# Automorphism groupoids in noncommutative projective geometry

Nicholas Cooney222Email<EMAIL_ADDRESS>
Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne,
63178 Aubière Cedex, France Jan E. Grabowski333Email:
<EMAIL_ADDRESS>Website: http://www.maths.lancs.ac.uk/~grabowsj/
Department of Mathematics and Statistics, Lancaster University,
Lancaster, LA1 4YF, United Kingdom

(31st July 2018)

###### Abstract

We address a natural question in noncommutative geometry, namely the rigidity
observed in many examples, whereby noncommutative spaces (or equivalently
their coordinate algebras) have very few automorphisms by comparison with
their commutative counterparts.

In the framework of noncommutative projective geometry, we define a groupoid
whose objects are noncommutative projective spaces of a given dimension and
whose morphisms correspond to isomorphisms of these. This groupoid is then a
natural generalization of an automorphism group. Using work of Zhang, we may
translate this structure to the algebraic side, wherein we consider
homogeneous coordinate algebras of noncommutative projective spaces. The
morphisms in our groupoid precisely correspond to the existence of a Zhang
twist relating the two coordinate algebras.

We analyse this automorphism groupoid, showing that in dimension $1$ it is
connected, so that every noncommutative $\mathbb{P}^{1}$ is isomorphic to
commutative $\mathbb{P}^{1}$. For dimension $2$ and above, we use the geometry
of the point scheme, as introduced by Artin-Tate-Van den Bergh, to relate
morphisms in our groupoid to certain automorphisms of the point scheme.

We apply our results to two important examples, quantum projective spaces and
Sklyanin algebras. In both cases, we are able to use the geometry of the point
schemes to fully describe the corresponding component of the automorphism
groupoid. This provides a concrete description of the collection of Zhang
twists of these algebras.

MSC (2010): 16S38 (Primary), 16D90, 16W50, 20G42 (Secondary)

##  1 Introduction

It is well-known that affine algebraic varieties can have many automorphisms:

######  Theorem 1 ([Jun42], [vdK53]).

Let $k=\overline{k}$ be an algebraically closed field. The group of
automorphisms of the affine plane $\mathbb{A}^{2}_{k}$ over $k$ is isomorphic
to $\operatorname{Aut}_{\text{alg}}(k[x,y])$, which is isomorphic to
$(GL_{2}(k)\ltimes k^{2})\ast_{S_{2}(k)}E_{2}(k)$ where

$E_{2}(k)=\\{f\colon k[x,y]\to k[x,y]\mid f\colon(x,y)\mapsto(\alpha
x+P(y),\beta y+\gamma),\alpha,\beta,\gamma\in k,P(y)\in k[y]\\}$

and $S_{2}(k)$ is the intersection of
$GA_{2}(k)\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}GL_{2}\ltimes
k^{2}$ and $E_{2}(k)$, and $\ast$ denotes the amalgamated free product. ∎

Furthermore, informally,

######  Theorem 2 ([Nag72], [SU04]).

$\operatorname{Aut}_{\text{alg}}(k[x,y,z])$ contains wild automorphisms - that
is, automorphisms that cannot be expressed in terms of elementary
automorphisms.

Of course, the situation is more tractable for projective rather than affine
varieties:

######  Theorem 3.

Let $\mathbb{P}_{k}^{n}=(\mathbb{A}_{k}^{n+1})/\sim$ be projective
($n$-)space. Then
$\operatorname{Aut}(\mathbb{P}_{k}^{n})\cong\operatorname{Aut}_{\mathbb{Z}-\text{gr}}(k[x_{0},\dotsc,x_{n}])\cong\operatorname{PGL}_{n+1}\\!\left(k\right)=GL_{n+1}(k)/k^{\ast}$.

However, noncommutative deformations of these spaces typically have very few
automorphisms:

######  Theorem 4 ([AC92]).

Let $q\in k^{\ast}$, $q$ not a root of unity. Denote by
$\mathcal{O}_{q}(\mathbb{P}_{k}^{n})$ the $k$-algebra

$\mathcal{O}_{q}(\mathbb{P}_{k}^{n})=k\langle
x_{0},\dotsc,x_{n}\rangle/\langle x_{i}x_{j}=qx_{j}x_{i}\ \forall\
i<j\rangle.$

Then
$\operatorname{Aut}(\mathcal{O}_{q}(\mathbb{P}_{k}^{n}))=\begin{cases}(k^{*})^{n+1}&n\neq
2,\\\ k\ltimes(k^{\ast})^{3}&n=2.\end{cases}$ ∎

The $(k^{\ast})^{n+1}$ is the torus action corresponding to the natural
$\mathbb{Z}^{n+1}$-grading. The automorphisms that act by scaling
corresponding to this torus preserve the total grading, which is the
$\mathbb{Z}$-grading of interest.

The same phenomenon of a significantly smaller automorphism group has been
observed for many other quantum algebras: quantized enveloping algebras and
related algebras, quantum matrices, quantum Weyl algebras, Nichols algebras,
and others. Contributors include Andruskiewitsch-Dumas [AD08], Fleury [Fle97],
Goodearl-Yakimov [GY15], Joseph [Jos76], Launois-Lenagan [LL13], Rigal [Rig96]
and Yakimov [Yak13],[Yak14].

This begs the (admittedly somewhat naive) question: where have all the
classical automorphisms gone? The ultimate goal of the approach we will
describe here is to understand how, given a projective variety $X$, one can
study families of noncommutative projective spaces that are close to $X$ in a
suitable sense, and how one may lift some symmetries of $X$ to symmetries of
and between these noncommutative spaces.

We start with a brief discussion of the underlying philosophy we will adopt,
with a view to making a more precise formulation of the previous paragraph.

###  1.1 Symmetries and groupoids

One classical notion of a geometry is that of a space equipped with an action
of a set of invertible, structure-preserving transformations. This “space” can
be a set, vector space, manifold, algebraic variety etc. and the set of
transformations forms a group of symmetries of it.

In categorical language, a group is a category with one object and all of
whose morphisms are invertible. An example relevant to our project is given by
taking a $\mathbb{Z}$-graded commutative $k$-algebra $A$ as the object and
elements of the group of degree $0$ automorphisms
$\operatorname{Aut}_{\mathrm{gr}}(A)$ as the morphisms.

However, in addressing the question of recovering automorphisms from the
commutative setting and of introducing a paradigm for studying automorphisms
in noncommutative projective geometry it is natural, and necessary, to work
with groupoids - that is, categories with more than one object but with all
morphisms invertible.

Noncommutative algebras “close enough” to a given commutative algebra are of
course not isomorphic. However, certain module categories associated to these
algebras will be equivalent and even isomorphic - induced by twists of these
algebras, explained below - and we wish to remember _how_ these categories are
equivalent. The morphisms between objects in a groupoid precisely retain this
information. We want to say that certain objects are the same without
identifying them. In Baez’s phrasing, “groupoids are like sets with
symmetries” ([Bae]). This point of view has also been expounded by Brown
[Bro87] and Weinstein [Wei96] among others.

As such, we take a collection of noncommutative spaces that model our chosen
classical space and, instead of allowing all possible morphisms between them,
concentrate on just the isomorphisms between them.

That is, we do something like

  * •

consider all spaces and all morphisms between them: a category of
noncommutative schemes over $k$,

  * •

throw away all objects not sufficiently like our chosen one and

  * •

throw away any non-invertible morphisms.

This groupoid typically has many connected components. This will be the basis
for the construction we describe. However, before that, we need to clarify
what we will mean by a noncommutative space. We will adopt the approach of
noncommutative projective geometry, the outline of which we will give below in
Section 2 for the reader unfamiliar with this area.

###  1.2 Contents

In Section 2 we recall both a theorem of Serre providing an equivalence
between the category of quasi-coherent sheaves on a projective scheme and a
certain module category over its homogeneous coordinate ring, and a
noncommutative analogue due to Artin and Zhang. These results motivate the
notion that such module categories should be considered as noncommutative
projective schemes. We define the class of algebras that we take to be
homogeneous coordinate rings of noncommutative projective spaces and then
recall the definition of a Zhang twist of a graded algebra. This is a family
of automorphisms of the underlying vector space which induce a new associative
multiplication, “twisting” the original multiplication. Finally, we recall the
definition of the point scheme, a classical scheme which is an important
geometric invariant of noncommutative projective schemes.

We introduce and define our main object of study, the groupoid
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$, in Section 3. This is a groupoid
whose objects are certain module categories for the coordinate rings of
noncommutative $\mathbb{P}^{n}$s and whose morphisms are particular
equivalences of these categories induced by Zhang twists. That this is a
groupoid is deduced from a result of Zhang, and that twisting preserves the
properties of a noncommutative $\mathbb{P}^{n}$. We also briefly introduce
slice categories associated to $\mathscr{NC}\left(\mathbb{P}^{n}\right)$,
which allow us to study all the twists - that is, all of the “generalized
automorphisms” - of a given noncommutative $\mathbb{P}^{n}$.

In Section 4 we begin studying the groupoid, considering the case $n=1$. Here
$\mathscr{NC}\left(\mathbb{P}^{1}\right)$ has only one connected component
(which of course contains the object associated to the commutative polynomial
ring).

We study $\mathscr{NC}\left(\mathbb{P}^{2}\right)$ in Section 5. Here, the
behaviour is considerably more varied and subtle. As such, more sophisticated
techniques are required to analyse the groupoid, in particular the geometry of
the point scheme. Using results of Artin-Tate-Van den Bergh and Mori, relating
twists of algebras in this dimension to isomorphisms between their point
schemes (Proposition 36 and Theorem 37), we study the connected components of
the commutative polynomial ring and those of certain quantum deformations of
it. In the latter case, we recover some automorphisms not arising from graded
algebra automorphisms of these deformations.

In Section 6 we consider the groupoid for general $n$. After introducing
Mori’s notion of a _geometric algebra_ , we generalize Proposition 36, now
connecting Zhang twists of such algebras with automorphisms of their
associated point schemes, for arbitrary $n$. Let $\mathcal{A}$ be an object in
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ associated to a noncommutative
$\mathbb{P}^{n}$ which is geometric in Mori’s sense. Let
$E\subset\mathbb{P}^{n}$ be its associated point scheme, $\sigma$ a certain
automorphism of $E$ and $\mathcal{L}$ the very ample line bundle giving the
embedding into $\mathbb{P}^{n}$. The main result of the paper is the
following, Theorem 45:

######  Theorem.

Let $\mathcal{A}$ be a geometric noncommutative $\mathbb{P}^{n}$ with
homogeneous coordinate ring $A$ and let $(E,\sigma,\mathcal{L})$ be its
associated triple. Then

$\uuline{\operatorname{Aut}}(\mathcal{A})\cong\operatorname{Aut}(E\uparrow\mathbb{P}^{n}).$

Here, $\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$ is the group of
automorphisms of the point scheme that extend to $\mathbb{P}^{n}$. The group
$\uuline{\operatorname{Aut}}(\mathcal{A})$ is that obtained from the slice
category associated to $\mathcal{A}$, modulo graded Morita equivalence. The
theorem thus says that, up to graded Morita equivalence, the generalized
automorphisms of a geometric noncommutative $\mathbb{P}^{n}$ are precisely
controlled by the geometry of the point scheme. We conclude with two examples
of the theorem: quantum deformations of polynomial rings in $n$ variables; and
Sklyanin algebras, an important class of noncommutative $\mathbb{P}^{2}$s,
before briefly discussing an alternative approach to considering
noncommutative $\mathbb{P}^{2}$s in an appendix.

### Acknowledgements

The authors would like to thank Yemon Choi, Andrew Davies, Sian Fryer, Sue
Sierra and Toby Stafford for helpful discussions during the development of
this work, as well as the many seminar audience members whose questions aided
our efforts to clarify the presentation of the results herein.

Financial support from the EPSRC (EP/M001148/1) and the ERC project ModRed is
gratefully acknowledged.

##  2 Preliminaries on noncommutative projective geometry

Let $k$ be an algebraically closed field of characteristic $0$ and let $A$ be
a right Noetherian $\mathbb{Z}$-graded $k$-algebra. Let $\operatorname{Gr}A$
denote the category of (not necessarily finitely generated)
$\mathbb{Z}$-graded right $A$-modules with morphisms of degree $0$.
Henceforth, by any graded module category over a $k$-algebra $A$, we mean the
category of right $A$-modules. We denote by $(1)$ the grading shift functor on
$\operatorname{Gr}A$. That is, for $M$ a $\mathbb{Z}$-graded right $A$-module,
and $n\in\mathbb{Z}$ denote by $M(n)$ the graded module whose degree $m$
component is given by $M(n)_{m}=M_{n+m}$. Recall from [AZ94] that, for $M$ in
$\operatorname{Gr}A$, an element $x\in M$ is called torsion if $xA_{\geq s}=0$
for some s. The set of torsion elements of $M$ forms a graded $A$-submodule,
$\tau(M)$. The module $M$ is called torsion if $\tau(M)=M$ and torsion-free if
$\tau(M)=0$. The submodule $\tau(M)$ is the smallest such that the quotient
$M/\tau(M)$ is torsion-free. The collection of all torsion modules forms a
Serre subcategory of $\operatorname{Gr}A$ which we denote
$\operatorname{Tors}(A)$.

Let $\pi:\operatorname{Gr}A\to\operatorname{Gr}A/\operatorname{Tors}(A)$ be
the canonical quotient functor and define
$\operatorname{QGr}A\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\operatorname{Gr}A/\operatorname{Tors}(A)$.
One can roughly think of $\operatorname{QGr}A$ as the category whose objects
are the same as those of $\operatorname{Gr}A$, but with torsion modules
isomorphic to the zero module. More precisely, the objects of
$\operatorname{QGr}A$ are those of $\operatorname{Gr}A$ and the morphisms can
be described as

$\operatorname{Hom}_{\operatorname{QGr}A}(\pi(M),\pi(N))\cong\varinjlim\operatorname{Hom}_{\operatorname{Gr}A}(M^{\prime},N/\tau(N))$

where the limit runs over the quasi-directed category of submodules
$M^{\prime}$ of $M$ such that $M/M^{\prime}$ is torsion. See [AZ94] for more
details.

To motivate the definition of a noncommutative projective space we want to
use, we recall a classical result of Serre and its noncommutative analogue.

Let $X$ be a projective scheme and $\mathcal{L}$ be a line bundle on $X$. To
this data one can associate the homogeneous coordinate ring
$B=B(X,\mathcal{L})=\bigoplus_{n\in\mathbb{N}}\Gamma(X,\mathcal{L}^{n})$.
Given a quasi-coherent sheaf $\mathcal{M}$ on $X$ we may similarly define
$\Gamma_{h}(\mathcal{M})\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\bigoplus_{n\in\mathbb{N}}\Gamma(X,\mathcal{M}\otimes\mathcal{L}^{n})$,
which is a graded $B$-module. Composing with $\pi$, we obtain a functor
$\overline{\Gamma}_{h}\colon\operatorname{QCoh}(X)\to\operatorname{QGr}B$.
Finally, an $\mathbb{N}$-graded $k$-algebra $A$ is called finitely graded if
$\dim_{k}A_{i}<\infty$ and connected if $A_{0}=k$. By describing such an
algebra as connected graded, we mean that it is both connected and finitely
graded.

Then we have:

######  Theorem 5 ([Ser55]).

  1. (i)

Let $\mathcal{L}$ be an ample line bundle on a projective scheme $X$. Then
$\overline{\Gamma}_{h}$ defines an equivalence of categories between
$\operatorname{QCoh}(X)$ and $\operatorname{QGr}B$.

  2. (ii)

If $A$ is a commutative connected graded $k$-algebra generated in degree 1
then there exists a line bundle $\mathcal{L}$ over $X=\operatorname{Proj}(A)$
such that $A=B(X,\mathcal{L})$, up to a finite-dimensional vector space.
Again, $\operatorname{QGr}A\simeq\operatorname{QCoh}(X)$.

That is, in the commutative setting, studying the category of quasi-coherent
sheaves on a projective variety is essentially the same as studying graded
modules modulo torsion over a graded ring.

