1 Introduction
The notion of twisting system was introduced by Zhang in [Reference Zhang14]. If there is a twisting system
$\theta =\{ \theta _n \}_{n \in \mathbb {Z}}$
for a graded algebra A, then we can define a new graded algebra
$A^{\theta }$
, called a twisted algebra. Zhang gave a necessary and sufficient algebraic condition via a twisting system when two categories of graded right modules are equivalent [Reference Zhang14, Theorem 3.5]. Although a twisting system plays an important role to study a graded algebra, it is often difficult to construct a twisting system on a graded algebra if it is given by generators and relations.
Mori introduced the notion of geometric algebra
$\mathcal {A}(E,\sigma )$
, which is determined by geometric data which consist of a projective variety E, called the point variety, and its automorphism
$\sigma $
. For these algebras, Mori gave a necessary and sufficient geometric condition when two categories of graded right modules are equivalent [Reference Mori11, Theorem 4.7]. By using this geometric condition, we can easily construct a twisting system.
Cooney and Grabowski defined a groupoid whose objects are geometric noncommutative projective spaces and whose morphisms are isomorphisms between them. By studying a correspondence between the morphisms of this groupoid and a twisting system, they showed that the morphisms of this groupoid are parametrized by a set of certain automorphisms of the point variety [Reference Cooney and Grabowski2, Theorem 28].
In this paper, we focus on studying a twisted algebra of a geometric algebra
$A=\mathcal {A}(E,\sigma )$
. For a twisting system
$\theta $
on A, we set
$\Phi (\theta ):=\overline {(\theta _{1}|_{A_{1}})^{\ast }} \in \mathrm {Aut}_{k}\,\mathbb {P}(A_1^{\ast })$
by dualization and projectivization. We find a subset
$M(E,\sigma )$
of
$\mathrm {Aut}_{k}\,\mathbb {P}(A_{1}^{\ast })$
parametrizing twisted algebras of A up to isomorphism. As an application to three-dimensional geometric Artin–Schelter regular algebras, we will compute
$M(E,\sigma )$
(see Theorems 4.8 and 4.11), which completes the classification of twisted algebras of three-dimensional geometric Artin–Schelter regular algebras.
Itaba and the author showed that for any three-dimensional quadratic Artin–Schelter regular algebra B, there are a three-dimensional quadratic Calabi–Yau Artin–Schelter regular algebra S and a twisting system
$\theta $
such that
$B \cong S^{\theta }$
[Reference Itaba and Matsuno8, Theorem 4.4]. Except for one case, a twisting system
$\theta $
can be induced by a graded algebra automorphism of S. By using
$M(E,\sigma )$
, we can recover this fact in the case that B is geometric (see Corollary 4.12).
2 Preliminary
Throughout this paper, we fix an algebraically closed field k of characteristic zero and assume that a graded algebra is an
$\mathbb {N}$
-graded algebra
$A=\bigoplus _{i \in \mathbb {N}} A_i$
over k. A graded algebra
$A=\bigoplus _{i \in \mathbb {N}} A_i$
is called connected if
$A_0=k$
. Let
$\mathrm {GrAut}_{k}\,A$
denote the group of graded algebra automorphisms of A. We denote by
$\mathrm {GrMod} A$
the category of graded right A-modules. We say that two graded algebras A and
$A'$
are graded Morita equivalent if two categories
$\mathrm {GrMod} A$
and
$\mathrm {GrMod} A'$
are equivalent.
2.1 Twisting systems and twisted algebras
Definition 2.1 Let A be a graded algebra. A set of graded k-linear automorphisms
$\theta =\{ \theta _n \}_{n \in \mathbb {Z}}$
of A is called a twisting system on A if
$$ \begin{align*} \theta_n(a\theta_m(b))=\theta_n(a)\theta_{n+m}(b) \end{align*} $$
for any
$l,m,n \in \mathbb {Z}$
and
$a \in A_m$
,
$b \in A_l$
. The twisted algebra of A by
$\theta $
, denoted by
$A^{\theta }$
, is a graded algebra A with a new multiplication
$\ast $
defined by
$$ \begin{align*}a \ast b =a\theta_m(b)\end{align*} $$
for any
$a \in A_m$
,
$b \in A_l$
. A twisting system
$\theta =\{ \theta _n \}_{n \in \mathbb {Z}}$
is called algebraic if
$\theta _{m+n}=\theta _m \circ \theta _n$
for every
$m,n \in \mathbb {Z}$
.
We denote by
$\mathrm {TS}^{\mathbb {Z}}(A)$
the set of twisting systems on A. Zhang [Reference Zhang14] found a necessary and sufficient algebraic condition for
$\mathrm {GrMod} A \cong \mathrm {GrMod} A'$
.
Theorem 2.2 [Reference Zhang14, Theorem 3.5]
Let A and
$A'$
be graded algebras finitely generated in degree
$1$
over k. Then
$\mathrm {GrMod} A \cong \mathrm {GrMod} A'$
if and only if
$A'$
is isomorphic to a twisted algebra
$A^{\theta }$
by a twisting system
$\theta \in \mathrm {TS}^{\mathbb {Z}}(A)$
.
Definition 2.3 For a graded algebra A, we define
$$ \begin{align*} &\mathrm{TS}^{\mathbb{Z}}_{0}(A):=\{ \theta \in \mathrm{TS}^{\mathbb{Z}}(A) \mid \theta_0=\mathrm{id}_A \}, \\ &\mathrm{TS}_{\mathrm{alg}}^{\mathbb{Z}}(A):=\{ \theta \in \mathrm{TS}^{\mathbb{Z}}_{0}(A) \mid \theta \text{ is algebraic} \}, \\ &\mathrm{Twist}(A):=\{ A^{\theta} \mid \theta \in \mathrm{TS}^{\mathbb{Z}}(A) \}/_{\cong},\\ &\mathrm{Twist}_{\mathrm{alg}}(A):=\{ A^{\theta} \mid \theta \in \mathrm{TS}^{\mathbb{Z}}_{\mathrm{alg}}(A) \}/_{\cong}. \end{align*} $$
Lemma 2.4 [Reference Zhang14, Proposition 2.4]
Let A be a graded algebra. For every
$\theta \in \mathrm {TS}^{\mathbb {Z}}(A)$
, there exists
$\theta ' \in \mathrm {TS}^{\mathbb {Z}}_{0}(A)$
such that
$A^{\theta } \cong A^{\theta '}$
.
