Zariski topology generated by finite unions of sets $V(J)$ defined by

$V(J):=\{M\in P(B)\mid Ann(M)\supseteq J\}$,

where $J$ ranges the semi-graded left ideals of $B$.

Proof.

See [10], Theorem 9. ∎

Definition 2.3**.**

Let $B=\bigoplus_{n\geq 0}B_{n}$ be a $FSG$ ring generated in degree $1$ such that $B_{0}$ is commutative and $B$ is a $B_{0}$-algebra. Let $S$ be a commutative $B_{0}$-algebra. A $S$-point module for $B$ is a $\mathbb{N}$-semi-graded $S\otimes_{B_{0}}B$-module $M$ which is cyclic, generated in degree [math], $M_{n}$ is a locally free $S$-module with ${\rm rank}_{S}(M_{n})=1$ for all $n\geq 0$, and $M_{0}=S$. $P(B;S)$ will denote the set of $S$-point modules for $B$.

Theorem 2.4**.**

Let $B=\bigoplus_{n\geq 0}B_{n}$ be a $FSG$ ring generated in degree $1$ such that $B_{0}$ is commutative and $B$ is a $B_{0}$-algebra. Let $\mathcal{B}_{0}$ be the category of commutative $B_{0}$-algebras and let $\mathcal{S}et$ be the category of sets. Then, $P$ defined by

[TABLE]

is a covariant functor called the point functor for $B$.

Proof.

See [10], Theorem 10. ∎

Definition 2.5**.**

Let $B=\bigoplus_{n\geq 0}B_{n}$ be a $FSG$ ring generated in degree $1$ such that $B_{0}$ is commutative and $B$ is a $B_{0}$-algebra. We say that a $B_{0}$-scheme $X$ parametrizes the point modules of $B$ if the point functor $P$ is naturally isomorphic to $h_{X}$.

Taking $S=B_{0}$ we get that $P(B)\subseteq P(B;B_{0})$. If $B_{0}=K$ is a field, then clearly $P(B)=P(B;K)$.

Theorem 2.6**.**

Let $B=\bigoplus_{n\geq 0}B_{n}$ be a $FSG$ ring generated in degree $1$ such that $B_{0}=K$ is a field and $B$ is a $K$-algebra. Let $X$ be a $K$-scheme that parametrizes $P(B)$. Then, there exists a bijective correspondence between the closed points of $X$ and $P(B)$.

Proof.

See [10], Theorem 11. ∎

Theorem 2.7**.**

Let $B=\bigoplus_{n\geq 0}B_{n}$ be a $FSG$ ring generated in degree $1$ such that $B_{0}=K$ is a field and $B$ is a $K$-algebra. Then,

1. (i)

There is an injective function

$(P(B)/\sim)\xrightarrow{\alpha}P(Gr(B))$,

where $\sim$ is the relation in $P(B)$ defined by

$[M]\sim[M^{\prime}]\Leftrightarrow Gr(M)\cong Gr(M^{\prime})$,

with $[M]$ the class of point modules isomorphic to the point module $M$. 2. (ii)

Let $X$ be a $K$-scheme that parametrizes $P(Gr(B))$. Then there is an injective function from $(P(B)/\sim)$ to the closed points of $X$:

$(P(B)/\sim)\xrightarrow{\widetilde{\alpha}}X$.

Proof.

(i) We divide the proof of this part in three steps.

Step 1. Note that $Gr(B)$ is a finitely graded $K$-algebra generated in degree $1$: According to Proposition 1.10, $Gr(B)$ is $\mathbb{N}$-graded with $Gr(B)_{n}\cong B_{n}$ for all $n\geq 0$ (isomorphism of $K$-vector spaces), in particular, $Gr(B)_{0}=K$; if $x_{1},\dots,x_{m}\in B_{1}$ generate $B$ as $K$-algebra, then $\overline{x_{1}},\dots,\overline{x_{m}}\in Gr(B)_{1}$ generate $Gr(B)$ as $K$-algebra.