This suggests that we could naturally consider noncommutative projective
schemes to be categories of the form $\operatorname{QGr}B$ for $B$ in some
suitable family of (not necessarily commutative) graded rings. For this
definition to be reasonable and useful, one would like to consider rings for
which an analogue of Serre’s theorem holds and for this to form a sufficiently
broad and interesting class of rings to study. The rings to which this
analogue pertains are characterized in [AZ94, Theorem 4.5].

In [AZ94], a categorical version of the homogeneous coordinate ring is
defined: namely, given the triple $(\mathcal{C},\mathcal{O},s)$ of a
Grothendieck category $\mathcal{C}$, a Noetherian object $\mathcal{O}$ in
$\mathcal{C}$ and an autoequivalence $s$, one can define the homogeneous
coordinate ring

$B=\Gamma_{h}(\mathcal{C},\mathcal{O},s)\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\bigoplus^{\infty}_{i=0}\operatorname{Hom}(\mathcal{O},s^{i}\mathcal{O})$

with the triple $(\operatorname{QCoh}(X),\mathcal{O}_{X},-\otimes\mathcal{L})$
for $X$ a projective scheme and $\mathcal{L}$ a line bundle being the key
example. Indeed, after giving a categorical definition for a pair
$(\mathcal{O},s)$ to be ample and, assuming certain additional technical
conditions on the category $\mathcal{C}$ and on a right Noetherian, finitely
graded $k$-algebra $A$, the authors show that:

  * •

Given a triple $(\mathcal{C},\mathcal{O},s)$ as above,
$\mathcal{C}\simeq\operatorname{QGr}(\Gamma_{h}(\mathcal{C},\mathcal{O},s))$

  * •

For such a graded $k$-algebra $A$, $A\cong\Gamma_{h}(\operatorname{QGr}A,\pi
A,(1))$ in sufficiently high degrees, and $(\pi A,(1))$ is ample.

(see [AZ94, §4] and [SVdB01, §2] for a more detailed exposition).

######  Remark 6.

Note that in [AZ94, Theorem 4.5(1)] the first result is stated as an
isomorphism of triples of the form $\left(\mathcal{C},\mathcal{O},s\right)$ as
above. Such an isomorphism of triples requires only that the relevant
categories, in our case $\mathcal{C}$ and
$\operatorname{QGr}(\Gamma_{h}(\mathcal{C},\mathcal{O},s))$ are equivalent and
not themselves isomorphic (see [AZ94, p.237]).

All of the $k$-algebras we consider henceforth satisfy the conditions of
Artin-Zhang’s theorem.

To further support the idea that studying module categories of the form
$\operatorname{QGr}A$ is as fruitful as studying their associated rings or
projective schemes - particularly with regard to our focus on automorphism
groups - recall the following result of Bondal-Orlov:

######  Theorem 7 (cf. [BO01, Theorem 3.1]).

If $X$ is a smooth irreducible projective variety with an ample canonical or
anticanonical sheaf, then the group of isomorphism classes of exact
autoequivalences is given by
$\operatorname{Aut}(D^{b}_{\operatorname{Coh}}(X))\cong\operatorname{Aut}(X)\ltimes(\operatorname{Pic}(X)\oplus\mathbb{Z})$.

It follows that
$\operatorname{Aut}(\operatorname{Coh}(X))\cong\operatorname{Aut}(X)\ltimes\operatorname{Pic}(X)$,
so that if we wish to study invertible morphisms of our noncommutative spaces
- meaning equivalences between categories of the form $\operatorname{QGr}B$ \-
then these do (essentially) correspond to considering automorphisms of
varieties in the commutative case.

Indeed, as is shown in [AZ94, Corollary 6.9] and [AVdB90, Proposition 2.15],
any automorphism of the category of coherent sheaves of $\mathcal{A}$-modules,
for $\mathcal{A}$ a coherent sheaf of $\mathcal{O}_{X}$-algebras, is of the
form $\sigma_{*}(-\otimes\mathcal{L})$ where $\sigma\in\operatorname{Aut}(X)$
and $\mathcal{L}$ is an invertible $(\mathcal{A},\mathcal{A})$-bimodule. As
such, following Artin-Zhang’s theorem, one can study the so-called _twisted_
homogeneous coordinate rings
$B=\Gamma_{h}(\operatorname{QCoh}(X),\mathcal{O}_{X},\sigma_{*}(-\otimes\mathcal{L}))$
and their module categories $\operatorname{QGr}B$ via an understanding of
$\operatorname{Pic}(X)$ and of the automorphisms of $X$.

Given that projective varieties embed into an ambient projective space, we now
restrict to considering noncommutative analogues of $\mathbb{P}^{n}$.

That is, we want to define the class of noncommutative $k$-algebras that we
consider as giving rise to noncommutative projective $n$-space \- pairs of the
form $\left(\operatorname{QGr}A,\pi A\right)$ where the $k$-algebra $A$ shares
important homological and algebraic properties with a graded polynomial ring
in $n+1$ variables. We adopt the definition given in section 4 of [Kee03]. We
first recall some basic definitions giving noncommutative analogues of
homological properties enjoyed by commutative polynomial rings. As such rings
are Gorenstein, we make the following:

######  Definition 8.

Let $A$ be a connected graded $k$-algebra. Then $A$ is called _Artin-Schelter
Gorenstein_ (AS-Gorenstein) if we have the following:

  1. (i)

$A$ has finite left and right injective dimension $n$;

  2. (ii)

for some shift $l$,

$\operatorname{Ext}^{i}_{\operatorname{Gr}A}\left(k,A\right)=\begin{cases}0&\text{if}~{}i\neq
n\\\ k(l)&\text{if}~{}i=n.\end{cases}$

######  Remark 9.

One reason for making this definition is that a Noetherian AS-Gorenstein ring
$A$ satisfies certain hypotheses required such that a noncommutative version
of Serre duality holds for $(\operatorname{QGr}A,\pi A)$. That is, there
exists an object $\omega^{\circ}\in D^{b}(\operatorname{QGr}A)$ and a natural
isomorphism

$\mathrm{RHom}(-,\omega^{\circ})\cong\mathrm{RHom}(\pi A,-)^{*}$

See [SVdB01, §7.4] and the references therein.

In keeping with the fact that polynomial rings also have finite global
dimension we define:

######  Definition 10.

Let $A$ be a connected graded $k$-algebra. Then $A$ is _Artin-Schelter
regular_ (AS-regular) of dimension $n$ if:

  1. (i)

$A$ has right and left global dimension $n$;

  2. (ii)

$\operatorname{GKdim}(A)<\infty$; and

  3. (iii)

$A$ is AS-Gorenstein of injective dimension $n$.

We direct the reader to §8 of [SVdB01] for more detailed intuition and
motivation behind this definition. With these notions in hand, we can define
the class of algebras we are interested in.

######  Definition 11 (cf. [Kee03, Definition 4.3]).

Let $A$ be a connected graded $k$-algebra. We say that the pair
$\left(\operatorname{QGr}A,\pi A\right)$ is a _noncommutative_
$\mathbb{P}^{n}$ and that $A$ is _the (homogeneous) coordinate ring of a
noncommutative_ $\mathbb{P}^{n}$ if

  1. (i)

$A$ is a Noetherian domain;

  2. (ii)

$A$ is AS-regular of dimension $n+1$;

  3. (iii)

if $l$ is the shift from Definition 8, then $l=n+1$;

  4. (iv)

$A$ is generated in degree 1; and

  5. (v)

$A$ has Hilbert series $\left(1-t\right)^{-n-1}$.

As Keeler notes, his characterization is more restrictive than is found
elsewhere in the literature.

######  Remark 12.

Note that we have reordered the conditions in Keeler’s definition so that $A$
being Noetherian is placed first. In this case, $A$ has equal left and right
global dimensions, and equal left and right injective dimensions. Moreover,
definitions 8 and 10 are left-right symmetric and so we can omit the
conditions placed on $A^{op}$ by Keeler.

Before proceeding further, we recall the definitions related to the notion of
a Zhang twist, the relevance of which will become clear shortly.

######  Definition 13 ([Zha96, Definition 2.1]).

Let $\mathbb{G}$ be a semigroup with identity $e$ and let $A$ be a
$\mathbb{G}$-graded $k$-algebra. A _twisting system_ for $A$ is a set of
graded $k$-linear $A$-automorphisms $\tau=\left\\{\tau_{g}\mid
g\in\mathbb{G}\right\\}$ such that

$\tau_{g}\left(y\tau_{h}\left(z\right)\right)=\tau_{g}\left(y\right)\tau_{gh}\left(z\right)$

for all $g,h,l\in\mathbb{G}$, $y\in A_{h}$ and $z\in A_{l}$.

Any semigroup homomorphism
$\mathbb{G}\rightarrow\operatorname{Aut}_{\mathbb{G}}\left(A\right)$ given by
$g\mapsto\tau_{g}$ produces a twisting system $\left\\{\tau_{g}\mid
g\in\mathbb{G}\right\\}$ where $\operatorname{Aut}_{\mathbb{G}}\left(A\right)$
denotes the group of graded algebra automorphisms of $A$. Twisting systems
arising from semigroup homomorphisms as above are called _algebraic_. For
example, for $\mathbb{G}=\mathbb{Z}$ and
$f\in\operatorname{Aut}_{\mathbb{Z}}\left(A\right)$, we have the homomorphism
$n\mapsto f^{n}$ and corresponding twisting system $\left\\{f^{n}\mid
n\in\mathbb{Z}\right\\}$.

In the case of a connected $\mathbb{N}$-graded algebra $A=\bigoplus A_{i}$
generated in degree 1, the fundamental defining relations for a Zhang twist
are equivalent to the following ([Zha96, p. 284]). A twisting system in this
case is a set $\tau=\\{\tau_{n}\mid n\in\mathbb{N}\\}$ of graded linear
isomorphisms satisfying

$\tau_{m}(ab)=\tau_{m}(a)\tau_{m+n}\tau_{n}^{-1}(b)$

for $a\in A_{n}$. Notice that if $\tau_{m+n}\tau_{n}^{-1}$ were equal to
$\tau_{m}$, we would have that $\tau_{m}$ was an algebra map (and in fact then
an automorphism). So this formulation makes visible another way in which
twisting systems are close to, but are more general than, algebra
automorphisms.

As with other notions of twisting, such as twisting by automorphisms or by
group 2-cocycles, given a twisting system $\tau$ for $A$, we can form a new
algebra $A^{\tau}$, whose underlying $k$-vector space is the same as that of
$A$ but whose multiplication is twisted by $\tau$.

######  Definition 14 ([Zha96, Definition and Proposition 2.3]).

Let $A$ be a $\mathbb{G}$-graded $k$-algebra and $\tau=\left\\{\tau_{g}\mid
g\in\mathbb{G}\right\\}$ be a twisting system for $A$. One defines the
_twisted algebra of $A$ by $\tau$_, denoted $A^{\tau}$, as the triple
$\left(\oplus_{g}A_{g},\star,1_{\tau}\right)$ where $\star$ is an associative,
graded multiplication given by

$y\star
z\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}y\tau_{h}\left(z\right)$

for $y\in A_{h}$ and $z\in A_{l}$ and where
$1_{\tau}\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\tau_{e}^{-1}\left(1_{A}\right)$
is the identity in $A^{\tau}$.

The following lemma concerns twisting systems for quadratic algebras and
generalizes an observation of Zhang, made in examining Example 5.12 in
[Zha96].

######  Lemma 15.

Let $A$ be a connected $\mathbb{N}$-graded finitely generated algebra,
generated in degree 1 and quadratic. Let $\tau=\\{\tau_{m}\mid
m\in\mathbb{N}\\}$ be a twisting system. Suppose the graded linear isomorphism
$\tau_{1}$ is additionally an algebra automorphism and set
$\tau^{\prime}=\\{\tau_{1}^{m}\mid m\in\mathbb{N}\\}$. Then we have an
isomorphism of algebras $A^{\tau}\cong A^{\tau^{\prime}}$.

###### Proof:.

Since $A$ is generated in degree 1, we may present $A$ as a quotient of the
tensor algebra on $A_{1}$: let $\rho\colon T(A_{1})\to A$ be the associated
canonical map, so that $\operatorname{ker}\rho\cap A_{2}$ generates the ideal
of (quadratic) relations defining $A$. Fix a basis $\\{x_{i}\mid 1\leq i\leq
r\\}$ for $A_{1}$.

Let $R\in A_{2}$ be a quadratic homogeneous relator, i.e.
$R\in\operatorname{ker}\rho\cap A_{2}$. Write
$R=\sum_{\underline{a}\in\mathbb{N}^{2}}\alpha_{\underline{a}}X^{\underline{a}}$,
where
$X^{\underline{a}}\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}x_{1}^{\underline{a}_{1}}x_{2}^{\underline{a}_{2}}\dotsm
x_{r}^{\underline{a}_{r}}$. Note that $\sum\underline{a}_{i}=2$, since $R$ is
chosen to be quadratic.

For each monomial $X^{\underline{a}}$, let $s(\underline{a})=\max\\{i\mid
a_{i}\neq 0\\}$ and set
$X^{\underline{a}}_{\tau}=x_{1}^{\underline{a}_{1}}\dotsm
x_{s(\underline{a})}^{\underline{a}_{s(\underline{a})}-1}\ast\tau_{1}^{-1}(x_{s(\underline{a})})$.

Arguing similarly to Example 5.12 of [Zha96], in $A^{\tau}$ we have

$\displaystyle R^{\tau}$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\sum_{\underline{a}\in\mathbb{N}^{r}}\alpha_{\underline{a}}X_{\tau}^{\underline{a}}$
$\displaystyle=\sum_{\underline{a}\in\mathbb{N}^{r}}\alpha_{\underline{a}}x_{1}^{\underline{a}_{1}}\dotsm
x_{s(\underline{a})}^{\underline{a}_{s(\underline{a})}-1}\ast\tau_{1}^{-1}(x_{s(\underline{a})})$
$\displaystyle=\sum_{\underline{a}\in\mathbb{N}^{r}}\alpha_{\underline{a}}x_{1}^{\underline{a}_{1}}\dotsm
x_{s(\underline{a})}^{\underline{a}_{s(\underline{a})}-1}\tau_{1}(\tau_{1}^{-1}(x_{s(\underline{a})}))$
(1)
$\displaystyle=\sum_{\underline{a}\in\mathbb{N}^{r}}\alpha_{\underline{a}}X^{\underline{a}}$
$\displaystyle=0.$

Here (1) is a consequence of $R$ being quadratic, so that
$x_{1}^{\underline{a}_{1}}\dotsm
x_{s(\underline{a})}^{\underline{a}_{s(\underline{a})}-1}$ has degree 1;
recall that $y\ast z=y\tau_{h}(z)$ with $h=\deg y$.

We see that $A^{\tau^{\prime}}$ is also generated by $A_{1}$ and $R^{\tau}=0$
in $A^{\tau^{\prime}}$; only the map $\tau_{1}$ is involved in the above
calculation. Hence $A^{\tau}$ is isomorphic as a graded algebra to
$A^{\tau^{\prime}}$. ∎

######  Remark 16.

Note that the condition on $\tau_{1}$ in the lemma is necessary, in the sense
that $\tau^{\prime}=\\{\tau_{1}^{m}\mid m\in\mathbb{N}\\}$ is a twisting
system if and only if $\tau_{1}$ is an algebra automorphism, under the other
assumptions of the lemma. This follows from the fact that the defining
property of a twisting system is equivalent to the above-stated identity,

$\tau_{m}(yz)=\tau_{m}(y)\tau_{m+n}\tau_{n}^{-1}(z)$

for all $y\in A_{m}$, $z\in A_{n}$.

If $\tau^{\prime}=\\{\tau_{1}^{m}\mid m\in\mathbb{N}\\}$ is a twisting system,
then $\tau_{m+n}^{\prime}=\tau_{1}^{m+n}$ satisfies

$\tau_{m}(yz)=\tau_{m}(y)\tau_{m}(z)$

and in particular $\tau_{1}$ is an algebra map. The converse is clear.