It follows from Lemma 2.4 that
$\mathrm {Twist}(A)=\{ A^{\theta } \mid \theta \in \mathrm {TS}^{\mathbb {Z}}_{0}(A) \}/_{\cong }$
, so we may assume that
$\theta \in \mathrm {TS}^{\mathbb {Z}}_{0}(A)$
to study
$\mathrm {Twist}(A)$
. By the definition of twisting system, it follows that
$\theta \in \mathrm {TS}^{\mathbb {Z}}_{\mathrm {alg}}(A)$
if and only if
$\theta _n=\theta _{1}^{n}$
for every
$n \in \mathbb {Z}$
and
$\theta _{1} \in \mathrm {GrAut}_{k}\,A$
, so
$$ \begin{align*}\mathrm{Twist}_{\mathrm{alg}}(A)=\{ A^{\phi} \mid \phi \in \mathrm{GrAut}_{k}\,A \}/_{\cong},\end{align*} $$
where
$A^{\phi }$
means the twisted algebra of A by
$\{ \phi ^{n} \}_{n \in \mathbb {Z}}$
.
2.2 Geometric algebra
Let V be a finite-dimensional k-vector space, and let
$A=T(V)/(R)$
be a quadratic algebra where
$T(V)$
is a tensor algebra over k and
$R \subset V \otimes V$
. Since an element of R defines a multilinear function on
$V^{\ast }\times V^{\ast }$
, we can define a zero set associated with R by
$$ \begin{align*}\mathcal{V}(R)=\{ (p,q) \in \mathbb{P}(V^{\ast}) \times \mathbb{P}(V^{\ast}) \mid g(p,q)=0 \text{ for any } g \in R \}.\end{align*} $$
Definition 2.5 Let
$A=T(V)/(R)$
be a quadratic algebra. A geometric pair
$(E,\sigma )$
consists of a projective variety
$E \subset \mathbb {P}(V^{\ast })$
and
$\sigma \in \mathrm {Aut}_{k} E$
.
-
(1) We say that A satisfies (G1) if there exists a geometric pair
$(E,\sigma )$
such that In this case, we write
$$ \begin{align*}\mathcal{V}(R)=\{ (p,\sigma(p)) \in \mathbb{P}(V^{\ast}) \times \mathbb{P}(V^{\ast}) \mid p \in E \}.\end{align*} $$
$\mathcal {P}(A)=(E,\sigma )$
, and call E the point variety of A.
-
(2) We say that A satisfies (G2) if there exists a geometric pair
$(E,\sigma )$
such that In this case, we write
$$ \begin{align*}R=\{ g \in V \otimes V \mid g(p,\sigma(p))=0 \text{ for all } p \in E \}.\end{align*} $$
$A=\mathcal {A}(E,\sigma )$
.
-
(3) We say that A is a geometric algebra if it satisfies both (G1) and (G2) with
$A=\mathcal {A}(\mathcal {P}(A))$
.
For geometric algebras, Mori [Reference Mori11] found a necessary and sufficient geometric condition for
$\mathrm {GrMod} A \cong \mathrm {GrMod} A'$
.
Theorem 2.6 [Reference Mori11, Theorem 4.7]
Let
$A=\mathcal {A}(E,\sigma )$
and
$A'=\mathcal {A}(E',\sigma ')$
be geometric algebras. Then
$\mathrm {GrMod} A \cong \mathrm {GrMod} A'$
if and only if there exists a sequence of automorphisms
$\{ \tau _n \}_{n \in \mathbb {Z}}$
of
$\mathbb {P}(V^{\ast })$
for
$n \in \mathbb {Z}$
, each of which sends E isomorphically onto
$E'$
, such that the diagram
commutes for every
$n \in \mathbb {Z}$
.
2.3 Artin–Schelter regular algebras and Calabi–Yau algebras
Definition 2.7 A connected graded algebra A is called a d-dimensional Artin–Schelter regular algebra (simply AS-regular algebra) if A satisfies the following conditions:
-
(1)
$\mathrm {gldim} A=d<\infty $
, -
(2)
$\mathrm {GKdim} A:=\inf \{ \alpha \in \mathbb {R} \mid \dim _{k}(\sum _{i=0}^{n}A_i) \leq n^{\alpha } \text { for all } n\gg 0 \} <\infty $
, and -
(3)
$\mathrm {Ext}^{i}_{A}(k,A)= \begin {cases} k, \,\,\,\text { if } i=d, \\ 0, \,\,\,\text { if } i \neq d. \end {cases}$
Artin and Schelter proved that a three-dimensional AS-regular algebra A finitely generated in degree
$1$
is isomorphic to one of the following forms:
$$ \begin{align*}k\langle x,y,z \rangle/(f_1,f_2,f_3) \text{ or } k\langle x,y \rangle/(g_1,g_2),\end{align*} $$
where
$f_i$
are homogeneous polynomials of degree
$2$
(the quadratic case) and
$g_j$
are homogeneous polynomials of degree
$3$
(the cubic case; see [Reference Artin and Schelter1, Theorem 1.5]).
We recall the definition of Calabi–Yau algebra introduced by [Reference Ginzburg4].
Definition 2.8 [Reference Ginzburg4]
A k-algebra S is called d-dimensional Calabi–Yau if S satisfies the following conditions:
-
(1)
$\mathrm {pd}_{S^e}\,S=d<\infty $
, and -
(2)
$\mathrm {Ext}^i_{S^e}(S,S^e) \cong \begin {cases} S, \,\,\,\text { if } i=d \\ 0, \,\,\,\text { if } i \neq d \end {cases}$
(as right
$S^e$
-modules),
where
$S^e=S^{\mathrm {op}} \otimes _{k} S$
is the enveloping algebra of S.
The following theorem tells us that we may assume that three-dimensional quadratic AS-regular algebra is Calabi–Yau up to graded Morita equivalence.
Theorem 2.9 [Reference Itaba and Matsuno8, Theorem 4.4]
For every three-dimensional quadratic AS-regular algebra A, there exists a three-dimensional quadratic Calabi–Yau AS-regular algebra S such that
$\mathrm {GrMod}\,A \cong \mathrm {GrMod}\,S$
.