Step 2. Let $[M]\in P(B)$, with $M=\bigoplus_{n\geq 0}M_{n}$, $M=Bm_{0}$, with $m_{0}\in M_{0}$, and $\dim_{K}(M_{n})=1$ for all $n\geq 0$. As in Proposition 1.10, we define

$Gr(M):=\bigoplus_{n\geq 0}Gr(M)_{n}$,

$Gr(M)_{n}:=\frac{M_{0}\oplus\cdots\oplus M_{n}}{M_{0}\oplus\cdots\oplus M_{n-1}}\cong M_{n}$ (isomorphism of $K$-vector spaces).

The structure of $Gr(B)$-module for $Gr(M)$ is given by bilinearity and the product

$\overline{b_{p}}\cdot\overline{m_{n}}:=\overline{b_{p}m_{n}}\in Gr(M)_{p+n}$.

This product is well-defined: If $\overline{b_{p}}=\overline{c_{p}}$, then $b_{p}-c_{p}\in B_{0}\oplus\cdots\oplus B_{p-1}$ and hence $(b_{p}-c_{p})m_{n}\in M_{0}\oplus\cdots\oplus M_{p-1+n}$, so in $Gr(M)_{p+n}$ we have $\overline{(b_{p}-c_{p})m_{n}}=\overline{0}$, i.e., $\overline{b_{p}m_{n}}=\overline{c_{p}m_{n}}$; now, if $\overline{m_{n}}=\overline{l_{n}}$, then $m_{n}-l_{n}\in M_{0}\oplus\cdots\oplus M_{n-1}$, whence $b_{p}(m_{n}-l_{n})\in M_{0}\oplus\cdots\oplus M_{p+n-1}$, so in $Gr(M)_{p+n}$ we have $\overline{b_{p}(m_{n}-l_{n})}=\overline{0}$, i.e., $\overline{b_{p}m_{n}}=\overline{b_{p}l_{n}}$. The associative law for this product holds:

$\overline{b_{q}}\cdot(\overline{b_{p}}\cdot\overline{m_{n}})=\overline{b_{q}}\cdot(\overline{b_{p}m_{n}})=\overline{b_{q}(b_{p}m_{n})}=\overline{(b_{q}b_{p})m_{n}}=\overline{(c_{0}+\cdots+c_{p+q})m_{n}}=\overline{c_{p+q}m_{n}}=(\overline{b_{q}}\ \overline{b_{p}})\overline{\cdot m_{n}}$.

Finally, $\overline{1}\cdot\overline{m_{n}}=\overline{m_{n}}$.

Observe that $Gr(M)=Gr(B)\cdot\overline{m_{0}}$: Let $\overline{m_{n}}\in Gr(M)_{n}$, with $m_{n}\in M_{n}$, we have $\overline{m_{n}}=\overline{bm_{0}}=\overline{(b_{0}+\cdots+b_{n})m_{0}}=\overline{b_{0}m_{0}+\cdots+b_{n-1}m_{0}+b_{n}m_{0}}=\overline{b_{n}m_{0}}=\overline{b_{n}}\cdot\overline{m_{0}}$.

Step 3. We define

[TABLE]

It is clear that $\alpha^{\prime}$ is well-defined, i.e., if $M\cong M^{\prime}$, then $Gr(M)\cong Gr(M^{\prime})$.

Note that the relation $\sim$ defined in the statement of the theorem is an equivalence relation, let $[[M]]$ be the class of $[M]\in P(B)$, then we define

[TABLE]

It is clear that $\alpha$ is a well-defined injective function.

(ii) This follows from (i) and applying Theorem 2.6 to $Gr(B)$. ∎

3 Point modules of skew $PBW$ extensions

In this final section we apply the results of the previous section to compute the point modules of some skew $PBW$ extensions, and hence, of some important examples of non $\mathbb{N}$-graded quantum algebras.