Given a coordinate ring of a noncommutative $\mathbb{P}^{n}$ (that is, an
algebra $A$ satisfying the conditions in Definition 11), Theorem 1.3 in
[Zha96] states that if $A$ is a Noetherian domain, then so is any twist of it.
Clearly, the degree of an element is preserved by twisting, and hence any
twist of $A$ is generated in degree 1 with Hilbert series
$\left(1-t\right)^{-n}$. From Theorem 5.11 of [Zha96], we have that any twist
of $A$ will be AS-regular of the same dimension. Finally, we have that for
some twisted algebra $B\cong A^{\tau}$,
$\operatorname{Ext}^{i}_{\operatorname{Gr}A}\left(k,A\right)\cong\operatorname{Ext}^{i}_{\operatorname{Gr}B}\left(k^{\tau},B\right)$
where $k$ is the $A$-module $A/A_{\geq 1}$. Hence $k^{\tau}\cong
A^{\tau}/A^{\tau}_{\geq 1}\cong B/B_{\geq 1}\cong k$ as $B$-modules. The class
of connected graded $k$-algebras defined above is thus closed under Zhang
twisting. With this in mind, we make the following definition.

######  Definition 17.

Let $A$ and $B$ be $\mathbb{G}$-graded $k$-algebras. We say that $A$ and $B$
are twist-equivalent if there exists a twisting system $\tau$ for $A$ such
that $B\cong A^{\tau}$ as graded $k$-algebras.

As in various incarnations of noncommutative algebraic geometry, it is useful
to define classical commutative objects which capture some of the information
of their noncommutative counterparts. We finish this section by recalling an
important invariant of noncommutative projective schemes - the point scheme -
introduced by Artin, Tate and Van den Bergh. To do so, we wish to define a
noncommutative analogue of a point on a projective scheme and so study
irreducible objects of $\operatorname{QGr}A$. As such, we make the following:

######  Definition 18.

Let $A$ satisfy the conditions of Definition 11. A _point module_ for $A$ is a
graded cyclic module with Hilbert series $1/\left(1-t\right)$.

######  Definition 19.

Let $A$ be as above and let $d\in\mathbb{N}$. A _truncated point module_ of
length $d+1$ is a graded cyclic $A$-module with Hilbert series
$\sum_{i=0}^{d}t^{i}$.

Before defining the point scheme, it remains to introduce the notion of a
_multilinearization_ of an element in the defining ideal of $A$.

We adopt the shorthand
$\overline{\left\\{n\right\\}}=\left\\{0,1,\ldots,n\right\\}$ and
$\underline{i}=\left\\{i_{0},\ldots,i_{d-1}\right\\}$ for a vector in
$\overline{\left\\{n-1\right\\}}^{d}$. Let $A=T(V)/I$ be as before, where
$V=kx_{0}+\cdots+kx_{n-1}$ and let $f\in I_{d}$ for some $d\in\mathbb{N}$.
Thus, $f=\sum_{I}\alpha_{\underline{i}}x_{i_{0}}\ldots x_{i_{d-1}}$ for some
$\alpha_{\underline{i}}\in k$.

######  Definition 20 ([ATVdB90]).

Define $\widetilde{f}$ to be the element of the homogeneous coordinate ring of
$\left(\mathbb{P}^{n-1}\right)^{\times d}$ given by

$\sum_{I}\alpha_{\underline{i}}x_{i_{0},0}\ldots x_{i_{d-1},d-1}$

where the coordinate ring of the $\left(j+1\right)^{st}$ copy of
$\mathbb{P}^{n-1}$ is given by $k\left[x_{0,j},\ldots,x_{n-1,j}\right]$. The
element $\widetilde{f}$ is the multilinearization of $f$.

As explained in [ATVdB90, §3], truncated point modules of length $d+1$ are
parametrized by a projective scheme $\Gamma_{d}$ \- that is, $\Gamma_{d}$
represents the functor of taking flat families of truncated point modules and
is thus independent of the chosen presentation of $A$. By sending a truncated
point module of length $d+1$ to one of length $d$ by factoring out the highest
degree component, a morphism of schemes $\Gamma_{d}\rightarrow\Gamma_{d-1}$ is
induced, corresponding to the forgetting of the last component in the
projection $\left(\mathbb{P}^{n-1}\right)^{\times
d}\rightarrow\left(\mathbb{P}^{n-1}\right)^{\times d-1}$. Finally, we come to:

######  Definition 21 ([ATVdB90]).

The _point scheme_ of $A$, $\Gamma(A)$, is defined as the inverse limit of the
diagram:

$\Gamma_{0}\leftarrow\Gamma_{1}\leftarrow\cdots\leftarrow\Gamma_{d-1}\leftarrow\Gamma_{d}\leftarrow\cdots$

The significance of the point scheme in our framework will become apparent
later.

######  Remark 22.

In general, $\Gamma(A)$ is a pro-scheme, representing the functor of taking
point modules of $A$. However, it is often the case that the projection
$\Gamma_{d}\rightarrow\Gamma_{d-1}$ is an isomorphism for some $d$. In that
case, the inverse system is constant for $i\geq d-1$ and
$\Gamma(A)\cong\Gamma_{d-1}$ is a scheme. This is true of regular algebras of
dimension 3, considered in [ATVdB90], and of the algebras we are concerned
with here.

##  3 Definition of $\mathscr{NC}\left(\mathbb{P}^{n}\right)$

The above description of Zhang twisting is rather concrete, which has
advantages when one wants to explicitly calculate specific twists, as we shall
later. However, even demonstrating some elementary properties, such as showing
that twist-equivalence is actually an equivalence relation, can be cumbersome
when using this definition.

Along with the definition, the other key insight of Zhang was that twist-
equivalence of $A$ and $B$ is equivalent to the existence of a well-behaved
functor between certain corresponding categories of graded modules for $A$ and
$B$. From this, it is easy to see that twisting gives an equivalence relation.
We briefly outline the results of Zhang in this direction, taken from [Zha96],
in order to motivate the definition of the groupoid we wish to study.

Assume that $A$ is a connected $\mathbb{Z}$-graded $k$-algebra. In Definition
11, we assume a lot more about algebras that are associated to noncommutative
$\mathbb{P}^{n}$s, but even this is already reasonably strong.

Let $\operatorname{GR}A$ denote the category of $\mathbb{Z}$-graded
$A$-modules with morphisms being graded morphisms of any degree - that is,
$\operatorname{Hom}_{\operatorname{GR}A}(M,N)=\bigoplus_{r\in\mathbb{Z}}\operatorname{Hom}_{\operatorname{Gr}A}(M,N(r))$
(note that Zhang writes $\underline{\operatorname{Hom}}(M,N)$).

We also note that $\operatorname{Hom}_{\operatorname{GR}A}(M,M)$, denoted
$\Gamma(M)$ by Zhang, is important in a key theorem of Zhang ([Zha96, Theorem
3.3]), most notably in the case $M=A_{A}$. We define
$\operatorname{GrEnd}(A_{A})\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\operatorname{Hom}_{\operatorname{GR}A}(A_{A},A_{A})$,
the graded endomorphism algebra of $A_{A}$ (we avoid the notation $\Gamma$,
which we used earlier for the point scheme of $A$). By the proof of [Zha96,
Theorem 3.4], with the above assumptions, $\operatorname{GrEnd}(A_{A})\cong
A$; that is, by analogy with the ungraded theory, $A$ is isomorphic to its
graded endomorphism algebra.

We say that $A$ and $B$ are graded Morita equivalent if $\operatorname{GR}A$
is equivalent to $\operatorname{GR}B$ by a graded functor, i.e. a functor
which induces a map of graded rings on $\operatorname{Hom}$ spaces. For
connected graded algebras, $A$ and $B$ are graded Morita equivalent if, and
only if, they are isomorphic.

As previously, we say $A$ and $B$ are twist-equivalent if $B$ is isomorphic to
a Zhang twist of $A$. By combining results of Artin-Zhang and Zhang, as
detailed in the proof, we obtain the following.

######  Proposition 23.

Let $A$ and $B$ be coordinate rings of noncommutative $\mathbb{P}^{n}$s, so
that $\left(\operatorname{QGr}A,\pi A\right)$ and
$\left(\operatorname{QGr}B,\pi B\right)$ are noncommutative $\mathbb{P}^{n}$s.
The following are equivalent:

  1. (i)

$A$ and $B$ are twist-equivalent.

  2. (ii)

$\operatorname{Gr}A$ is isomorphic to $\operatorname{Gr}B$.

  3. (iii)

$\operatorname{QGr}A$ is equivalent to $\operatorname{QGr}B$ via a functor
$\mathcal{F}$ and

$\mathcal{F}((\pi A)(n))\cong(\pi B)(n)\ \text{for all}\ n$ (SSS)

(the “preserves shifts of the structure sheaf” condition).

Furthermore $A$ and $B$ are graded Morita equivalent, i.e.
$\operatorname{GR}A$ is equivalent to $\operatorname{GR}B$ via a graded
functor if, and only if, $B\cong A^{\tau}\cong A$.

###### Proof:.

Consider the following assertions:

  1. (1)

$B$ is isomorphic to a Zhang twist of $A$;

  2. (2)

$\operatorname{Gr}A$ is equivalent to $\operatorname{Gr}B$;

  3. (3)

$\operatorname{Gr}A$ is isomorphic to $\operatorname{Gr}B$;

  4. (4)

$\operatorname{Gr}A$ is equivalent to $\operatorname{Gr}B$ via a functor $F$
and

$F(A(n))\cong B(n)\ \text{for all}\ n;$ (GrSSS)

  5. (5)

$\operatorname{Gr}A$ is isomorphic to $\operatorname{Gr}B$ and (GrSSS) holds;

  6. (6)

$\operatorname{QGr}A$ is equivalent to $\operatorname{QGr}B$;

  7. (7)

$\operatorname{QGr}A$ is equivalent to $\operatorname{QGr}B$ and (SSS) holds
for $\pi A$ and $\pi B$;

  8. (8)

$\operatorname{QGr}A$ is isomorphic to $\operatorname{QGr}B$ and (SSS) holds
for $\pi A$ and $\pi B$

Clearly,

  * •

(4) implies (2),

  * •

(5) implies (3) implies (2) and

  * •

(8) implies (7) implies (6).

The first fundamental result of Zhang’s paper is [Zha96, Theorem 3.1], which
proves that (1) implies (3). Next, [Zha96, Theorem 3.4] proves that (1) holds
if and only if (4). Notice that in the course of proving (4) implies (1), one
has that $\operatorname{GrEnd}(A_{A})\cong A$,
$\operatorname{GrEnd}(B_{B})\cong B$ and
$\operatorname{GrEnd}(B)\cong\operatorname{GrEnd}(A)^{\tau}$, and hence
$B\cong A^{\tau}$.

Now, by [Zha96, Theorem 3.5], recalling that we are assuming that $A$ and $B$
are connected graded, we have (1) if and only if (2). The proof of [Zha96,
Theorem 3.5] (Case 1) also tells us that (2) implies (4), that is, the
condition (SSS) holds automatically for connected $\mathbb{Z}$-graded
algebras. Correspondingly [Zha96, Theorem 3.5] works for $\mathbb{N}$-graded
algebras also.

Assuming further that $A$, $B$ are $\mathbb{N}$-graded right Noetherian AS-
Gorenstein algebras - as we do, as these are among the conditions of
Definition 11 \- we have the following. The AS-Gorenstein condition implies,
by Artin-Zhang ([AZ94]), that

$\operatorname{GrEnd}(\pi
A)\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\bigoplus_{n\in\mathbb{Z}}\operatorname{Hom}_{\operatorname{QGr}A}(\pi
A,(\pi A)(n))\cong A$

which very much need not be the case in general.

Then by the proof of [Zha96, Theorem 3.7], one has

  * •

(5) implies (7) and, most importantly,

  * •

(1) if and only if (7).

The key point is that [Zha96, Theorem 3.7(2)] gives that the existence of an
equivalence of $\operatorname{QGr}A$ with $\operatorname{QGr}B$ satisfying
(SSS) implies $\operatorname{GrEnd}(\pi A)$ twist-equivalent to
$\operatorname{GrEnd}(\pi B)$, for not necessarily AS-Gorenstein algebras. As
noted above, having AS-Gorenstein as well yields
$A\cong\operatorname{GrEnd}(\pi A)$, $B\cong\operatorname{GrEnd}(\pi B)$ and
hence twist-equivalence of $A$ and $B$, that is, (7) implies (1).

That is,

$\ref{te1}\Leftrightarrow\ref{te2}\Leftrightarrow\ref{te3}\Leftrightarrow\ref{te4}\Leftrightarrow\ref{te7}$

and (5) and (8) both imply all of these equivalent statements. In particular
we have that (1) if and only if (3) if and only if (7), as claimed.

The final claim of the proposition is immediate from the discussion preceding
its statement. ∎

We wish to highlight that the list of conditions in Definition 11 is extremely
strong. In particular, as noted in Example 3.10 of [Zha96], for a
$\mathbb{Z}$-graded algebra $A$ with right progenerator $P$, $A$ and
$B\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\oplus_{n}\operatorname{Hom}_{\operatorname{Gr}A}(P,P(n))$
are graded Morita equivalent, and hence $\operatorname{Gr}A$ is equivalent to
$\operatorname{Gr}B$. However, in this generality $B$ is not always isomorphic
to a twist of $A$ whereas in our setting, where we have $\mathbb{N}$-graded
AS-Gorenstein algebras, these statements are equivalent.

The proof of Proposition 23 uses the following result due to Artin and Zhang
([AZ94])

######  Proposition 24.

Let $A$ be a coordinate ring of a noncommutative $\mathbb{P}^{n}$ and let
$\left(\operatorname{QGr}A,\pi A\right)$ be the associated noncommutative
projective space.

The graded endomorphism algebra of $\pi A$ in $\operatorname{QGr}A$,

$\operatorname{GrEnd}(\pi
A)\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\bigoplus_{n\in\mathbb{Z}}\operatorname{Hom}_{\operatorname{QGr}A}(\pi
A,(\pi A)(n))$

is isomorphic as graded $k$-algebras to $A$. ∎

Hence, in our setting the algebra $A$ can be recovered from the data
$(\operatorname{QGr}A,\pi A)$. We mention this result separately to highlight
that regarding the pair $\left(\operatorname{QGr}A,\pi A\right)$ as a
noncommutative projective space mirrors Grothendieck’s insight that to study a
scheme, one should study its category of quasi-coherent sheaves.

We thus take condition (iii) in Proposition 23, $\operatorname{QGr}A$ being
equivalent to $\operatorname{QGr}B$ and (SSS) holding, as the most appropriate
formulation of a noncommutative projective space of the three equivalent
conditions of the proposition, as it is closest to the statement in Serre’s
theorem. That is, given Grothendieck’s philosophy and Serre’s result, we
regard $\operatorname{QGr}A$ as the category of quasi-coherent sheaves on a
noncommutative space and regard such spaces as being equivalent if the
categories $\left(\operatorname{QGr}A,\pi A\right)$ and
$\left(\operatorname{QGr}B,\pi B\right)$ are equivalent.

It is useful to phrase certain statements in terms of properties of the
algebras that are coordinate rings of noncommutative $\mathbb{P}^{n}$s, as we
have being doing, only because not all of the conditions of Definition 11 have
a convenient expression internal to $\operatorname{QGr}A$.

Given that an algebra $A$ and its appropriate module category determine one
another, we now define our main object of study, the groupoid
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$:

######  Definition 25.

Let $\mathscr{NC}\left(\mathbb{P}^{n}\right)$ be the category whose objects
are the pairs $\left(\operatorname{QGr}A,\pi A\right)$ where $A$ is an algebra
satisfying the conditions of definition 11 and whose morphisms are
equivalences of categories
$\mathcal{F}:\operatorname{QGr}A\to\operatorname{QGr}B$ such that
$\mathcal{F}\left(\pi A(m)\right)\cong\pi B(m)$ for all $m\in\mathbb{N}$.

By Proposition 23, the groupoid $\mathscr{NC}\left(\mathbb{P}^{n}\right)$
partitions into connected components corresponding precisely to all of the
Zhang twists of any given algebra $A$ whose pair
$\left(\operatorname{QGr}A,\pi A\right)$ is in that component. For example,
for $n\geq 2$,
$\mathcal{O}(\mathbb{P}^{n})=k\left[x_{0},x_{1},\ldots,x_{n}\right]$ and
$\mathcal{O}_{q}(\mathbb{P}^{n}_{k})$ correspond to pairs in different
connected components. Henceforth, we call the component containing
$(\operatorname{QGr}\mathcal{O}(\mathbb{P}^{n}),\pi\mathcal{O}(\mathbb{P}^{n}))$
the _commutative component_.