Lemma 2.10 ([Reference Hu, Matsuno and Mori6, Lemma 2.8], [Reference Itaba and Matsuno7, Theorem 3.2], and [Reference Mori and Ueyama12, Lemma 3.8])
Every three-dimensional geometric AS-regular algebra A is graded Morita equivalent to
$S=k \langle x,y,z \rangle /(f_1,f_2,f_3)=\mathcal {A}(E,\sigma )$
in Table 1 .
Defining relations and geometric pairs.
Remark 2.11 The original definition of geometric algebra given by Mori [Reference Mori11] is different from our definition. In the sense of Definition 2.5, there exists a three-dimensional quadratic AS-regular algebra which is not a geometric algebra. Strictly speaking, a three-dimensional quadratic AS-regular algebra is a geometric algebra in our sense if and only if the “point scheme” is reduced.
3 Twisted algebras of geometric algebras
In this section, we study twisted algebras of geometric algebras. Let
$E \subset \mathbb {P}(V^{\ast })$
be a projective variety where V is a finite-dimensional k-vector space. We use the following notations introduced in [Reference Cooney and Grabowski2].
Definition 3.1 Let
$E \subset \mathbb {P}(V^{\ast })$
be a projective variety and
$\sigma \in \mathrm {Aut}_{k}\,E$
. We define
$$ \begin{align*} &\mathrm{Aut}_{k}(E \uparrow \mathbb{P}(V^{\ast})):= \{ \tau \in \mathrm{Aut}_{k}\,E \mid \tau=\overline{\tau}|_{E} \text{ for some } \overline{\tau} \in \mathrm{Aut}_{k}\,\mathbb{P}(V^{\ast}) \}, \\ &\mathrm{Aut}_{k}(\mathbb{P}(V^{\ast}) \downarrow E):= \{ \tau \in \mathrm{Aut}_{k}\,\mathbb{P}(V^{\ast}) \mid \tau|_{E} \in \mathrm{Aut}_{k}\,E \}, \\ &Z(E,\sigma):=\{ \tau \in \mathrm{Aut}_{k}(\mathbb{P}(V^{\ast}) \downarrow E) \mid \sigma\tau|_{E}\sigma^{-1}=\tau|_{E}\}, \\ &M(E,\sigma):=\{ \tau \in \mathrm{Aut}_{k}(\mathbb{P}(V^{\ast}) \downarrow E) \mid (\tau|_{E}\sigma)^{i}\sigma^{-i} \in \mathrm{Aut}_{k}(E \uparrow \mathbb{P}(V^{\ast}))\,\,\, \forall i\in\mathbb{Z} \}, \\ &N(E,\sigma):=\{ \tau \in \mathrm{Aut}_{k}(\mathbb{P}(V^{\ast}) \downarrow E) \mid \sigma\tau|_{E}\sigma^{-1} \in \mathrm{Aut}_{k}(E \uparrow \mathbb{P}(V^{\ast})) \}. \end{align*} $$
Note that
$ Z(E,\sigma ) \subset M(E,\sigma ) \subset N(E,\sigma ) \subset \mathrm {Aut}_{k}(\mathbb {P}(V^{\ast }) \downarrow E)$
, and
$Z(E,\sigma )$
,
$N(E,\sigma )$
are subgroups of
$\mathrm {Aut}_{k}(\mathbb {P}(V^{\ast }) \downarrow E)$
.
Lemma 3.2 Let
$E \subset \mathbb {P}(V^{\ast })$
be a projective variety and
$\sigma \in \mathrm {Aut}_{k}\,E$
. If
$\sigma \mathrm {Aut}_{k}(E \uparrow \mathbb {P}(V^{\ast }))=\mathrm {Aut}_{k}(E \uparrow \mathbb {P}(V^{\ast }))\sigma $
, then
$M(E,\sigma )=N(E,\sigma )=\mathrm {Aut}_{k}(\mathbb {P}(V^{\ast }) \downarrow E)$
.
Proof Since
$M(E,\sigma ) \subset N(E,\sigma ) \subset \mathrm {Aut}_{k}(\mathbb {P}(V^{\ast }) \downarrow E)$
in general, it is enough to show that
$\mathrm {Aut}_{k}(\mathbb {P}(V^{\ast }) \downarrow E) \subset M(E,\sigma )$
. We will show that for any
$\tau \in \mathrm {Aut}_{k}(\mathbb {P}(V^{\ast })\downarrow E)$
,
$(\tau |_E\sigma )^{i}\sigma ^{-i} \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}(V^{\ast }))$
for every
$i \in \mathbb {Z}$
by induction so that
$\tau \in M(E,\sigma )$
. The claim is trivial for
$i=0$
. If
$(\tau |_E\sigma )^{i}\sigma ^{-i} \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}(V^{\ast }))$
for some
$i \ge 0$
, then
$(\tau |_E\sigma )^{i+1}\sigma ^{-i-1}= \tau |_E\sigma ((\tau |_E\sigma )^i\sigma ^{-i})\sigma ^{-1} \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}(V^{\ast }))$
. If
$(\tau |_E\sigma )^{-i}\sigma ^i \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}(V^{\ast }))$
for some
$i \ge 0$
, then
$$ \begin{align*}(\tau|_E\sigma)^{-(i+1)}\sigma^{i+1}=\sigma^{-1}\tau|_E^{-1}((\tau|_E\sigma)^{-i}\sigma^i)\sigma \in \mathrm{Aut}_{k}(E \uparrow \mathbb{P}(V^{\ast})).\\[-34pt]\end{align*} $$
Let
$A=\mathcal {A}(E,\sigma )$
be a geometric algebra. The map
$\Phi : \mathrm {TS}^{\mathbb {Z}}_{0}(A) \to \mathrm {Aut}_{k}\,\mathbb {P}(A_1^{\ast })$
is defined by
$\Phi (\theta ):=\overline {(\theta _{1}|_{A_{1}})^{\ast }}$
.