Theorem 3.1**.**

Let $K$ be a field and $A=\sigma(K)\langle x_{1},\dots,x_{n}\rangle$ be a bijective skew $PBW$ extension such that $A$ is a $K$-algebra. Then, there exists $m_{0}\geq 1$ such that there is an injective function from $P(A)/\sim$ to the closed points of $X_{m_{0}}$:

$(P(A)/\sim)\longrightarrow X_{m_{0}}$.

Proof.

From Proposition 1.20, $A$ is a $FSG$ ring generated in degree $1$ with $A_{0}=K$, and by hypothesis, $A$ is a $K$-algebra. Observe that the filtration in Theorem 1.16 coincides with the one in (1.5), thus, from Theorem 1.17 we get that $Gr(A)$ is the $n$-multiparametric quantum space $K_{\textbf{q}}[x_{1},\dots,x_{n}]$ (in $A$, $x_{i}r=rx_{i}$, for all $r\in K$ and $1\leq i\leq n$). It is clear that $Gr(A)$ is a finitely graded algebra with finite presentation as in Proposition 1.4, moreover, $Gr(A)$ is strongly noetherian (Theorem 1.19), so the claimed follows from Corollary 1.5 and Theorem 2.7. ∎

For the exact definition of any of the quantum algebras in the following examples see [12] and [14].

Example 3.2**.**

The following examples of skew $PBW$ extensions satisfy the hypothesis of Theorem 3.1, and hence, for them there is $m_{0}\geq 1$ such that there exists an injective function from $P(A)/\sim$ to the closed points of $X_{m_{0}}$.

The enveloping algebra of a Lie algebra $\mathcal{G}$ of dimension $n$: In this case $Gr(A)=K[x_{0},\dots,x_{n-1}]$, so $m_{0}=1$ and $X_{1}=\mathbb{P}^{n-1}$ (Example 1.1). We have a similar situation for the algebra $S_{h}$ of shift operators: In this case $Gr(A)=K[x_{0},x_{1}]$, so $m_{0}=1$ and $X_{1}=\mathbb{P}^{1}$. 2. 2.

Quantum algebra $\mathcal{U}^{\prime}(so(3,K))$, with $q\in K-\{0\}$: Since in this case $Gr(A)$ is a $3$-multiparametric quantum affine space, then from Example 1.6 we get that $m_{0}=2$ and $X_{2}$ is as in (1.1). We have similar descriptions for the dispin algebra \mbox{{\cal U}}(osp(1,2)), the Woronowicz algebra \mbox{{\cal W}}_{\nu}(\mathfrak{sl}(2,K)), where $\nu\in K-\{0\}$ is not a root of unity, nine types of $3$-dimensional skew polynomial algebras, and the multiplicative analogue of the Weyl algebra with $3$ variables. 3. 3.

For the following algebras, $Gr(A)=K_{\textbf{q}}[x_{1},\dots,x_{n}]$, so from Example 1.7 we conclude that $m_{0}=2$ and $X_{2}$ is as in (1.4): The $q$-Heisenberg algebra, the algebra of linear partial $q$-dilation operators, the algebra $D$ for multidimensional discrete linear systems, the algebra of linear partial shift operators.

Example 3.3**.**

The following algebras are not skew $PBW$ extensions of the base field $K$, however, they satisfy the hypothesis of Corollary 1.5, and hence, for them there is $m_{0}\geq 1$ such that $X_{m_{0}}$ parametrizes $P(A)$ and there exists a bijective correspondence between the closed points of $X_{m_{0}}$ and $P(A)$: The algebras of diffusion type, the algebra U, the Manin algebra, or more generally, the algebra $\mathcal{O}_{q}(M_{n}(K))$ of quantum matrices, some quadratic algebras in $3$ variables, the quantum symplectic space $\mathcal{O}_{q}(\mathfrak{sp}(K^{2n}))$.