Our initial project is to attempt to understand these connected components for
the cases $n=1$ and $n=2$ and, as such, to systematically introduce the study
of generalized automorphisms into this strand of noncommutative geometry, in
keeping with the classical literature of Artin, Stafford, Tate, Van den Bergh,
Zhang et al. and the more recent work of Pym ([Pym15]).

By considering the groupoid formed from noncommutative spaces and those
equivalences between them induced by Zhang twists, we can give categorical
interpretations of the twists of a given algebra $A$ germane to our project of
defining and studying an object that generalizes the graded automorphism group
of $A$. As such, we will soon consider the slice category associated to
$(\operatorname{QGr}A,\pi A)$, which allows us to focus attention on the
connected component of $\mathscr{NC}\left(\mathbb{P}^{n}\right)$ containing
$(\operatorname{QGr}A,\pi A)$.

This object will not be a group but in the two approaches we outline it can be
thought of as either a groupoid or a functor. This object does not form a
group because in general taking a Zhang twist of a given algebra does not
produce an isomorphic algebra. Composing Zhang twists thus involves passing
between different domains and codomains, and so between different module
categories. However, the groupoid precisely encodes the compositions that are
defined.

######  Remark 26.

The proof of Proposition 23 shows that we do not have a lot of scope to relax
our definition of $\mathscr{NC}\left(\mathbb{P}^{n}\right)$, to - for example
- obtain a groupoid with fewer components. (We will see later that for $n>1$,
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ has many components.) In particular,
the condition $\operatorname{Gr}A$ isomorphic to $\operatorname{Gr}B$ is seen
to imply and be implied by $\operatorname{Gr}A$ equivalent to
$\operatorname{Gr}B$, so weakening “isomorphic” to “equivalent” has no effect
here. We consider it to be an interesting but wide open question as to whether
there is a weaker form of equivalence than twist-equivalence yielding fewer
equivalence classes but retaining sufficient strength to obtain results
comparable with our main theorem below.

######  Remark 27.

We note that an alternative approach would be to use $\mathbb{Z}$-algebras, as
introduced by Bondal-Polishchuk ([BP93]). In that setting, twist-equivalence
corresponds precisely to isomorphism of the associated $\mathbb{Z}$-algebras.
However, as we wish to highlight the geometric aspects in our approach and
especially since point schemes play a pivotal role in our main theorem, we
have opted to follow closely the approaches of Artin-Tate-Van den Bergh and
Mori.

######  Remark 28.

An additional property on the algebras we consider that one might wish for,
with its roots in geometry, is the Calabi-Yau property. In work of Reyes-
Rogalski-Zhang ([RRZ14],[RRZ17]), a definition of a skew Calabi-Yau $A$ is
given (the term twisted Calabi-Yau being used elsewhere), part of which asks
for the existence of an algebra automorphism $\mu_{A}$ of $A$, known as the
Nakayama automorphism. Then the algebra is Calabi-Yau if $\mu_{A}$ is an inner
automorphism; the original definition of Calabi-Yau is due to Ginzburg
[Gin06].

We refer the reader to [RRZ14] for a comprehensive discussion but note that
our assumptions in Definition 11 in particular imply that homogeneous
coordinate rings of noncommutative $\mathbb{P}^{n}$s are skew Calabi-Yau
([RRZ14, Lemma 1.2]). In [RRZ14], the authors establish a relationship between
the Nakayama automorphism of a skew Calabi-Yau algebra and that of a twist by
a graded algebra automorphism, noting that not every skew Calabi-Yau algebra
can be twisted by an automorphism to a Calabi-Yau one.

Given that we have additional assumptions on our algebras and also that we
wish to consider all Zhang twists, not just those coming from automorphisms,
the question of whether or not each connected component of
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ contains a noncommutative projective
space whose homogeneous coordinate ring is Calabi-Yau remains open. We note
that Pym [Pym15] has an analogous statement in the setting of deformations,
but the analytic argument used there is not available in our framework.

We also note that in [RRZ17], the authors prove that we can detect whether the
algebras we consider are Calabi-Yau by examining the algebra
$E=\operatorname{Ext}^{\bullet}_{A}(k,k)$, specifically showing that $A$ is
Calabi-Yau if and only if $E$ is graded-symmetric.

We now adopt the calligraphic font
$\mathcal{A}\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\left(\operatorname{QGr}A,\pi
A\right)$ for objects of $\mathscr{NC}\left(\mathbb{P}^{n}\right)$. Given an
object $\mathcal{A}$, we wish to focus attention on the morphisms to
$\mathcal{A}$. The connected components of
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ do not in general have a terminal
object but, for a given $\mathcal{A}$, we can form the slice category
$\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A}$, whose terminal object
is the morphism $\text{id}_{\mathcal{A}}:\mathcal{A}\rightarrow\mathcal{A}$.
The objects of $\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A}$ are maps
$\mathcal{B}\rightarrow\mathcal{A}$ and the morphisms are the triangles:

$\mathcal{B}$$\mathcal{C}$$\mathcal{A}$$f$

where $f$ is a morphism in $\mathscr{NC}\left(\mathbb{P}^{n}\right)$.

The slice category $\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A}$ comes
with a forgetful functor $\Phi$ to $\mathscr{NC}\left(\mathbb{P}^{n}\right)$,
given on objects by taking the domain,
$\Phi(\mathcal{B}\to\mathcal{A})=\mathcal{B}$ and on morphisms by taking the
map $f$ in the triangle above. In our situation, since
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ is a groupoid, one can easily show
that this functor is full. However, it is not faithful;
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ and its slices are closely related
but do not hold identical information. Part of the reason for this is that
“unique” is a very strong statement in this set-up whereas “unique up to
isomorphism” is very weak.

######  Remark 29.

The slice category has a close relationship with the functor
$\operatorname{Hom}_{\mathscr{NC}\left(\mathbb{P}^{n}\right)}(-,\mathcal{A})$,
the latter also being a natural way to focus on morphisms in
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ whose codomain is $\mathcal{A}$. The
relationship between these is akin to that between a function $f\colon X\to Y$
and its graph $\Gamma(f)\subseteq X\times Y$. In this instance we will find
that statements regarding the structure of the slice category (and hence of
twists of $A$) are easier to state for the category than for the associated
functor.

However at other times it is fruitful to consider the functor
$\operatorname{Hom}_{\mathscr{NC}\left(\mathbb{P}^{n}\right)}(-,\mathcal{A})$,
not least because the form of this functor permits its interpretation as a
(representable) presheaf. We note many potentially interesting morphisms
between noncommutative projective spaces are not contained in the groupoid
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ and as such this presheaf is rather
special.

The presheaf model and the corresponding Yoneda embedding
$Y\colon\mathscr{NC}\left(\mathbb{P}^{n}\right)\to[\mathscr{NC}\left(\mathbb{P}^{n}\right)^{\mathrm{op}},\mathrm{Set}]$
has a “test space” interpretation: $\mathscr{NC}\left(\mathbb{P}^{n}\right)$
is our category of noncommutative projective spaces and the presheaf
$Y(\mathcal{A})=\operatorname{Hom}_{\mathscr{NC}\left(\mathbb{P}^{n}\right)}(-,\mathcal{A})$
captures the information of all of the noncommutative spaces _isomorphic to_
$\mathcal{A}$ and how they are isomorphic. That is, we are “testing for
sameness”, which as we suggested earlier should correspond to the generalized
notion of symmetry that we need in this generality.

By Proposition 23, morphisms $\mathcal{F}:\left(\operatorname{QGr}B,\pi
B\right)\rightarrow\left(\operatorname{QGr}A,\pi A\right)$ in
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ correspond to twists $\tau$ such
that $B\cong A^{\tau}$. As such, the slice category
$\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A}$ gives a categorical
interpretation of Zhang twists. Indeed, we could define
$\operatorname{Twists}\left(A\right)\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A}$,
as the twists of the algebra $A$ corresponding precisely to the morphisms in
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ with codomain
$\left(\operatorname{QGr}A,\pi A\right)$. That is,
$\operatorname{Twists}\left(A\right)=\operatorname{Hom}_{\mathscr{NC}\left(\mathbb{P}^{n}\right)}\left(-,\left(\operatorname{QGr}A,\pi
A\right)\right)$.

This makes clear the fact that Zhang twists under composition form a groupoid,
as the slice of the groupoid $\mathscr{NC}\left(\mathbb{P}^{n}\right)$ is a
groupoid. This supports the contention referred to earlier that the correct
replacement of automorphisms in the commutative setting should be morphisms in
a groupoid. More precisely, we have the following elementary results.

######  Lemma 30.

Let $A$, $B$ be coordinate rings of noncommutative $\mathbb{P}^{n}$s and let
$f\colon B\to A$ be a graded algebra isomorphism. Then the pushforward
$f_{\star}\colon\operatorname{QGr}B\to\operatorname{QGr}A$ is an equivalence
of categories satisfying $f_{\star}(\pi B(m))\cong(\pi A)(m)$ for all
$m\in\mathbb{N}$, and hence defines an object in
$\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A}$, where
$\mathcal{A}=(\operatorname{QGr}A,\pi A)$. ∎

######  Remark 31.

Note that if $B$ is isomorphic to $A$, then certainly $B$ is isomorphic to a
twist of $A$, namely the identity twist. For connected graded algebras,
isomorphism is equivalent to graded Morita equivalence, as explained above.

In general, not every twist will arise from an automorphism of algebras, but
certainly every automorphism of algebras gives rise to an algebraic twist (p.
2). If we take $B=A$ in the above - that is, $f$ an automorphism - then the
pushforward functor $f_{\star}$ yields an automorphism of
$\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A}$, in the following sense,
this being a categorical version of the algebraic twist construction.

For a category $\mathcal{C}$, let $\operatorname{Iso}(\mathcal{C})$ denote the
group of auto-isomorphisms of $\mathcal{C}$.

######  Proposition 32.

Let $\operatorname{Aut}_{\mathrm{gr}}(A)$ denote the group of $k$-algebra
automorphisms of $A$ that are $\mathbb{Z}$-graded of degree 0. For each
coordinate ring of a noncommutative $\mathbb{P}^{n}$, $A$, there is an
injection

$\operatorname{Aut}_{\mathrm{gr}}(A)\hookrightarrow\operatorname{Iso}(\mathcal{A})\hookrightarrow\operatorname{Iso}\left(\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A}\right)$

given by $f\mapsto f_{\star}\circ-$ which on objects is given by
$f_{\star}\circ-:\left(\mathcal{B}\rightarrow\mathcal{A}\right)\rightarrow\left(\mathcal{B}\rightarrow\mathcal{A}\overset{f_{\star}}{\rightarrow}\mathcal{A}\right)$,
where $f_{\star}$ is the equivalence of module categories induced via push-
forward along an isomorphism of algebras. On morphisms, one sends the diagram

$\mathcal{B}$$\mathcal{C}$$\mathcal{A}$$h$$g$$g^{\prime}$to the
diagram$\mathcal{B}$$\mathcal{C}$$\mathcal{A}$$\mathcal{A}$$h$$g$$g^{\prime}$$f_{\star}$

###### Proof:.

Let us briefly abuse notation and write simply $f_{\star}$, as opposed to
$f_{\star}\circ-$. By the properties of composition of homomorphisms,
$f_{\star}$ is clearly a functor. In the same manner, it can also be readily
verified that $x_{\star}\circ y_{\star}=\left(x\circ y\right)_{\star}$, that
$\left(f_{\star}\right)^{-1}=\left(f^{-1}\right)_{\star}$, and that
$\left(1_{\mathcal{A}}\right)_{\star}=\operatorname{Id}_{\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A}}$.
Injectivity follows by definition. ∎

Note that we could have used the pull-back of modules along a morphism
$f^{\star}$ in the previous proposition. The pull-back applies more generally
to any morphism of modules, though would produce a contravariant functor. As
we are concerned only with push-forward and pull-back under isomorphisms of
modules, we choose to work with the push-forward. Accepting the various
notions of a group action on a category, in our context, the above definition
is sufficient.

We also note that the above injection is not usually surjective. For in the
commutative case, the Picard group also acts by auto-equivalences on
$\operatorname{Coh}X$, by the result of Bondal and Orlov recalled earlier.

We now turn our attention to a more detailed analysis for
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ for small $n$, beginning with $n=1$.

##  4 Noncommutative $\mathbb{P}^{1}$s

We now consider the case of $\mathscr{NC}\left(\mathbb{P}^{1}\right)$. To
describe the algebras $A$ such that $\left(\operatorname{QGr}A,\pi
A\right)\in\mathscr{NC}\left(\mathbb{P}^{1}\right)$, we mimic the discussion
in Section 1 of [AS87] and more specifically the proof of their Theorem 1.5.
We are concerned with a special case of their results but we consider it
helpful to include a detailed exposition here for the particular case at hand,
where in some parts we can be more explicit.

Let $A$ be a connected graded $k$-algebra generated in degree 1, and write

$A\cong k\langle
x_{1}\ldots,x_{r_{1}}\rangle/\left(f_{1},\ldots,f_{r_{2}}\right)$

where $r_{1}$ is the minimal number of generators and $r_{2}$ is the minimal
number of homogeneous relations. Without loss of generality, we can write the
relations as

$f_{i}=\sum_{j=1}^{r_{1}}m_{ij}x_{j}.$

We then define an exact sequence of left $A$-modules

$\displaystyle A^{r_{2}}\overset{M}{\rightarrow}A^{r_{1}}\rightarrow
A\rightarrow_{A}k\rightarrow 0$ (2)

where $M=\left(m_{ij}\right)$ with $m_{ij}\in k\langle
x_{1},\ldots,x_{r_{1}}\rangle$.

######  Proposition 33 (cf. [AS87],[Zha96]).

Let $A$ be a coordinate ring of a noncommutative $\mathbb{P}^{1}$. Then we
have $r_{1}=2$ and $r_{2}=1$ so that $A$ has a single quadratic relation $f$.
Hence, one obtains a graded resolution of left $A$-modules of the form

$\displaystyle 0\rightarrow A(-2)\overset{M}{\rightarrow}A(-1)\oplus
A(-1)\rightarrow A\rightarrow_{A}k\rightarrow 0$ (3)

###### Proof:.

Consider a minimal resolution of ${}_{A}k$

$\displaystyle 0\rightarrow A^{r_{2}}\rightarrow A^{r_{1}}\rightarrow
A\rightarrow_{A}k\rightarrow 0$ (4)

with $r_{1}$ and $r_{2}$ as before. The ranks of the free modules are given by
$r_{i}=\dim_{k}\operatorname{Tor}_{i}^{A}\left(k,k\right)$. From the AS-
Gorenstein property with shift $l=2$ (Definition 8), we know that applying the
functor $\operatorname{Hom}(-,A)$ to this resolution will produce a resolution
of the right $A$ module $k_{A}=\operatorname{Ext}_{A}^{2}\left(k,A\right)$

$\displaystyle 0\leftarrow k_{A}\leftarrow A^{r_{2}}\leftarrow
A^{r_{1}}\leftarrow A\leftarrow 0$ (5)

and this is minimal, since (4) is minimal. Since
$\dim_{k}\operatorname{Tor}_{i}^{A}\left(k,k\right)$ is symmetric in both left
and right, we conclude that $r_{2}=1$. Henceforth set $r_{1}=r$.

We have therefore obtained a resolution

$\displaystyle 0\rightarrow
A(-s)\overset{M}{\rightarrow}\left(A(-1)\right)^{\oplus r}\rightarrow
A\rightarrow_{A}k\rightarrow 0$ (6)

where $s$ is the degree of $f$. Setting
$a_{n}\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\dim_{k}A_{n}$,
the above exact sequence gives rise to the recurrence relation
$a_{n}-ra_{n-1}+a_{n-s}=0$. The characteristic polynomial of this relation is
$p(t)=t^{s}-rt^{s-1}+1$.