Lemma 3.3 Let
$A=\mathcal {A}(E,\sigma )$
be a geometric algebra. Then
$$ \begin{align*}\Phi(\mathrm{TS}^{\mathbb{Z}}_{0}(A))=M(E,\sigma).\end{align*} $$
Proof Let
$\theta \in \mathrm {TS}^{\mathbb {Z}}_{0}(A)$
. We set
$V:=A_1=(A^{\theta })_1$
. Then
$\theta _n$
is also a graded k-linear isomorphism from
$A^{\theta }$
to A and satisfies
$\theta _n(a \ast b)=\theta _n(a)\theta _{n+m}(b)$
for every
$n, m, l\in \mathbb {Z}$
and
$a \in A^{\theta }_m$
,
$b \in A^{\theta }_l$
. Let
$\tau _n : \mathbb {P}(V^{\ast }) \to \mathbb {P}(V^{\ast })$
be automorphisms induced by the duals of
$\theta _n|_V : V \to V$
. By [Reference Cooney and Grabowski2, Remark 15],
$\tau _n \in \mathrm {Aut}_{k}(\mathbb {P}(V^{\ast }) \downarrow E)$
and the diagram of automorphisms
commutes for every
$n \in \mathbb {Z}$
. Then
$(\tau _{1}|_E\sigma )^{n}\sigma ^{-n}=\tau _n|_E \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}(V^{\ast }))$
for every
$n \in \mathbb {Z}$
, so it holds that
$\Phi (\theta )=\tau _{1} \in M(E,\sigma )$
.
Conversely, let
$\tau \in M(E,\sigma )$
. Since
$(\tau |_E\sigma )^{n}\sigma ^{-n} \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}(V^{\ast }))$
, there is an automorphism
$\tau _n \in \mathrm {Aut}_{k}\,\mathbb {P}(V^{\ast })$
such that
$\tau _n|_E=(\tau |_E\sigma )^{n}\sigma ^{-n}$
for every
$n \in \mathbb {Z}$
. By [Reference Cooney and Grabowski2, Remark 15], there exists
$\theta \in \mathrm {TS}^{\mathbb {Z}}_{0}(A)$
such that
$\overline {(\theta _{n}|_{A_{1}})^{\ast }}=\tau _n$
for every
$n \in \mathbb {Z}$
. Hence, it follows that
$\Phi (\theta )=\overline {(\theta _{1}|_{A_{1}})^{\ast }}=\tau $
.
Let
$A=T(V)/I$
be a connected graded algebra. Let
$\Psi : \mathrm {GrAut}_{k}\,A \to \mathrm {PGL}(V)$
be a group homomorphism defined by
$\Psi (\phi )=\overline {\phi |_V}$
. We set
$$ \begin{align*}\mathrm{PGrAut}_{k}\,A:=\mathrm{GrAut}_{k}\,A/\mathrm{Ker\,} \Psi.\end{align*} $$
Lemma 3.4 Let
$A=\mathcal {A}(E,\sigma )$
be a geometric algebra.
-
(1)
$\Phi (\mathrm {TS}^{\mathbb {Z}}_{\mathrm {alg}}(A))=Z(E,\sigma )$
. -
(2)
$\mathrm {PGrAut}_{k}\,A \cong Z(E,\sigma )^{\mathrm {op}}.$
Proof (1) Let
$\theta =\{ \theta _{1}^{n} \}_{n \in \mathbb {Z}} \in \mathrm {TS}^{\mathbb {Z}}_{\mathrm {alg}}(A)$
. We set
$V:=A_1=(A^{\theta })_1$
. Let
$\tau _n : \mathbb {P}(V^{\ast }) \to \mathbb {P}(V^{\ast })$
be automorphisms induced by the duals of
$\theta _{1}^n|_V : V \to V$
. Then we can write
$\tau _n=\tau _{1}^n$
for every
$n \in \mathbb {Z}$
. By the proof of Lemma 3.3, it follows that
$(\tau _{1}|_E)^n=(\tau _1|_E\sigma )^n\sigma ^{-n}$
for every
$n \in \mathbb {Z}$
. If
$n=2$
, then
$\tau _{1}|_E\sigma =\sigma \tau _{1}|_E$
, so
$\Phi (\theta )=\tau _{1} \in Z(E,\sigma )$
.
Conversely, let
$\tau \in Z(E,\sigma )$
. Since
$\tau |_E\sigma =\sigma \tau |_E$
,
$(\tau |_E\sigma )^{n}\sigma ^{-n}=(\tau |_E)^{n}$
for every
$n \in \mathbb {Z}$
. By [Reference Cooney and Grabowski2, Remark 15], there exists
$\theta =\{ \theta _{1}^n \}_{n \in \mathbb {Z}} \in \mathrm {TS}^{\mathbb {Z}}_{\mathrm {alg}}(A)$
such that
$\overline {(\theta _{1}|_{V})^{\ast }}^n=\tau ^n$
for every
$n \in \mathbb {Z}$
. Hence, it follows that
$\Phi (\theta )=\tau $
.
(2) By the following commutative diagram
it follows that
$\mathrm {PGrAut}_{k}\,A \cong \Phi (\mathrm {TS}^{\mathbb {Z}}_{\mathrm {alg}}(A))=Z(E,\sigma )^{\mathrm {op}}$
.
Theorem 3.5 Let
$A=\mathcal {A}(E,\sigma )$
be a geometric algebra.
-
(1)
$\mathrm {Twist}(A)=\{ \mathcal {A}(E,\tau |_E\sigma ) \mid \tau \in M(E,\sigma ) \}/_{\cong }$
. -
(2)
$\mathrm {Twist}_{\mathrm {alg}}(A)=\{ \mathcal {A}(E,\tau |_E\sigma ) \mid \tau \in Z(E,\sigma ) \}/_{\cong }$
.