We know that $A$ has Hilbert series
$1/\left(1-t\right)^{2}=1+2t+3t^{2}+4t^{3}+\ldots$ Setting
$\left(r,s\right)=\left(2,2\right)$ gives the correct dimensions and it is
clear that $r,s\geq 2$. We claim that this is the only solution.

To show this, we demonstrate that if $r+s>4$, then $p(t)$ has a real root
strictly greater than $1$. If $s\geq 2$ and $r>2$, $p(1)=2-r<0$ but $p(t)>0$
for $t\gg 1$ and we are done.

Otherwise, if $r=2$ and $s>2$, $p(1)=0$. Since $p(t)>0$ for $t\gg 1$ still, it
suffices to show that the derivative

$p^{\prime}(t)=t^{s-2}\left(st-r\left(s-1\right)\right)$

is negative at $t=1$. But for $r=2$, $p^{\prime}(1)=s-2(s-1)=2-s<0$ and so we
are done in this case too. ∎

We have shown that $A\cong k\langle x,y\rangle/\left(f\right)$ where
$\deg(f)=2$. One can show that, in order for $A$ to satisfy the condition on
$\operatorname{Ext}^{i}_{\operatorname{Gr}A}(k,A)$ in Definition 8, $f$ must
be of the form $x(ax+by)-y(cx+dy)=0$ for $ad-bc\neq 0$. By Example 3.6 of
[Zha96], these are precisely the algebras which are Zhang twists of
$k\left[x,y\right]$ and two such algebras are isomorphic if, and only if, the
linear automorphisms corresponding to two relations $f_{1}$ and $f_{2}$ are in
the same $\operatorname{PGL}_{2}\\!\left(k\right)$-conjugacy class. Thus, we
have that $\mathscr{NC}\left(\mathbb{P}^{1}\right)$ has one connected
component, whose morphisms are controlled by
$\operatorname{PGL}_{2}\\!\left(k\right)$-conjugacy class.

More categorically, via Serre’s theorem, we then see that every
$\operatorname{QGr}A\in\mathscr{NC}\left(\mathbb{P}^{1}\right)$ is equivalent
to $\operatorname{QCoh}\mathbb{P}^{1}$. We may interpret this as saying that
there are no non-trivial noncommutative $\mathbb{P}^{1}$’s, and the trivial
ones all have the classical automorphism group as their twists.

The above theoretical considerations are complemented by the fact that in low
dimensions we can proceed very concretely.

In the case of a connected $\mathbb{N}$-graded algebra $A=\bigoplus A_{i}$
generated in degree 1, recall that the fundamental defining relations for a
Zhang twist are equivalent to the following ([Zha96, p. 284]): a twisting
system in this case is a set $\tau=\\{\tau_{n}\mid n\in\mathbb{N}\\}$ of
graded linear isomorphisms satisfying

$\tau_{m}(ab)=\tau_{m}(a)\tau_{m+n}\tau_{n}^{-1}(b)$

for $a\in A_{n}$.

Since $A$ is assumed to be generated in degree 1, we can view the above as
giving us a way to inductively construct twisting systems. Let $\tau_{m}^{i}$
denote $\tau_{m}|_{A_{i}}\colon A_{i}\to A_{i}$. By results of Zhang, without
loss of generality $\tau_{0}$ may be taken to be the identity, so we start by
choosing $\tau_{1}^{1}\colon A_{1}\to A_{1}$([Zha96, Proposition 2.4]). This
is equivalent to choosing an invertible matrix acting on a set of generators
of the algebra.

Then since $\tau_{1}^{2}$ will need to respect the relations of our algebra we
may try to define $\tau_{1}^{2}$ by solving to find $\tau_{2}^{1}$ such that
the twisting relation above is satisfied. If we can do this, we can then try
to find $\tau_{2}^{2}$, $\tau_{3}^{1}$ and so on, building up $\tau$ piece by
piece.

If we know more about our algebra, such as that $A$ is quadratic, then we
might expect this process to terminate and the higher components of the maps
to be determined. Note though that we will need to specify $\tau_{m}^{1}$ for
all $m$.

Conversely, we may view the above process as a series of obstructions for a
collection of linear maps to form a twisting system. A failure of the
existence of solutions to the relevant equations at some point can indicate
that a particular choice of $\tau_{1}^{1}$ cannot lead to a twisting system,
for example. That is, we can consider twisting as a type of deformation
problem. In the case of noncommutative $\mathbb{P}^{n}$s corresponding to
_geometric_ algebras (see Definition 39), we can partially resolve this
problem by computing certain twists via geometric data (see Theorem 45).

By writing down a matrix $T$ representing the projection from the tensor
algebra over $A_{1}$ to $A$, we can produce the following linear algebra
computations corresponding to this process, and for dimension 1 actually solve
these, as follows.

######  Example 34.

We consider first $A=\mathcal{O}(\mathbb{P}^{1})=k[x,y]$, with $\deg x=\deg
y=1$.

Let

$\displaystyle T$ $\displaystyle=\begin{pmatrix}1&0&0&0\\\ 0&1&1&0\\\
0&0&0&0\\\ 0&0&0&1\end{pmatrix}$ whose kernel is spanned by $v=(0\ 1\ -1\
0)^{T}$ corresponding to $x\otimes y-y\otimes x$, and $\displaystyle S$
$\displaystyle=A\otimes BA^{-1}=\lambda^{-1}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\otimes\begin{pmatrix}\alpha d-\beta c&\beta a-\alpha b\\\
\gamma d-\delta c&\delta a-\gamma b\end{pmatrix}$ where $A=\tau_{1}^{1}$,
$B=\tau_{2}^{1}$ and $\lambda=\det A$. Then $A$ and $B$ will extend to a
twisting system precisely if
$\operatorname{ker}TS\subseteq\operatorname{ker}T$: we may calculate and see
that $\displaystyle TSv$ $\displaystyle=0\qquad\Leftrightarrow\qquad
A^{2}\begin{pmatrix}\beta&\delta\\\
-\alpha&-\gamma\end{pmatrix}=\begin{pmatrix}0&\mu\\\
-\mu&0\end{pmatrix}\qquad\Leftrightarrow\qquad\text{up to scaling,}\ B=A^{2}$

By means of this computation, we see that all twists of the commutative
polynomial ring are algebraic: they come from automorphisms as
$\\{\tau_{m}=f^{m}\\}$.

######  Example 35.

Consider next $A=\mathcal{O}_{q}(\mathbb{P}^{1})$ (p.4), also with $\deg
x=\deg y=1$.

Let $q\in k^{*}$ and let

$\displaystyle T$ $\displaystyle=\begin{pmatrix}1&0&0&0\\\ 0&q&1&0\\\
0&0&0&0\\\ 0&0&0&1\end{pmatrix}$ whose kernel is spanned by $v=(0\ 1\ -q\
0)^{T}$ corresponding to $x\otimes y-qy\otimes x$, and $\displaystyle S$
$\displaystyle=A\otimes BA^{-1}=\lambda^{-1}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\otimes\begin{pmatrix}\alpha d-\beta c&\beta a-\alpha b\\\
\gamma d-\delta c&\delta a-\gamma b\end{pmatrix}$ as before. Then $A$ and $B$
will extend to a twisting system precisely if
$\operatorname{ker}TS\subset\operatorname{ker}T$: we may calculate and see
that $\displaystyle TSv$ $\displaystyle=0\qquad\Leftrightarrow\qquad
A\begin{pmatrix}1&0\\\ 0&q\end{pmatrix}A\begin{pmatrix}\beta&\delta\\\
-\alpha&-\gamma\end{pmatrix}=\begin{pmatrix}0&\mu\\\
-\mu&0\end{pmatrix}\qquad\Leftrightarrow\qquad\text{up to scaling,}\
B=A\begin{pmatrix}1&0\\\ 0&q\end{pmatrix}A.$

That is, $S=\nu A\otimes A\begin{pmatrix}1&0\\\ 0&q\end{pmatrix}$. This ought
not to be a surprise: $\begin{pmatrix}1&0\\\ 0&q\end{pmatrix}$ is precisely
what we twist the polynomial ring $\mathcal{O}(\mathbb{P}^{1})=k[x,y]$ by to
get $\mathcal{O}_{q}(\mathbb{P}^{1})$. So we see that
$\mathcal{O}_{q}(\mathbb{P}^{1})$ “has the same twists” as
$\mathcal{O}(\mathbb{P}^{1})$ \- which we knew from a more theoretical
analysis above - and we see this concretely, as a combination of a general
twist of the polynomial ring and the particular twist taking us to the quantum
plane.

However, two significant differences occur in higher dimensions, even
dimension 2. Firstly, it is straightforward to show explicitly that the
polynomial ring and the algebra $\mathcal{O}_{q}(\mathbb{P}^{2})$ (often less
formally called “quantum $\mathbb{P}^{2}$”) are no longer twist-equivalent.
Secondly, the dimension of $A_{1}$ and thus the size of the matrices $T$, $S$
increases significantly (from $(1+1)^{2}=4$ to $(2+1)^{2}=9$) and the
dimension of the kernel of $T$ becomes 3 rather than just 1. So the above
explicit linear-algebraic approach becomes less viable.

Fortunately in the case of $\mathbb{P}^{2}$ \- that is, algebras that are AS-
regular of dimension 3, so of global dimension 3 - we have the seminal work of
Artin-Tate-Van den Bergh ([ATVdB90],[ATVdB91]) available to us, in which the
authors introduced point schemes and associated concepts. In the next section,
we will utilize this to analyse the groupoid
$\mathscr{NC}\left(\mathbb{P}^{2}\right)$.

##  5 Noncommutative $\mathbb{P}^{2}$s

In [ATVdB90] and [ATVdB91] the authors study AS-regular algebras $A$ of
dimension $3$, among which are the noncommutative analogues of
$\mathbb{P}^{2}$ we wish to consider. To such an algebra $A$ they associate a
triple $\mathscr{T}=\left(E,\sigma,\mathcal{L}\right)$ where $E$ is the point
scheme of $A$ (Definition 21), $\sigma$ is an automorphism of $E$ and
$\mathcal{L}\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}j^{*}\mathcal{O}\left(1\right)$
where $j$ is the embedding of $E$ in $\mathbb{P}^{2}$.

Let $r$ be the size of a minimal generating set for $A$. Let $\mathscr{T}$ be
a triple associated to a regular algebra as above. In [ATVdB90, Theorem 2] it
is shown that these triples satisfy

$\left(\sigma-1\right)^{2}\left[\mathcal{L}\right]=0\quad\text{if}\
r=3\quad\text{and}\quad\left(\sigma^{2}-1\right)\left(\sigma-1\right)\left[\mathcal{L}\right]=0\quad\text{if}\
r=2$

where $\left[\mathcal{L}\right]$ is the class of $\mathcal{L}$ in
$\operatorname{Pic}(E)$ and such triples are also called regular. Let $s$ be
the degree of the defining relations of $A$.

For regular algebras of dimension $3$ generated in degree $1$, there are two
possibilities: $A$ is generated by three elements and has three quadratic
relations; or $A$ is generated by two elements and has two cubic relations. As
noted in [AS87, p. 175], $A$ has the same Hilbert series as the polynomial
ring if and only if the algebra has three generators and is quadratic, so
henceforth we will focus on this case, where
$\mathcal{L}^{(\sigma-1)^{2}}\cong\mathcal{O}_{E}$ (here $\mathcal{L}^{\rho}$
denotes the pullback $\rho^{*}\mathcal{L}$ along an automorphism $\rho$).

In §8 of [ATVdB91] the authors define the notion of twisting a regular
$\mathbb{Z}$-graded algebra $A$ by a graded $A$-automorphism. This twisting
process was later generalized by Zhang to that described above. They then
define an action of graded $A$-automorphisms on the set of regular triples and
demonstrate a correspondence between twists of a given algebra and images of
its regular triple under the action of $A$-automorphisms. For completeness, we
give a slight generalization of their argument ([ATVdB91, Proposition 8.9]) in
the case $r=3$ for Zhang twists.

######  Proposition 36.

Let $A$ be a regular algebra of dimension three determined by the regular
triple $\mathscr{T}=\left(E,\sigma,\mathcal{L}\right)$ and let $\tau$ be a
twisting system on $A$. By dualization and projectivization, there is an
automorphism $\overline{\tau_{1}}^{\ast}$ of $\mathbb{P}(A_{1})$ induced by
$\tau_{1}$ and $\overline{\tau_{1}}^{\ast}(E)=E$. Denote by $\tau_{1}^{E}$ the
induced automorphism of $E$ and let $\mathscr{T}_{\tau_{1}^{E}}$ be the triple
$\left(E,\tau_{1}^{E}\sigma,\mathcal{L}\right)$. Then $A^{\tau}$ is the
algebra determined (up to isomorphism) by $\mathscr{T}_{\tau_{1}^{E}\sigma}$.

###### Proof:.

Since Zhang twisting preserves regularity, $A^{\tau}$ is regular. Let
$\varphi=\left(1,\tau\right)$ and let
$\Gamma^{\prime}\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\varphi(\Gamma(E))$.
Since $\Gamma(E)$ is the locus of common zeroes of the multilinearized
relations of $A$, $\left\\{\widetilde{f_{i}}\right\\}$, we have that
$\Gamma^{\prime}$ is the locus of common zeroes of the multilinearized twisted
relations $\left\\{\widetilde{f_{\tau,i}}\right\\}$ and hence,
$\Gamma^{\prime}=\Gamma(A^{\tau})$.

Since $s=2$, we have that $E=\mathrm{pr}_{1}\Gamma^{\prime}$. Since $A^{\tau}$
is regular, $\Gamma^{\prime}$ defines an automorphism $\sigma^{\prime}$ of
$E$. Hence, $A^{\tau}$ defines a triple
$\mathscr{T}^{\prime}=\left(E,\sigma^{\prime},\mathcal{L}\right)$. By
analysing the construction, one can see that
$\sigma^{\prime}=\tau_{1}^{E}\sigma$ and so
$\mathscr{T}^{\prime}=\mathscr{T}_{\tau_{1}^{E}\sigma}=\left(E,\tau_{1}^{E}\sigma,\mathcal{L}\right)$.
∎

To proceed further, we will use a combination of algebraic and geometric
methods, as were illustrated for $\mathscr{NC}\left(\mathbb{P}^{1}\right)$. On
the geometric side, we have the following theorem of Mori ([Mor06]); the
version below has been adapted to the specific case at hand.

######  Theorem 37 (cf. [Mor06, Theorem 4.7]).

Let $A$ and $A^{\prime}$ be coordinate rings of noncommutative
$\mathbb{P}^{2}$s, with associated triples
$\mathscr{T}=(E,\sigma,\mathcal{L})$ and
$\mathscr{T}^{\prime}=(E^{\prime},\sigma^{\prime},\mathcal{L}^{\prime})$
respectively. Then $A$ is twist-equivalent to $A^{\prime}$ if and only if
there exists a sequence of automorphisms $\\{\rho_{m}\mid
m\in\mathbb{N}\\}\subset\operatorname{Aut}\mathbb{P}^{2}$ such that, setting
$\rho_{m}^{E}\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\rho_{m}|_{E}$,
we have $\rho_{m}^{E}(E)\cong E^{\prime}$ and
$\rho_{n+1}^{E}\sigma=\sigma^{\prime}\rho_{n}^{E}$ for all $n\in\mathbb{N}$.

Diagrammatically,

$E$$E^{\prime}$$E$$E^{\prime}$$E$$E^{\prime}$$E$$E^{\prime}$$E$$E^{\prime}$$\rho_{0}^{E}$$\sigma$$\sigma^{\prime}$$\rho_{1}^{E}$$\sigma$$\sigma^{\prime}$$\rho_{2}^{E}$$\sigma$$\sigma^{\prime}$$\rho_{3}^{E}$$\sigma$$\sigma^{\prime}$$\rho_{4}^{E}$$\sigma$$\sigma^{\prime}$

Mori’s original version of the result makes fewer assumptions on $A$ and
$A^{\prime}$, including that the algebras concerned are only
$\mathbb{Z}$-graded and not necessarily $\mathbb{N}$-graded. By use of [Zha96,
Proposition 2.8] for the $\mathbb{N}$-graded case and an examination of Mori’s
proof, one sees that for $\mathbb{N}$-graded algebras, a family
$\\{\rho_{m}\mid m\in\mathbb{N}\\}$ is sufficient, rather than a family
$\\{\rho_{m}\mid m\in\mathbb{Z}\\}$ as Mori requires.