Proof By [Reference Cooney and Grabowski2, Proposition 13], for every
$\theta \in \mathrm {TS}^{\mathbb {Z}}_{0}(A)$
,
$A^{\theta } \cong \mathcal {A}(E,\Phi (\theta )|_E\sigma )$
. By Lemma 3.3, it follows that
$$ \begin{align*} \mathrm{Twist}(A)&:=\{ A^{\theta} \mid \theta \in \mathrm{TS}^{\mathbb{Z}}_{0}(A) \}/_{\cong} \\ &=\{ \mathcal{A}(E,\Phi(\theta)|_E\sigma) \mid \theta \in \mathrm{TS}^{\mathbb{Z}}_{0}(A) \}/_{\cong} \\ &=\{ \mathcal{A}(E,\tau|_E\sigma) \mid \tau \in \Phi(\mathrm{TS}^{\mathbb{Z}}_{0}(A)) \}/_{\cong} \\ &=\{ \mathcal{A}(E,\tau|_E\sigma) \mid \tau \in M(E,\sigma) \}/_{\cong}. \end{align*} $$
By Lemma 3.4, we can similarly show that
$$ \begin{align*}\mathrm{Twist}_{\mathrm{alg}}(A)= \{ \mathcal{A}(E,\tau|_E\sigma) \mid \tau \in Z(E,\sigma) \}/_{\cong}.\\[-34pt]\end{align*} $$
4 Twisted algebras of three-dimensional geometric AS-regular algebras
In this section, we classify twisted algebras of three-dimensional geometric AS-regular algebras. We recall that for connected graded algebras A and
$A'$
generated in degree 1,
$\mathrm {GrMod}\,A \cong \mathrm {GrMod}\,A'$
if and only if
$A' \in \mathrm {Twist}(A)$
, so
$$ \begin{align*}\mathrm{Twist}(A)=\{ A' \mid \mathrm{GrMod}\,A' \cong \mathrm{GrMod}\,A \}/_{\cong}. \end{align*} $$
By Lemma 2.10, we may assume that A is a three-dimensional geometric Calabi–Yau AS-regular algebra in Table 1 to compute
$\mathrm {Twist}(A)$
. The algebras in Table 1 are called standard in this paper. For three-dimensional standard AS-regular algebras, we will compute the subsets
$Z(E,\sigma )$
and
$M(E,\sigma )$
of
$\mathrm {Aut}_{k}(\mathbb {P}^2 \downarrow E)$
. We remark that some of the computations were given in [Reference Cooney and Grabowski2, Section 4].
For a three-dimensional geometric AS-regular algebra
$\mathcal {A}(E,\sigma )$
, the map
$$ \begin{align*}\mathrm{Aut}_{k}(\mathbb{P}^{2} \downarrow E) \to \mathrm{Aut}_{k}(E \uparrow \mathbb{P}^{2}); \tau \mapsto \tau|_E\end{align*} $$
is a bijection, so we identify
$\tau \in \mathrm {Aut}_{k}(\mathbb {P}^2 \downarrow E)$
with
$\tau |_E \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}^2)$
if there is no potential confusion.
Let E be an elliptic curve in
$\mathbb {P}^2$
. We use a Hesse form
$$ \begin{align*}E=\mathcal{V}(x^3+y^3+z^3-3 \lambda xyz),\end{align*} $$
where
$\lambda \in k$
with
$\lambda ^3 \neq 1$
. It is known that every elliptic curve in
$\mathbb {P}^2$
can be written in this form (see [Reference Frium3, Corollary 2.18]). The j-invariant of a Hesse form E is given by
$j(E)=\frac {27\lambda ^3(\lambda ^3+8)^3}{(\lambda ^3-1)}$
(see [Reference Frium3, Proposition 2.16]). The j-invariant
$j(E)$
classifies elliptic curves in
$\mathbb {P}^2$
up to projective equivalence (see [Reference Hartshorne5, Theorem IV.4.1(b)]). We fix the group structure on E with the zero element
$o:=(1,-1,0) \in E$
(see [Reference Frium3, Theorem 2.11]). For a point
$p \in E$
, a translation by p, denoted by
$\sigma _{p}$
, is an automorphism of E defined by
$\sigma _{p}(q)=p+q$
for every
$q \in E$
. We define
$\mathrm {Aut}_{k}(E,o):=\{ \sigma \in \mathrm {Aut}_{k}\,E \mid \sigma (o)=o \}$
. It is known that
$\mathrm {Aut}_{k}(E,o)$
is a finite cyclic subgroup of
$\mathrm {Aut}_{k}\,E$
(see [Reference Hartshorne5, Corollary IV.4.7]).
Lemma 4.1 [Reference Itaba and Matsuno7, Theorem 4.6]
A generator of
$\mathrm {Aut}_{k}(E,o)$
is given by:
-
(1)
$\tau _E(a,b,c):=(b,a,c)$
if
$j(E) \neq 0,12^3$
, -
(2)
$\tau _E(a,b,c):=(b,a,\varepsilon c)$
if
$\lambda =0$
(so that
$j(E)=0$
), -
(3)
$\tau _E(a,b,c):=(\varepsilon ^2a+\varepsilon b+c,\varepsilon a+\varepsilon ^2b+c,a+b+c)$
if
$\lambda =1+\sqrt {3}$
(so that
$j(E)=12^3$
),
where
$\varepsilon $
is a primitive third root of unity. In particular,
$\mathrm {Aut}_{k}(E,o)$
is a subgroup of
$\mathrm {Aut}_{k}(E \uparrow \mathbb {P}^{2})=\mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
.
Remark 4.2 When
$j(E)=0, 12^3$
, we may fix
$\lambda =0,1+\sqrt {3}$
, respectively, as in Lemma 4.1, because if two elliptic curves E and
$E'$
in
$\mathbb {P}^2$
are projectively equivalent, then for every
$\mathcal {A}(E,\sigma )$
, there exists an automorphism
$\sigma ' \in \mathrm {Aut}_{k}\,E'$
such that
$\mathcal {A}(E,\sigma ) \cong \mathcal {A}(E',\sigma ')$
(see [Reference Mori and Ueyama12, Lemma 2.6]).
It follows from [Reference Itaba and Matsuno7, Proposition 4.5] that every automorphism
$\sigma \in \mathrm {Aut}_{k}\,E$
can be written as
$\sigma =\sigma _{p}\tau _E^i$
where
$\sigma _{p}$
is a translation by a point
$p \in E$
,
$\tau _E$
is a generator of
$\mathrm {Aut}_{k}(E,o)$
, and
$i \in \mathbb {Z}_{|\tau _E|}$
. For any
$n \ge 1$
, we call a point
$p \in E$
n-torsion if
$np=o$
. We set
$E[n]:=\{ p \in E \mid np=o \}$
and
$T[n]:=\{ \sigma _{p} \in \mathrm {Aut}_{k}\,E \mid p \in E[n] \}$
.