Also, the conclusion of the theorem is expressed as $\operatorname{Gr}A$ being
isomorphic to $\operatorname{Gr}A^{\prime}$ if and only if $\\{\rho_{m}\mid
m\in\mathbb{N}\\}$ exists with the stated properties. Of course, we have
already seen that for coordinate rings of noncommutative projective spaces,
this is the same as twist-equivalence. Indeed, Mori’s proof is not categorical
but goes via the existence of a Zhang twist, for which Mori has sufficiently
many assumptions.

In particular, Mori’s result works in higher dimensions: he gives a definition
of a _geometric algebra_ , this being a quadratic algebra that satisfies some
conditions relating to the point scheme. For dimension 2, it is known that
every coordinate ring of a noncommutative projective $\mathbb{P}^{2}$ is
geometric (see the references given in [Mor06]).

It is natural to ask whether this is true for all noncommutative
$\mathbb{P}^{n}$s with $n>2$; given what is known of the rich landscape of
algebras, with quite diverse examples, we expect that the answer is no in
general. Specifically, a generic AS-regular algebra of dimension 4 has a
finite point scheme, whereas geometricity asks that the point scheme is
sufficiently large to encode the relations of the algebra (see [Van15, §1.4]).

However, Mori’s results do show that if we have a geometric algebra then any
twist of this is also geometric. So provided we know that one algebra in a
given connected component of $\mathscr{NC}\left(\mathbb{P}^{n}\right)$ is
geometric, we immediately know this for all algebras in the component. For
analysis of twists of some particular algebra, this can allow us to use Mori’s
results.

Staying with dimension 2, let us use Mori’s result to analyse twist-
equivalence for the 2-dimensional algebras analogous to
$\mathcal{O}(\mathbb{P}^{1})=k[x,y]$ and $\mathcal{O}_{q}(\mathbb{P}^{1})$
that we treated in the previous section.

Consider the component of $\mathscr{NC}\left(\mathbb{P}^{2}\right)$ containing
$(\operatorname{QGr}A,\pi A)$ for $A=\mathcal{O}(\mathbb{P}^{2})$, whose
associated triple is $(\mathbb{P}^{2},\mathrm{id},\mathcal{O}(1))$. Any triple
for an algebra twist-equivalent to $A$ must necessarily have point scheme
$\mathbb{P}^{2}$, so without loss of generality we may let $A^{\prime}$ be an
algebra associated to the triple $(\mathbb{P}^{2},\sigma,\mathcal{O}(1))$ for
$\sigma\in\operatorname{PGL}_{3}\\!\left(k\right)$ arbitrary.

In this situation, Mori’s theorem may be rephrased (as in [Mor06, Remark 4.8])
to say that $A$ and $A^{\prime}$ are twist-equivalent if and only if there
exists $\rho_{0}\in\operatorname{PGL}_{3}\\!\left(k\right)$ such that

  * •

$\rho_{0}(\mathbb{P}^{2})=\mathbb{P}^{2}$ and

  * •

$\rho_{m}\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\sigma^{m}\rho_{0}(\mathrm{id})^{-m}$
can be extended to an automorphism of $\mathbb{P}^{2}$ for all
$m\in\mathbb{N}$.

But $\rho_{0}=\mathrm{id}$ clearly satisfies this, so we deduce that
$(\operatorname{QGr}A^{\prime},\pi A^{\prime})$ is also in the connected
component and $A^{\prime}\cong A^{\sigma}$, the twist of $A$ by the
automorphism induced by $\sigma$.

Mori ([Mor06, Remark 4.9]) also shows that such $A$ and $A^{\prime}$ are
isomorphic if and only if $\\{\rho_{m}\\}$ can be taken to be constant, i.e.
$\rho_{m}=\rho$ for all $m$. Then the condition
$\rho_{m+1}\sigma=\sigma^{\prime}\rho_{m}$ for all $m\in\mathbb{N}$ reduces to
conjugacy:

$A^{\sigma}\cong A^{\sigma^{\prime}}\quad\Leftrightarrow\quad\exists\ \rho\
\text{such that}\
\sigma^{\prime}\rho=\rho\sigma\quad\Leftrightarrow\quad\sigma^{\prime}\
\text{is conjugate to}\ \sigma$

This generalizes the earlier statement for noncommutative $\mathbb{P}^{1}$s,
where there is only one component whose morphisms are controlled by
$\operatorname{PGL}_{2}\\!\left(k\right)$ conjugacy.

For every $\left(\operatorname{QGr}A^{\prime},\pi A^{\prime}\right)$ in the
commutative component of $\mathscr{NC}\left(\mathbb{P}^{n}\right)$, we have
that $\operatorname{QGr}A^{\prime}\simeq\operatorname{QCoh}(\mathbb{P}^{n})$.
Furthermore, by exactly the same argument as above for $\mathbb{P}^{2}$, every
element of $\operatorname{PGL}_{n+1}\\!\left(k\right)$ gives rise to a twist,
and graded Morita equivalence is controlled by conjugacy in
$\operatorname{PGL}_{n+1}\\!\left(k\right)$.

We may use this method to analyse other components too. The following is an
extended presentation of [Mor06, Example 4.10], which we include both for the
reader’s convenience, as an illustration of the method, and also in
preparation for its extension later.

Consider the algebra $\mathcal{O}_{Q}(\mathbb{P}^{2})$ for
$Q=(\alpha,\beta,\gamma)\in(k^{\ast})^{3}$ with presentation

$\mathcal{O}_{Q}(\mathbb{P}^{2})\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}k\langle
x,y,z\rangle/\langle zy=\alpha yz,xz=\beta zx,yx=\gamma xy\rangle.$

This is the natural multi-parameter generalization of
$\mathcal{O}_{q}(\mathbb{P}^{2})=\mathcal{O}_{(q^{-1},q,q^{-1})}(\mathbb{P}^{2})$.

Provided $\alpha\beta\gamma\notin\\{0,1\\}$, all algebras of the form
$\mathcal{O}_{Q}(\mathbb{P}^{2})$ have the same point scheme
$E\subset\mathbb{P}^{2}$, namely the union of the three lines in
$\mathbb{P}^{2}$ given by the vanishing of the coordinates $x$, $y$ and $z$;
let $l_{1}=\mathbb{V}(x)$ etc. The associated $\sigma=\sigma(Q)$ is given by
$\sigma(Q)=\mu(\alpha,\beta,\gamma)$ where

$\displaystyle\mu|_{l_{1}}(0,b,c)$ $\displaystyle=(0,\alpha b,c)$
$\displaystyle\mu|_{l_{2}}(a,0,c)$ $\displaystyle=(a,0,\beta c)$
$\displaystyle\mu|_{l_{3}}(a,b,0)$ $\displaystyle=(\gamma a,b,0).$

One may easily check that such a $\mu(\alpha,\beta,\gamma)$ extends to all of
$\mathbb{P}^{2}$ if and only if $\alpha\beta\gamma=1$.

Furthermore, the subgroup of elements of
$\operatorname{PGL}_{3}\\!\left(k\right)$ that preserve this triangle $E$ is
isomorphic to $(k^{\ast})^{2}\rtimes S_{3}$, where the 2-torus action
precisely comes from the action of maps of the above form
$\mu(\alpha,\beta,\gamma)$ with $\alpha\beta\gamma=1$ and the $S_{3}$ permutes
the three lines.

In [Mor06, Example 4.10], Mori uses his theorem to give necessary and
sufficient conditions for the twist-equivalence of $A$ and $A^{\prime}$
associated to $(E,\sigma=\mu(\alpha,\beta,\gamma))$ and
$(E,\sigma^{\prime}=\mu(\alpha^{\prime},\beta^{\prime},\gamma^{\prime}))$.

For example, take $\rho_{0}=(1\ 2)\in S_{3}$. Then one may straightforwardly
check that

$(\sigma^{\prime}\rho_{0}\sigma^{-1})|_{l_{1}}(0,b,c)=(\alpha^{-1}b,0,\beta^{\prime}c)=(b,0,\alpha\beta^{\prime}c)\in
l_{2}$

with similar expressions for $\sigma^{\prime}\rho_{0}\sigma^{-1}$ on $l_{2}$
and $l_{3}$. From these, one sees that
$\sigma^{\prime}\rho_{0}\sigma^{-1}=\mu(\alpha^{\prime}\beta,\alpha\beta^{\prime},\gamma\gamma^{\prime})\circ\rho_{0}$.

Now $\rho_{0}$ extends to $\mathbb{P}^{2}$ by definition, hence
$\sigma^{\prime}\rho_{0}\sigma^{-1}$ does if and only if
$\mu(\alpha^{\prime}\beta,\alpha\beta^{\prime},\gamma\gamma^{\prime})$ does,
which happens if and only if
$\alpha^{\prime}\beta^{\prime}\gamma^{\prime}=(\alpha\beta\gamma)^{-1}$.
Furthermore this condition suffices to ensure that
$(\sigma^{\prime})^{m}\rho_{0}(\sigma)^{-m}$ extends to $\mathbb{P}^{2}$ for
all $m$ and hence $A$ and $A^{\prime}$ are twist-equivalent.

This phenomenon - of the sufficiency of one condition, independent of $m$, to
establish twist-equivalence - can also be seen in other examples (Theorems 5.2
and 5.4 of [Mor06]) and we will return to this later.

Iterating through the other choices for $\rho_{0}$, one may show that
$A=\mathcal{O}_{(\alpha,\beta,\gamma)}(\mathbb{P}^{2})$ and
$A^{\prime}=\mathcal{O}_{(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})}(\mathbb{P}^{2})$
are twist equivalent if and only if
$\alpha^{\prime}\beta^{\prime}\gamma^{\prime}=(\alpha\beta\gamma)^{\pm 1}$.

Recalling that $Q=(q^{-1},q,q^{-1})$ gives the single parameter quantum
$\mathbb{P}^{2}$, $\mathcal{O}_{q}(\mathbb{P}^{2})$, we see as a special case
that $\mathcal{O}_{q}(\mathbb{P}^{2})$ is not twist equivalent to
$\mathcal{O}_{q^{\prime}}(\mathbb{P}^{2})$ for all but one other choice of
$q^{\prime}\in k^{\ast}$:

$((q^{\prime})^{-1})\cdot q^{\prime}\cdot((q^{\prime})^{-1})=(q^{-1})\cdot
q\cdot(q^{-1})^{\pm 1}\qquad\Leftrightarrow\qquad q^{\prime}=q^{\pm 1}$

This makes it very clear that $\mathscr{NC}\left(\mathbb{P}^{2}\right)$ has
very many connected components. It also shows that in the case of quantum
algebras defined with respect to a parameter $q\in k^{\ast}$ (or indeed,
multiple parameters) varying $q$ does not give rise to twist equivalences. So
while twist equivalence has some features of a deformation-theoretic problem,
it is too strong and somewhat too discrete to be related to smooth variation
of deformation parameters and the associated geometry of the latter.

This completely answers the question of when two algebras of the form
$\mathcal{O}_{Q}(\mathbb{P}^{2})$, that is, multi-parameter quantum
$\mathbb{P}^{2}$s, have associated objects in
$\mathscr{NC}\left(\mathbb{P}^{2}\right)$ in the same connected component.
However, more work is needed to completely describe these components in full:
they contain objects associated to algebras $A$ not of the form
$\mathcal{O}_{Q}(\mathbb{P}^{2})$.

The above example shows that there exist non-trivial twists of the standard
quantum $\mathbb{P}^{2}$, $\mathcal{O}_{q}(\mathbb{P}^{2})$, induced by
automorphisms of the point scheme and associated to elements of $S_{3}$, which
is not a subgroup of
$\operatorname{Aut}(\mathcal{O}_{q}(\mathbb{P}^{2}))=k\ltimes(k^{\ast})^{3}$
(see Theorem 4, recalled earlier). That is, we have recovered some classical
automorphisms.

######  Remark 38.

Recent work of Itaba-Matsuno ([IM]) gives tables of defining relations for
3-dimensional quadratic AS-regular algebras. Note that certain statements
there and here are not directly comparable, due to differences in the
definitions and of terminology. For example, we reserve the term “graded
Morita equivalence” for equivalence of the full graded module categories (with
morphisms of all degrees), rather than equivalence of graded modules
categories with morphisms only of degree zero (what we call twist-
equivalence).

##  6 Geometric noncommutative $\mathbb{P}^{n}$s

To proceed further, we use a careful analysis of Mori’s proof of the above
theorem to generalize Proposition 36. First we give the precise definition of
a geometric algebra.

######  Definition 39 ([Mor06, Definition 4.3]).

A quadratic algebra $A=T(V)/R$ is called geometric if there is a pair
$(E,\sigma)$ where $j\colon E\to\mathbb{P}(V^{\ast})$ is an embedding of $E$
as a closed $k$-subscheme and $\sigma$ is a $k$-automorphism of $E$ such that

  1. 1.

$\Gamma_{2}=\mathcal{V}(R)\subset\mathbb{P}(V^{\ast})\times\mathbb{P}(V^{\ast})$
is the graph of $E$ under $\sigma$, and

  2. 2.

setting $\mathcal{L}=j^{\ast}\mathcal{O}_{\mathbb{P}(V^{\ast})}(1)$, the map

$\mu\colon H^{0}(E,\mathcal{L})\otimes H^{0}(E,\mathcal{L})\to
H^{0}(E,\mathcal{L}\otimes_{\mathcal{O}_{E}}\sigma^{\ast}\mathcal{L})$

defined by $v\otimes w\mapsto v\otimes(w\circ\sigma)$ has
$\operatorname{ker}\mu=R$, with the identification

$H^{0}(E,\mathcal{L})=H^{0}(\mathbb{P}(V^{\ast}),\mathcal{O}(\mathbb{P}(V^{\ast})(1))=V$

as $k$-vector spaces.

If $A$ is geometric as above, we associate the triple $(E,\sigma,\mathcal{L})$
to both $A$ and the pair $\operatorname{QGr}A,\pi A)$.

Then we may give Mori’s original theorem in its greater generality (with some
adjustments in notation to fit with our own).

######  Theorem 40 ([Mor06, Theorem 4.7]).

Let $A=T(V)/I$ and $A^{\prime}=T(V)/I^{\prime}$ be graded algebras finitely
generated in degree 1 over $k$.

  1. 1.

If $A=\mathcal{A}(E,\sigma)$ is geometric and
$\operatorname{Gr}A\cong\operatorname{Gr}A^{\prime}$ then
$A^{\prime}=\mathcal{A}(E^{\prime},\sigma^{\prime})$ is also geometric and
there is a sequence of automorphisms $\\{\tau_{n}^{\ast}\\}$ of
$\mathbb{P}(V^{\ast})$, each of which sends $E$ isomorphically onto
$E^{\prime}$ such that
$(\tau_{n+1}^{\ast}|_{E})\sigma=\sigma^{\prime}(\tau_{n}^{\ast}|_{E})$ for
every $n\in\mathbb{Z}$.

  2. 2.

Conversely if $A=\mathcal{A}(E,\sigma)$ and
$A^{\prime}=\mathcal{A}(E^{\prime},\sigma^{\prime})$ are geometric and there
is a sequence of automorphisms $\\{\tau_{n}^{\ast}\\}$ of
$\mathbb{P}(V^{\ast})$, each of which sends $E$ isomorphically onto
$E^{\prime}$ such that
$(\tau_{n+1}^{\ast}|_{E})\sigma=\sigma^{\prime}(\tau_{n}^{\ast}|_{E})$ for
every $n\in\mathbb{Z}$, then
$\operatorname{Gr}A\cong\operatorname{Gr}A^{\prime}$.

We note that if $A$, $A^{\prime}$ are $\mathbb{N}$-graded, the index $n$ may
be taken to range over $\mathbb{N}$ rather than $\mathbb{Z}$, with no change
to the conclusions.

We will insert the adjective “geometric” into our terminology for
noncommutative projective spaces in the natural way, referring to
$(\operatorname{QGr}A,\pi A)$ as a geometric noncommutative $\mathbb{P}^{n}$
if $(\operatorname{QGr}A,\pi A)$ is a noncommutative $\mathbb{P}^{n}$ for a
geometric algebra $A$, and similarly refer to such an $A$ as a coordinate ring
for a geometric noncommutative $\mathbb{P}^{n}$.

######  Proposition 41.

Let $A$ be a homogeneous coordinate ring for a geometric noncommutative
$\mathbb{P}^{n}$ and let $(E,\sigma,\mathcal{L})$ be its associated (regular)
triple. Let $\tau$ be a twisting system for $A$.

By dualization and projectivization, there is an automorphism
$\overline{\tau_{1}}^{\ast}$ of $\mathbb{P}(A_{1}^{\ast})$ induced by
$\tau_{1}$ and $\overline{\tau_{1}}^{\ast}(E)=E$. Denote by $\tau_{1}^{E}$ the
induced automorphism of $E$ and let $\mathscr{T}_{\tau_{1}^{E}\sigma}$ be the
triple $\left(E,\tau_{1}^{E}\sigma,\mathcal{L}\right)$. Then $A^{\tau}$ is the
algebra determined (up to isomorphism) by $\mathscr{T}_{\tau_{1}^{E}\sigma}$.

That is, by comparison with Proposition 36, we are able to remove the
assumption on the dimension at the expense of requiring geometricity.

###### Proof:.

By Theorem 40, $A^{\tau}$ is geometric and has an associated triple
$(E^{\prime},\sigma^{\prime},\mathcal{L}^{\prime})$. Now since $A^{\tau}$ is a
twist of $A$, there are graded linear isomorphisms $\phi_{n}\colon A^{\tau}\to
A$ such that $\phi_{n}(a\ast_{A^{\tau}}b)=\phi_{n}(a)\ast_{A}\phi_{n+l}(b)$
for $a\in A^{\tau}_{l}$, where $\ast_{A^{\tau}}$ and $\ast_{A}$ are the
multiplications in $A^{\tau}$ and $A$ respectively. These graded linear
isomorphisms are obtained via [Zha96, Proposition 2.8(1)] and a careful
examination of the proof of that result - which is stated for an algebra $B$
_isomorphic to_ a twist of $A$ \- shows that if $B$ is _equal_ to a twist of
$A$, i.e. we take $B=A^{\tau}$ in Zhang’s proposition, then in fact
$\phi_{n}=\tau_{n}$. (In general, $\phi_{n}=\tau_{n}\circ f$ for $f\colon B\to
A^{\tau}$.)

Consequently, in Mori’s theorem (40), the key to the proof of the first part
is the dualization of the maps $\tau_{n}|_{A_{1}}$, namely
$\overline{\tau_{n}}^{\ast}\colon\mathbb{P}(A_{1}^{\ast})\to\mathbb{P}(A_{1}^{\ast})$.
(Since the $\tau_{n}|_{A_{1}}$ are isomorphisms, it follows that the maps
$(\tau_{n}|_{A_{1}})^{\ast}$ are also isomorphisms and hence, being injective,
they descend to the projectivization of $A_{1}^{\ast}$.)

Then Mori shows that $E^{\prime}=(\overline{\tau_{0}}^{\ast}|_{E})(E)$ and
$\sigma^{\prime}=(\overline{\tau_{1}}^{\ast}|_{E})\circ\sigma\circ(\overline{\tau_{0}}^{\ast}|_{E})^{-1}$.
As above, set $\tau_{1}^{E}=\overline{\tau_{1}}^{\ast}|_{E}$.

In principle, $\tau_{0}$ can be any graded $k$-linear automorphism of $A$ and
then these claims would be best possible. However, by [Zha96, Proposition
2.4], we may if necessary - with an acceptable loss of generality - replace
$\tau$ by a twisting system $\tau^{\prime}$ for which
$\tau^{\prime}_{0}=\operatorname{id}$, and have $A^{\tau}\cong
A^{\tau^{\prime}}$.

Doing this, and bearing in mind that we have potentially introduced a “hidden”
isomorphism in the course of doing so, we conclude that the triples
$(E^{\prime},\sigma^{\prime},\mathcal{L}^{\prime})$ and
$(E,\tau_{1}^{E}\circ\sigma,\mathcal{L})$, with
$\tau_{1}^{E}\in\operatorname{Aut}E$, yield isomorphic algebras, one of these
being $A^{\tau}$. ∎

######  Remark 42.

We note that [IM, Proposition 2.6] proves the same result in the special case
of $\tau$ being algebraic.

######  Remark 43.

Mori’s theorem also tells us how to construct the full twisting system. For as
in [Mor06, Remark 4.8], the condition
$(\overline{\tau_{n+1}}^{\ast}|_{E})\sigma=\sigma^{\prime}(\overline{\tau_{n}}^{\ast}|_{E})$
implies that given $\overline{\tau_{1}}^{\ast}$, we may inductively define

$\overline{\tau_{n}}^{\ast}\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}((\sigma^{\prime})^{n}\sigma^{-n})^{\ast}=((\tau_{1}^{E}\sigma)^{n}\sigma^{-n})^{\ast}$

and dualize and extend these from automorphisms of $\mathbb{P}(A_{1}^{\ast})$
to automorphisms of $T(A_{1})$, which will define a twisting system $\tau$ on
$A$.

We note two consequences. The first is that the twisting system $\tau$
obtained is completely determined by $\tau_{1}$, corresponding to $A$ being
generated in degree 1.

Secondly, if $\tau_{1}^{E}$ commutes with $\sigma$ then the twisting system
$\tau$ is algebraic: $\overline{\tau_{n}}^{\ast}=(\tau_{1}^{E})^{n}$ and hence
$\tau_{n}=\tau_{1}^{n}$ for all $n$. As in [ATVdB91, Proposition 8.8] this is
precisely the situation of $A^{\tau}$ being a twist by an automorphism.
Conversely, non-algebraic twists arise when we can find some $\tau_{1}^{E}$
that does not commute with $\sigma$ in $\operatorname{Aut}(E)$.

Let $E$ be a closed subscheme of $\mathbb{P}^{n}$ and define the following two
subgroups:

$\displaystyle\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\\{\sigma\in\operatorname{Aut}E\mid(\exists\hat{\sigma}\in\operatorname{Aut}\mathbb{P}^{n})(\hat{\sigma}|_{E}=\sigma)\\}\subseteq\operatorname{Aut}E,$
$\displaystyle\operatorname{Aut}(\mathbb{P}^{n}\downarrow E)$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\\{\rho\in\operatorname{Aut}\mathbb{P}^{n}\mid\rho(E)=E\\}\subseteq\operatorname{Aut}\mathbb{P}^{n}.$

Restriction defines a surjective group homomorphism
$\mathrm{Res}_{E}\colon\operatorname{Aut}(\mathbb{P}^{n}\downarrow
E)\to\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$.

We may then interpret the previous proposition as follows. Given a geometric
noncommutative $\mathbb{P}^{n}$ with associated coordinate ring $A$, we have a
triple $(E,\sigma,\mathcal{L})$ with $\sigma\in\operatorname{Aut}E$. If $\tau$
is a twisting system for $A$, then
$\tau_{1}^{E}=\mathrm{Res}_{E}(\overline{\tau_{1}}^{\ast})\in\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$
is such that $(E,\tau_{1}^{E}\sigma,\mathcal{L})$ determines $A^{\tau}$ up to
isomorphism.

Furthermore, by symmetry, every twist is of this form.

So we conclude that

######  Corollary 44.

Let $A$ be a coordinate ring of a geometric noncommutative $\mathbb{P}^{n}$
with associated triple $(E,\sigma,\mathcal{L})$. The isomorphism classes of
algebras twist-equivalent to $A$ are in bijection with the elements of the
coset
$\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\sigma\subset\operatorname{Aut}E$.
∎

The coset $\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\sigma$ is not a group,
unless the identity of $\operatorname{Aut}E$ is contained in
$\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\sigma$, in which case we must in
fact have $E=\mathbb{P}^{n}$,
$\operatorname{Aut}(E\uparrow\mathbb{P}^{n})=\operatorname{Aut}E=\operatorname{Aut}\mathbb{P}^{n}=\operatorname{Aut}(\mathbb{P}^{n}\downarrow
E)$ and $A$ twist-equivalent to $\mathcal{O}(\mathbb{P}^{n})$.

Note that there is a bijection between any two cosets of
$\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$ so that any two algebras having
the same point scheme $E$, twist-equivalent or not, can be viewed as having
the same number of twists.

The group $\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$ has a natural left
action on $\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\sigma$, giving rise to
a groupoid structure on $\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\sigma$ by
defining the set of objects to be the elements of
$\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\sigma$ and the morphisms to be
pairs $(\tau^{\prime},\tau\sigma)\colon\tau\sigma\to\tau^{\prime}\tau\sigma$.
This is precisely the action groupoid
$\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\sigma\mathbin{/\mkern-3.0mu/}\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$,
also known as the weak quotient.

It is well-known that the action groupoid gives rise to a faithful functor

$\mathcal{G}\colon\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\sigma\mathbin{/\mkern-3.0mu/}\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\to\operatorname{Aut}(E\uparrow\mathbb{P}^{n}),$

the latter being the group $\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$
considered as a one-object groupoid. The functor $\mathcal{G}$ is given by
sending every object to the unique object of
$\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$ and each morphism
$(\tau^{\prime},\tau\sigma)$ to $\tau^{\prime}$. In the instance at hand, with
the action groupoid arising from the left action of a group on a coset, we
have a transitive action, or equivalently a connected groupoid. Then
$\mathcal{G}$ is the functor giving rise to the equivalence of a connected
groupoid with a group.

That is, while it is conceptually helpful to consider
$\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\sigma\mathbin{/\mkern-3.0mu/}\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$,
as it keeps the role of $\sigma$ explicit, this groupoid is in fact equivalent
to the group $\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$.

We fix some $\mathcal{A}=(\operatorname{QGr}A,\pi A)$ with $A$ having
associated triple $(E,\sigma,\mathcal{L})$. We may now define a functor
$\mathcal{F}\colon\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A}\to\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\sigma$,
by $\mathcal{F}(\mathcal{B})=\sigma^{\prime}$ for
$(E,\sigma^{\prime},\mathcal{L})$ the triple associated to $B$. We may see by
examining the definitions of morphisms in the two categories that given a
twisting system $\tau$, we have the associated map
$\tau_{1}^{E}\sigma=\sigma^{\prime}$, and by the proposition (asserting that
twists precisely correspond to $\sigma\mapsto\tau_{1}^{E}\sigma$), this is
functorial.

However while this functor $\mathcal{F}$ is both full and essentially
surjective, it not an equivalence: it fails to be faithful because we may have
graded Morita equivalences in $\mathscr{NC}\left(\mathbb{P}^{n}\right)$, which
give rise to isomorphic twists that the triples cannot detect. We saw this
when we noted that we were suppressing some isomorphisms in the proof of
Proposition 41 above - indeed, precisely those coming from graded Morita
equivalences.

Since $\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A}$ is a connected
groupoid, it too is equivalent to a group, which we denote
$\underline{\operatorname{Aut}}(\mathcal{A})$. Note that this group is
isomorphic to the same such group for any object in the connected component of
$\mathcal{A}$. Via $\mathcal{G}$, it follows that $\mathcal{F}$ induces a
surjection
$\underline{\operatorname{Aut}}(\mathcal{A})\twoheadrightarrow\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$.

From this, we can obtain an isomorphism by first taking the quotient of
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$ by graded Morita equivalence, which
we denote $\overline{\mathscr{NC}\left(\mathbb{P}^{n}\right)}$. The
corresponding functors $\overline{\mathcal{F}}$ and $\overline{\mathcal{G}}$
then induce an equivalence
$\overline{\mathscr{NC}\left(\mathbb{P}^{n}\right)}/\overline{\mathcal{A}}\simeq\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$.

Let us denote by $\uuline{\operatorname{Aut}}(\mathcal{A})$ the group obtained
from the connected groupoid
$\overline{\mathscr{NC}\left(\mathbb{P}^{n}\right)}/\overline{\mathcal{A}}$
which is, by definition, the automorphism group of $\overline{\mathcal{A}}$ in
$\overline{\mathscr{NC}\left(\mathbb{P}^{n}\right)}/\overline{\mathcal{A}}$.

This gives our main result, which identifies the Zhang twists of homogeneous
coordinate rings of noncommutative $\mathbb{P}^{n}$s (up to graded Morita
equivalence) with the automorphisms of the associated point scheme that extend
to $\mathbb{P}^{n}$.

######  Theorem 45.

Let $\mathcal{A}=(\operatorname{QGr}A,\pi A)$ be a geometric noncommutative
$\mathbb{P}^{n}$ with homogeneous coordinate ring $A$ and let
$(E,\sigma,\mathcal{L})$ be its associated triple. Then

$\uuline{\operatorname{Aut}}(\mathcal{A})\cong\operatorname{Aut}(E\uparrow\mathbb{P}^{n}).$

That is, the automorphism group $\uuline{\operatorname{Aut}}(\mathcal{A})$ of
$\overline{\mathcal{A}}$ in the connected component
$\overline{\mathscr{NC}\left(\mathbb{P}^{n}\right)}/\overline{\mathcal{A}}$
containing the equivalence class of $\mathcal{A}$ of the groupoid of
noncommutative projective spaces up to graded Morita equivalence,
$\overline{\mathscr{NC}\left(\mathbb{P}^{n}\right)}$, is isomorphic to the
group $\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$ of automorphisms of the
point scheme $E$ of $A$ that extend to $\mathbb{P}^{n}$. ∎

We reiterate that the group $\operatorname{Aut}(E\uparrow\mathbb{P}^{n})$ and
hence the group $\uuline{\operatorname{Aut}}(\mathcal{A})$ depends only on
$E$, so that any two components of $\mathscr{NC}\left(\mathbb{P}^{n}\right)$
with the same associated point schemes have the same twists.

Also note that the group on the left hand side is defined entirely
algebraically, whereas the group on the right hand side is defined entirely
geometrically.

######  Remark 46.

Some care needs to be taken with respect to twists by graded algebra
automorphisms of $A$, $\operatorname{Aut}_{\mathrm{gr}}(A)$. As we saw in
Proposition 32, $\operatorname{Aut}_{\mathrm{gr}}(A)$ embeds in
$\mathrm{Iso}(\mathscr{NC}\left(\mathbb{P}^{n}\right)/\mathcal{A})$, but this
does not imply that $\operatorname{Aut}_{\mathrm{gr}}(A)$ embeds in
$\uuline{\operatorname{Aut}}(\mathcal{A})$. To form the latter group, we take
a quotient by graded Morita equivalences, or equivalently by twists yielding
isomorphic algebras. A twist by a graded algebra automorphism may or may not
yield an isomorphic algebra, so that in general we only know that a quotient
of $\operatorname{Aut}_{\mathrm{gr}}(A)$ embeds into
$\uuline{\operatorname{Aut}}(\mathcal{A})$ (this quotient possibly being the
trivial group).

Thus, Theorem 45 does not necessarily give any information about the graded
automorphism groups of homogeneous coordinate rings of geometric
noncommutative $\mathbb{P}^{n}$s.

Zhang twists of a given algebra are rarely isomorphic to one another. The
group $\uuline{\operatorname{Aut}}(\mathcal{A})$ characterizes the generalized
automorphisms arising as morphisms in the groupoid
$\mathscr{NC}\left(\mathbb{P}^{n}\right)$, and not those coming from
isomorphisms of the algebra $A$ itself. That is,
$\uuline{\operatorname{Aut}}(\mathcal{A})$ describes the non-algebraic twists
of $A$, without containing the classical automorphism group
$\operatorname{Aut}(A)$ as a subgroup.

We conclude with two examples, the first being
$\mathcal{O}_{q}(\mathbb{P}^{n})$, generalizing Example 35 and the discussion
in Section 5. The second example concerns 3-dimensional Sklyanin algebras,
whose point schemes are smooth elliptic curves.

######  Example 47.

Recall that we define quantum projective $n$-space to be the $k$-algebra

$\mathcal{O}_{q}(\mathbb{P}^{n})=k\langle x_{0},\dotsc,x_{n}\rangle/\langle
x_{i}x_{j}=qx_{j}x_{i}\ \forall\ i<j\rangle$

where $q\in k^{\ast}$ is assumed not a root of unity.