If
$A=\mathcal {A}(E,\sigma )$
is a three-dimensional standard AS-regular algebra, we write
$$ \begin{align*}\mathrm{Aut}_{k}(\mathbb{P}^{2} \downarrow E)=Z(E) \rtimes G(E)\end{align*} $$
as in Table 2 where
$Z(E) \leq \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
with
$Z(E) \subset Z(E,\sigma )$
and
$G(E) \leq \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
so that
$$ \begin{align*}N(E,\sigma)=Z(E) \rtimes (G(E) \cap N(E,\sigma)).\end{align*} $$
The list of Z(E) and G(E).
Table 2 can be checked by the following three steps:
-
• Step
$1$
: Calculate
$\mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
. -
• Step
$2$
: Find
$Z(E) \leq \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
with
$Z(E) \subset Z(E,\sigma ) \cong (\mathrm {PGrAut}_{k}\,A)^{\mathrm {op}}$
(see Lemma 3.4(2)). -
• Step
$3$
: Find
$G(E) \leq \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
.
$\mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
and
$\mathrm {PGrAut}_{k}\,A$
were computed in [Reference Yamamoto13]. We explain these steps for Type S. By Lemma 2.10,
$E=\mathcal {V}(x) \cup \mathcal {V}(y) \cup \mathcal {V}(z)$
and
$$ \begin{align*}\begin{cases} \sigma(0,b,c)=(0,b,\alpha c), \\ \sigma(a,0,c)=(\alpha a,0,c), \\ \sigma(a,b,0)=(a,\alpha b,0), \end{cases}\end{align*} $$
where
$\alpha ^3 \neq 0,1$
. By [Reference Yamamoto13, Lemma 3.2.1],
$$ \begin{align*}\mathrm{Aut}_{k}(\mathbb{P}^{2} \downarrow E)=\left\{ \begin{pmatrix} 1 & 0 & 0 \\ 0 & e & 0 \\ 0 & 0 & i \end{pmatrix} \middle| ei \neq 0 \right\} \rtimes \left\langle \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \right\rangle.\end{align*} $$
By [Reference Yamamoto13, Theorem 3.3.1],
$$ \begin{align*} Z(E,\sigma)&=(\mathrm{PGrAut}_{k}\,A)^{\mathrm{op}} \\ &= \begin{cases} \left\{ \begin{pmatrix} 1 & 0 & 0 \\ 0 & e & 0 \\ 0 & 0 & i \end{pmatrix} \middle| ei \neq 0 \right\} \rtimes \left\langle \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \right\rangle, &\text{if } \sigma^2 \neq \mathrm{id}, \\ \mathrm{Aut}_{k}(\mathbb{P}^{2} \downarrow E), &\text{if } \sigma^2=\mathrm{id}, \end{cases} \end{align*} $$
so we may take
$$ \begin{align*}Z(E)=\left \{ \begin {pmatrix} 1 & 0 & 0 \\ 0 & e & 0 \\ 0 & 0 & i \end {pmatrix} \middle | ei \neq 0 \right \} \rtimes \left \langle \begin {pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end {pmatrix} \right \rangle\ \mathrm{and}\ G(E)=\left \langle \begin {pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end {pmatrix} \right \rangle. \end{align*} $$
Remark 4.3 By Table 2:
-
(1)
$|G(E)|<\infty $
if and only if A is of Types P, S, S’, NC, and EC, and, in this case, there exists
$\tau _E \in \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
such that
$G(E)=\langle \tau _E \rangle $
is a finite cyclic group. -
(2)
$|G(E)|<\infty $
but
$|G(E)| \neq 2$
if and only if A is of Type P (
$|G(E)|=1$
), or Type EC with
$j(E)=0$
(
$|G(E)=6|$
), or Type EC with
$j(E)=12^3$
(
$|G(E)|=4$
).
Theorem 4.4 If
$A=\mathcal {A}(E,\sigma )$
is a three-dimensional quadratic AS-regular algebra of Types T, T’, and CC (so that
$|\sigma |=\infty $
(cf. [Reference Itaba and Mori9])), then
$Z(E,\sigma )=M(E,\sigma )=N(E,\sigma )$
.
Proof Writing
$\mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)=Z(E) \rtimes G(E)$
as in Table 2, it is enough to show that
$G(E) \cap N(E,\sigma )=\{ \mathrm {id} \}$
.
Type T: For every
$\tau =\begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & i \end {pmatrix} \in G(E)$
,
$$ \begin{align*}\begin{cases} \sigma\tau|_E\sigma^{-1}(a,-a,c)=(a,-a,(1-i)a+ic), \\ \sigma\tau|_E\sigma^{-1}(a,-\varepsilon a,c)=(a,-\varepsilon a,\varepsilon^2(1-i)a+ic), \\ \sigma\tau|_E\sigma^{-1}(a,-\varepsilon^2 a,c)=(a,-\varepsilon^2a,\varepsilon(1-i)a+ic). \end{cases}\end{align*} $$
If
$\tau \in N(E,\sigma )$
, then there exists
$\overline {\tau } \in \mathrm {Aut}_{k}\,\mathbb {P}^2$
such that
$\sigma \tau |_E\sigma ^{-1}=\overline {\tau }|_E$
. Then
$1-i=\varepsilon ^2(1-i)=\varepsilon (1-i)$
, so
$i=1$
.
Type T
$'$
: For every
$\tau = \begin {pmatrix} 1 & 0 & 0 \\ 0 & e & 0 \\ 0 & 0 & e^2 \end {pmatrix} \in G(E)$
,
$$ \begin{align*}\begin{cases} \sigma\tau|_E\sigma^{-1}(0,b,c)=(0,eb,e(1-e)b+e^2c), \\ \sigma\tau|_E\sigma^{-1}(a,b,c)=(a,(e-1)a+eb,(e-1)^2a+2e(e-1)b+e^2c). \end{cases}\end{align*} $$
If
$\tau \in N(E,\sigma )$
, then there exists
$\overline {\tau } \in \mathrm {Aut}_{k}\,\mathbb {P}^2$
such that
$\sigma \tau |_E\sigma ^{-1}=\overline {\tau }|_E$
. Then
$e(1-e)=2e(e-1)$
, so
$e=1$
.