Define
$\genfrac{[}{]}{0.0pt}{1}{n+1}{2}=\\{I\subseteq\\{0,\dotsc,n\\}\mid|I|=2\\}$.
By work of De Laet-Le Bruyn111See also Belmans-De Laet-Le Bruyn, [BDLLB16].
([DLLB15, Proposition 1]), the point scheme of
$\mathcal{O}_{q}(\mathbb{P}^{n})$ is the union of the lines

$\ell_{I}\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\mathbb{V}(\\{x_{j}\mid
j\notin I\\}),\ \text{for}\ I\in\genfrac{[}{]}{0.0pt}{1}{n+1}{2}$

in $\mathbb{P}^{n}$. Set
$E=\bigcup_{I\in\genfrac{[}{]}{0.0pt}{1}{n+1}{2}}\ell_{I}$.

The associated automorphism of $E$ is defined on each line as

$\sigma|_{\ell_{\\{i_{1},i_{2}\\}}}(0:\dotsm:0:p_{i_{1}}:0:\dotsm:0:p_{i_{2}}:0:\dotsm:0)=(0:\dotsm:0:p_{i_{1}}:0:\dotsm:0:qp_{i_{2}}:0:\dotsm:0),$

corresponding to the relator $x_{i_{1}}\otimes x_{i_{2}}-qx_{i_{2}}\otimes
x_{i_{1}}$ via the correspondence indicated in the definition of geometricity
(39). Indeed, the existence of such data $(E,\sigma)$ affirms that
$\mathcal{O}_{q}(\mathbb{P}^{n})$ is geometric.

This automorphism does not extend to all of $\mathbb{P}^{n}$. Indeed, let
$\underline{\lambda}=(\lambda_{I})_{I\in\genfrac{[}{]}{0.0pt}{1}{n+1}{2}}\in(k^{\ast})^{\genfrac{[}{]}{0.0pt}{1}{n+1}{2}}$
and define $\mu(\underline{\lambda})\in\operatorname{Aut}E$ by

$\mu(\underline{\lambda})|_{\ell_{\\{i_{1},i_{2}\\}}}((0:\dotsm:0:p_{i_{1}}:0:\dotsm:0:p_{i_{2}}:0:\dotsm:0)=(0:\dotsm:0:\lambda_{\\{i_{1},i_{2}\\}}p_{i_{1}}:0:\dotsm:0:p_{i_{2}}:0:\dotsm:0).$

Then one may check that a purported extension of $\mu(\underline{\lambda})$ to
$\mathbb{P}^{n}$ would have to be represented by the image of the diagonal
matrix

$\mathrm{diag}(\prod_{k=0}^{n-1}\lambda_{k(k+1)},\dotsc,\prod_{k=i}^{n-1}\lambda_{k(k+1)},\dotsc,\lambda_{(n-1)n},1)$

in $\operatorname{PGL}_{n+1}\\!\left(k\right)$ subject to the conditions

$\lambda_{i_{1},i_{2}}=\prod_{k=i_{1}}^{i_{2}-1}\lambda_{k(k+1)}$

for all $i_{1}<i_{2}$. Now $\sigma=\mu(\underline{\lambda})$ with
$\lambda_{i_{1},i_{2}}=q^{-1}$ for all $i_{1},i_{2}$, which therefore does not
extend to $\mathbb{P}^{n}$ since $q$ is not a root of unity.

Noting that any element $\nu$ of $\operatorname{PGL}_{n+1}\\!\left(k\right)$
preserving $E\subseteq\mathbb{P}^{n}$ must be projectively linear, send each
line $\ell_{I}$ to another line $\ell_{J}$ and each intersection
$e_{i}\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}\ell_{i,i^{\prime}}\cap\ell_{i,i^{\prime\prime}}$
to another such intersection, we see that $\nu$ is determined by its values on
the intersections $e_{i}$, $i\in\\{0,\dotsc,n\\}$. From a consideration of the
remaining possibilities, it follows that

$\operatorname{Aut}(\mathbb{P}^{n}\downarrow
E)\cong\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\cong
S_{n+1}\ltimes((k^{\ast})^{n+1}/k^{\ast})$

with $S_{n+1}$ acting on $(k^{\ast})^{n+1}$ by the natural permutation action
$\triangleright$: that is, for $\sigma\in S_{n+1}$,
$\mu\in(k^{\ast})^{n+1}/k^{\ast}$ and $\underline{p}\in\mathbb{P}^{n}$ we have
$(\sigma\triangleright\mu)(\underline{p})=(\sigma\circ\mu\circ\sigma^{-1})(\underline{p})$.

We therefore conclude that the automorphism group
$\uuline{\operatorname{Aut}}(\operatorname{QGr}\mathcal{O}_{q}(\mathbb{P}^{n}),\pi\mathcal{O}_{q}(\mathbb{P}^{n}))$
is isomorphic to the group $\operatorname{Aut}(E\uparrow\mathbb{P}^{n})\cong
S_{n+1}\ltimes((k^{\ast})^{n+1}/k^{\ast})$ of automorphisms of the point
scheme $E=\bigcup_{I\in\genfrac{[}{]}{0.0pt}{1}{n+1}{2}}\ell_{I}$ that extend
to $\mathbb{P}^{n}$.

Notice in particular that elements of $(S_{n+1},\underline{0})\subseteq
S_{n+1}\ltimes((k^{\ast})^{n+1}/k^{\ast})$ do not induce graded algebra
automorphisms of $\mathcal{O}_{q}(\mathbb{P}^{n})$ and therefore correspond to
non-algebraic twists.

For example, the transposition $(0\,1)$ gives rise to the triple
$(E,(0\,1)\sigma,\mathcal{L})$ and we can compute the relations in its
associated algebra as per Definition 39. For example, on $\ell_{01}$,
$(0\,1)\sigma$ maps $(p_{0}:p_{1}:0:\dotsm:0)$ to $(qp_{1}:p_{0}:0:\dotsm:0)$,
for which

$(x_{0}\otimes x_{0}-qx_{1}\otimes
x_{1})((p_{0}:p_{1}:0:\dotsm:0),(qp_{1}:p_{0}:0:\dotsm:0))=p_{0}(qp_{1})-qp_{1}p_{0}=0,$

leading to the relation $x_{0}^{2}=qx_{1}^{2}$.

Continuing in this way yields the algebra

$k\langle x_{0},\dotsc,x_{n}\rangle/\langle x_{0}^{2}=qx_{1}^{2},\
x_{0}x_{j}=qx_{j}x_{1},\ x_{i}x_{j}=qx_{j}x_{i}\ \forall\ 1\leq i<j\leq
n\rangle.$

######  Example 48.

Let $k$ be of characteristic not equal to 2 or 3. Consider the 3-dimensional
Sklyanin algebras

$\mathrm{Skl}_{3}(a,b,c)\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}k\langle
x_{0},x_{1},x_{2}\rangle/\langle
ax_{i}x_{i+1}+bx_{i+1}x_{i}+cx_{i+2}^{2}\rangle$

where all indices are taken modulo 3. For all but a known finite set of points
$(a,b,c)$, this algebra is a geometric noncommutative $\mathbb{P}^{2}$ with
associated point scheme the smooth elliptic curve

$E\colon abc(x^{3}+y^{3}+z^{3})-(a^{3}+b^{3}+c^{3})xyz=0$

and automorphism $\sigma$ given by translation by a point (with respect to the
group law on $E$).

A summary of relevant results and references concerning the 3-dimensional
Sklyanin algebras (and their relationship with mathematical physics) may be
found in work of Walton ([Wal12]). For completeness, we also note that De Laet
([DL17]) has identified an action by graded algebra automorphisms of the
Heisenberg group $H_{3}$ of order 27 on Sklyanin algebras.

By general results on elliptic curves (see for example, [Sil09, §III]), every
automorphism of $E$ is the composition of a translation and an isogeny of the
curve with itself. Thus $\operatorname{Aut}(E)$ is the semidirect product of
the curve itself with the auto-isogeny group, the isogenies preserving a
chosen base point and the translations changing the base point.

The auto-isogeny groups of elliptic curves are well-known: depending on the
$j$-invariant of the curve, the auto-isogeny group is cyclic of order $n=2$, 4
or 6, realized as $\mu_{n}$ acting as
$[\zeta](x:y:z)=(\zeta^{2}x:\zeta^{3}y:z)$ (cf. [Sil09, Corollary III.10.2]).
It is common to call the auto-isogeny group $\operatorname{Aut}(E)$, but we
shall not; rather we will denote it $\mathcal{I}(E)$. Note that the
aforementioned action is projectively linear and hence
$\mathcal{I}(E)\subseteq\operatorname{Aut}(E\uparrow\mathbb{P}^{2})$.

Furthermore, it is known when a translation extends to an automorphism of the
ambient $\mathbb{P}^{2}$. This is given explicitly in [Mor06, Lemma 5.3],
where it is shown that a translation $\tau_{p}$ by a point $p$ extends to
$\mathbb{P}^{2}$ if and only if $p$ is 3-torsion, i.e. $\tau_{p}$ has order 3.
The group $E[3]$ of 3-torsion elements of an elliptic curve $E$ is also known:
it is isomorphic to $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$, a
group of order 9 ([Sil09, Corollary III.6.4]).

Combining these results, we conclude that
$\operatorname{Aut}(E\uparrow\mathbb{P}^{2})\cong(\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z})\rtimes\mathcal{I}(E)$,
a finite group of order $9n$. One may interpret this as saying that Sklyanin
algebras are extremely rigid, or very noncommutative, as they have very few
twists.

Note that the auto-isogenies act as graded algebra automorphisms of
$\mathrm{Skl}_{3}(a,b,c)$; indeed
$\mathrm{Skl}_{3}(a,b,c)^{[\zeta]}=\mathrm{Skl}_{3}(\zeta^{2}a,\zeta^{3}b,c)$.
Furthermore, for a point $p$, $\tau_{p}\sigma$ is again a translation by a
point, so that twists of Sklyanin algebras by translations are again Skylanin
algebras. Thus, we see that $\mathscr{NC}(\mathbb{P}^{2})$ contains countably
many connected components which, modulo graded Morita equivalence, are finite
and consist of Sklyanin algebras.

##  Appendix A An alternative approach to analysing noncommutative
$\mathbb{P}^{2}$s

As explained in the introduction to [ATVdB91] and explored in more detail in
§7 of that work, there is another approach to analysing noncommutative
$\mathbb{P}^{2}$s that one can take, by studying the “open complement” of the
“closed subscheme” defined by the point scheme of a noncommutative
$\mathbb{P}^{2}$. We briefly recall the general construction and then give
some more details in the specific case of $\mathcal{O}_{Q}(\mathbb{P}^{2})$.
We will not prove the assertions we make: proofs may be found in [ATVdB91].

Given a homogeneous coordinate ring of a noncommutative $\mathbb{P}^{2}$, $A$
say, we have seen the construction of its point scheme $E$, parameterizing
point modules in $\operatorname{QGr}A$, and the associated triple
$(E,\sigma,\mathcal{L}=\mathcal{O}_{E}(1))$. The coordinate ring of $E$ is in
fact given by a quotient of $A$ by a unique (up to scalar) normal regular
degree 3 element $g\in A_{3}$. Specifically, $B=A/gA$ is a twisted homogeneous
coordinate ring $B=\bigoplus_{n\geq
0}\operatorname{Hom}(\mathcal{O}_{E},\mathcal{L}\otimes\mathcal{L}^{\sigma}\otimes\dotsm\otimes\mathcal{L}^{\sigma^{n-1}})$.

Furthermore, $\operatorname{QGr}B\simeq\operatorname{QCoh}{E}$ and $g$
annihilates point modules in $\operatorname{QGr}A$. Indeed, via the surjection
$A\twoheadrightarrow B=A/gA$, we have
$\operatorname{QGr}B\subset\operatorname{QGr}A$. By a natural extension of the
usual terminology, we regard $(\operatorname{QGr}B,\pi B)$ as defining a
closed subscheme of $(\operatorname{QGr}A,\pi A)$.

It is then also natural to ask about the open complement. This is given by the
degree zero part $\Lambda_{0}$ of the graded localized algebra
$\Lambda=A[g^{-1}]$.

One has a natural functor $\operatorname{Gr}A\to\operatorname{Gr}\Lambda_{0}$,
given by localization: $M\mapsto M[g^{-1}]_{0}$. Furthermore, the category of
finite-dimensional $\Lambda_{0}$-modules is equivalent to the category of
$g$-torsion-free normalized $A$-modules (see [ATVdB91, p.371] for details).

Then the dichotomy between $g$-torsion and $g$-torsion-free $A$-modules
corresponds to the complementarity of the closed subscheme and its open
complement.

Lastly, we note that it follows from [Zha96, Proposition 5.4] that one may
directly generalize [ATVdB91, Proposition 8.12], to show that if $A^{\tau}$ is
a twist of $A$ then
$\Lambda_{0}=A[g^{-1}]_{0}\cong(A^{\tau})[g_{\tau}^{-1}]_{0}$. That is,
twisting is compatible with the decomposition of $\operatorname{QGr}A$ into
the closed subscheme (corresponding to the point scheme) and its open
complement (given by $\Lambda_{0}$).

For the case of $A=\mathcal{O}_{Q}(\mathbb{P}^{2})$, we may calculate
$\Lambda_{0}$ explicitly, as follows.

Recall that we have

$\mathcal{O}_{Q}(\mathbb{P}^{2})\stackrel{{\scriptstyle\scriptscriptstyle{\mathrm{def}}}}{{=}}k\langle
x,y,z\rangle/\langle zy=\alpha yz,xz=\beta zx,yx=\gamma xy\rangle$

so that $g=(1-\alpha\beta\gamma)xyz$ is the canonical normalizing element
described above. As previously, we assume that
$\alpha\beta\gamma\notin\\{0,1\\}$.

From the quasi-commutation relations in $\mathcal{O}_{Q}(\mathbb{P}^{2})$, it
is straightforward to verify the following relations in $\Lambda$:

$\displaystyle gx^{i}y^{j}z^{k}$
$\displaystyle=\alpha^{i-j}\beta^{i-k}\gamma^{j-k}x^{i}y^{j}z^{k}g$
$\displaystyle g^{-1}x^{i}y^{j}z^{k}$
$\displaystyle=\alpha^{j-i}\beta^{k-i}\gamma^{k-j}x^{i}y^{j}z^{k}g^{-1}$
$\displaystyle(x^{i}y^{j}z^{k}g^{-1})(x^{l}y^{m}z^{n}g^{-1})$
$\displaystyle=\alpha^{-\left|\begin{smallmatrix}i-1&j-1\\\
l-1&m-1\end{smallmatrix}\right|}\beta^{-\left|\begin{smallmatrix}i-1&k-1\\\
l-1&n-1\end{smallmatrix}\right|}\gamma^{-\left|\begin{smallmatrix}j-1&k-1\\\
m-1&n-1\end{smallmatrix}\right|}(x^{l}y^{m}z^{n}g^{-1})(x^{i}y^{j}z^{k}g^{-1})$

Indeed, we note that computationally it is easiest to work in the fully-
localized algebra $A[x^{-1},y^{-1},z^{-1}]$, into which $\Lambda$ naturally
embeds.

A generating set for $\Lambda_{0}$ is given by the cubic monomials in $x$, $y$
and $z$ that span $A_{3}$, multiplied by the inverse of $g$:

$\\{x^{3}g^{-1},y^{3}g^{-1},z^{3}g^{-1},x^{2}yg^{-1},x^{2}zg^{-1},y^{2}zg^{-1},xy^{2}g^{-1},xz^{2}g^{-1},yz^{2}g^{-1},xyzg^{-1}=(1-\alpha\beta\gamma)^{-1}\\}.$

From the above relations, we see that every pair of these satisfies a quasi-
commutation relation and since $\mathcal{O}_{Q}(\mathbb{P}^{2})$ is quadratic,
we deduce that $\Lambda_{0}$ is in this case isomorphic to what is commonly
referred to in the literature as a quantum polynomial algebra on nine
generators. This contrasts starkly with the commutative case, where $g=0$,
$E=\mathbb{P}^{2}$, $B=A$ and there is an empty open complement.

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