Type CC: For every
$\tau = \begin {pmatrix} 1 & 0 & 0 \\ 0 & e & 0 \\ 0 & 0 & e^2 \end {pmatrix} \in G(E)=\mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
,
$$ \begin{align*} &\sigma\tau|_E\sigma^{-1}(a,b,c) \\ &= \left( a+(1-e)b,eb,-3\frac{(a+b)^2}{eb}+3(a+b)-eb+e^{-2}\left( 3\frac{a^2}{b}+3a+b+c \right) \right). \end{align*} $$
If
$\tau \in N(E,\sigma )$
, then there exists
$\overline {\tau } \in \mathrm {Aut}_{k}\,\mathbb {P}^2$
such that
$\sigma \tau |_E\sigma ^{-1}=\overline {\tau }|_E$
. Then there exists
$0 \neq e' \in k$
such that
$$ \begin{align*}(1+(1-e)b,eb)=(1,e'b) \text{ in } \mathbb{P}^1\end{align*} $$
for every
$(1,b,c) \in E$
, so
$e'=e=1$
.
Lemma 4.5 Let
$A=\mathcal {A}(E,\sigma )$
be a three-dimensional standard AS-regular algebra of Type S, S’, NC, or EC. For every
$i \ge 1$
,
$\sigma ^{i} \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}^{2})=\mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
if and only if
$\sigma ^{3i}=\mathrm {id}$
.
Proof This lemma follows from [Reference Itaba and Mori9, Theorem 3.4].
Lemma 4.6 Let
$A=\mathcal {A}(E,\sigma )$
be a three-dimensional standard AS-regular algebra. If
$\sigma \tau \sigma ^{-1}, \sigma ^{-1}\tau \sigma \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}^{2})$
for every
$\tau \in G(E)$
, then
$M(E,\sigma )=N(E,\sigma )=\mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
.
Proof Every
$\tau \in \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
can be written as
$\tau =\tau _1\tau _2$
where
$\tau _1 \in Z(E)$
,
$\tau _2 \in G(E)$
. Since
$\sigma \tau \sigma ^{-1}=\sigma \tau _1\tau _2\sigma ^{-1}=\tau _1\sigma \tau _2\sigma ^{-1}$
, it holds that
$\sigma \tau \sigma ^{-1} \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}^{2})$
. Similarly, every
$\tau \in \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
can be written as
$\tau =\tau _2\tau _1$
where
$\tau _1 \in Z(E)$
,
$\tau _2 \in G(E)$
, so
$\sigma ^{-1}\tau \sigma =\sigma ^{-1}\tau _2\tau _1\sigma = \sigma ^{-1}\tau _2\sigma \tau _{1}$
, and hence
$\sigma ^{-1}\tau \sigma \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}^{2})$
. The result now follows from Lemma 3.2.
Theorem 4.7 Let
$A=\mathcal {A}(E,\sigma )$
be a three-dimensional standard AS-regular algebra such that
$|G(E)|=2$
.
-
(1) If
$\sigma ^2=\mathrm {id}$
, then
$Z(E,\sigma )=M(E,\sigma )=N(E,\sigma )=\mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
. -
(2) If
$\sigma ^{6}=\mathrm {id}$
, then
$M(E,\sigma )=N(E,\sigma )=\mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
. -
(3) If
$\sigma ^{6} \neq \mathrm {id}$
, then
$Z(E)=Z(E,\sigma )=M(E,\sigma )=N(E,\sigma )$
.
Proof (1) We will give a proof for Type S’. The other types are proved similarly. By Lemma 2.10 and Table 2,
$E=\mathcal {V}(x) \cup \mathcal {V}(x^2-\lambda yz)$
,
$$ \begin{align*}\begin{cases} \sigma(0,b,c)=(0,b,\alpha c), \\ \sigma(a,b,c)=(a,\alpha b,\alpha^{-1}c), \end{cases}\end{align*} $$
where
$\lambda =\frac {\alpha ^3-1}{\alpha }$
and
$\alpha ^3 \neq 0,1$
, and
$\tau _E= \begin {pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end {pmatrix}$
. In general,
$Z(E) \subset Z(E,\sigma ) \subset M(E,\sigma ) \subset N(E,\sigma ) \subset \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
. In this case,
$\sigma ^2=\mathrm {id}$
if and only if
$\alpha ^{2}=1$
. Since
$$ \begin{align*}\begin{cases} \sigma\tau_E\sigma^{-1}(0,b,c)=(0,c,\alpha^2 b), \\ \sigma\tau_E\sigma^{-1}(a,b,c)=(a,\alpha^{2}c,\alpha^{-2}b), \end{cases}\end{align*} $$
if
$\sigma ^{2}=\mathrm {id}$
, then
$\sigma \tau _E\sigma ^{-1}=\tau _E$
. Since
$\tau _E \in Z(E,\sigma )$
,
$Z(E,\sigma )=\mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
.
(2) By direct calculations,
$(\tau _E\sigma )^2=\mathrm {id}$
, so
$\sigma \tau _E\sigma ^{-1}\tau _E^{-1}=\sigma ^2=\tau _E^{-1}\sigma ^{-1}\tau _E\sigma $
. By Lemma 4.5,
$\sigma \tau _E\sigma ^{-1}\tau _E^{-1}, \tau _E^{-1}\sigma ^{-1}\tau _E\sigma \in \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
if and only if
$\sigma ^6=\mathrm {id}$
. In particular, if
$\sigma ^6=\mathrm {id}$
, then
$\sigma \tau _E\sigma ^{-1}, \sigma ^{-1}\tau _E\sigma \in \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
. By Lemma 4.6,
$M(E,\sigma )=N(E,\sigma )=\mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
, and hence (2) holds.
(3) If
$\sigma ^6 \neq \mathrm {id}$
, then
$\tau _E \notin N(E,\sigma )$
. Since
$G(E) \cap N(E,\sigma )=\{ \mathrm {id} \}$
,
$N(E,\sigma )=Z(E)$
, and hence (3) holds.
Theorem 4.8 Let
$A=\mathcal {A}(E,\sigma )$
be a three-dimensional standard AS-regular algebra except for Type EC. Then Table 3 gives
$Z(E,\sigma )$
and
$M(E,\sigma )$
for each type.
Proof By Theorems 4.4 and 4.7, the result holds.
$Z(E,\sigma)$
and
$M(E,\sigma)$
except for Type EC.
Definition 4.9 Let
$E=\mathcal {V}(x^3+y^3+z^3) \subset \mathbb {P}^2$
so that
$j(E)=0$
, and define
$$ \begin{align*}\mathcal{E}:=\{ (a,b,c) \in E \mid a^9=b^9=c^9 \} \subset E[9] \setminus E[6].\end{align*} $$
In this paper, we say that a three-dimensional quadratic AS-regular algebra is exceptional if it is graded Morita equivalent to
$\mathcal {A}(E,\sigma _{p})$
for some
$p \in \mathcal {E}$
.
Lemma 4.10 Let
$A=\mathcal {A}(E,\sigma _{p})$
be a three-dimensional standard AS-regular algebra of Type EC and
$\sigma _{q}\tau _{E}^{i} \in \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
where
$q \in E[3]$
,
$i \in \mathbb {Z}_{|\tau _E|}$
. Then:
-
(1)
$\sigma _{q}\tau _{E}^i \in Z(E,\sigma _{p})$
if and only if
$p-\tau _{E}^i(p)=o$
, -
(2)
$\sigma _{q}\tau _{E}^i \in N(E,\sigma _{p})$
if and only if
$p-\tau _{E}^i(p) \in E[3]$
, and -
(3)
$M(E,\sigma _{p})=N(E,\sigma _{p})$
.
Proof (1) Since
$\sigma _{p}(\sigma _{q}\tau _{E}^i)\sigma _{p}^{-1}=\sigma _{q+p-\tau _{E}^{i}(p)}\tau _{E}^i$
,
$\sigma _{q}\tau _{E}^{i} \in Z(E,\sigma _{p})$
if and only if
$p-\tau _{E}^{i}(p)=o$
.
(2) Since
$\sigma _{p}(\sigma _{q}\tau _{E}^i)\sigma _{p}^{-1}=\sigma _{q+p-\tau _{E}^{i}(p)}\tau _{E}^i$
, by [Reference Mori11, Lemma 5.3],
$\sigma _{q}\tau _{E}^{i} \in N(E,\sigma _{p})$
if and only if
$p-\tau _{E}^{i}(p) \in E[3]$
.
(3) In general,
$M(E,\sigma _{p}) \subset N(E,\sigma _{p})$
, so it is enough to show that
$N(E,\sigma _{p}) \subset M(E,\sigma _{p})$
. Let
$\sigma _{q}\tau _E^{i} \in N(E,\sigma _{p}) \subset \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
where
$q \in E[3]$
and
$i \in \mathbb {Z}_{|\tau _E|}$
. Since
$\sigma _{p}(\sigma _{q}\tau _E^{i})\sigma _{p}^{-1}=\sigma _{q+p-\tau _E^{i}(p)}\tau _E^{i} \in \mathrm {Aut}_{k}(\mathbb {P}^{2} \downarrow E)$
,
$p-\tau _E^{i}(p) \in E[3]$
. For any
$j \ge 1$
, we can write
$$ \begin{align*}(\sigma_{q}\tau_E^{i}\sigma_{p})^{j}(\sigma_{p})^{-j}=\sigma_{r_{j}}\tau_E^{ji},\end{align*} $$
where
$r_{j}=\sum _{l=0}^{j-1}\tau _E^{li}(q)+\sum _{l=1}^{j}\tau _E^{li}(p-\tau _E^{(j-l)i}(p))$
, and
$$ \begin{align*}(\sigma_{q}\tau_E^{i}\sigma_{p})^{-j}(\sigma_{p})^{j}=\sigma_{s_{j}}\tau_E^{-ji},\end{align*} $$
where
$s_{j}=\sum _{l=1}^{j}(-\tau _E^{-li}(q))+\sum _{l=0}^{j-1}\tau _E^{-li}(p-\tau _E^{(l-j)i}(p))$
. By [Reference Itaba and Matsuno7, Lemma 4.19], for any
$j \ge 1$
,
$r_{j}, s_{j} \in E[3]$
, so
$$ \begin{align*}(\sigma_{q}\tau_E^{i}\sigma_{p})^{j}(\sigma_{p})^{-j}, (\sigma_{q}\tau_E^{i}\sigma_{p})^{-j}(\sigma_{p})^{j} \in \mathrm{Aut}_{k}(\mathbb{P}^{2} \downarrow E),\end{align*} $$
and hence (3) holds.
Theorem 4.11 Let
$A=\mathcal {A}(E,\sigma _{p})$
be a three-dimensional standard AS-regular algebra of Type EC. Then Table 4 gives
$Z(E,\sigma _{p})$
and
$M(E,\sigma _{p})$
.
$Z(E,\sigma)$
and
$M(E,\sigma)$
for Type EC.
Proof By Lemma 4.10(3), it is enough to calculate
$Z(E,\sigma _{p})$
and
$N(E,\sigma _{p})$
. By Lemma 3.4(2),
$Z(E,\sigma _{p})$
was given in [Reference Matsuno10, Proposition 4.7]. The set of points satisfying
$p-\tau _{E}^i(p) \in E[3]$
was given in [Reference Matsuno10, Theorem 3.8]. By Lemma 4.10(1) and (2), the result follows.
Corollary 4.12 shows that in most cases a twisting system can be replaced by an automorphism to compute a twisted algebra.
Corollary 4.12 Let
$A=\mathcal {A}(E,\sigma )$
be a three-dimensional nonexceptional standard AS-regular algebra. If
$\sigma ^6 \neq \mathrm {id}$
or
$\sigma ^2=\mathrm {id}$
, then
$Z(E,\sigma )=M(E,\sigma )$
, so
$\mathrm {Twist}_{\mathrm {alg}}(A)=\mathrm {Twist}(A)$
.
Proof By Theorems 4.4 and 4.7, it is enough to show the case that
$A=\mathcal {A}(E,\sigma _{p})$
is of Type EC such that
$j(E)=0$
,
$p \notin \mathcal {E}$
, or
$j(E)=12^3$
.
(1)
$j(E)=0$
,
$p \notin \mathcal {E}$
: Let
$E=\mathcal {V}(x^3+y^3+z^3) \subset \mathbb {P}^2$
. By Theorem 4.11, if
$p \in E[2]$
or
$p \notin E[6]$
, then
$N(E,\sigma _{p})=M(E,\sigma _{p})=Z(E,\sigma _{p})$
.
(2)
$j(E)=12^3$
: Let
$E=\mathcal {V}(x^3+y^3+z^3-3\lambda xyz) \subset \mathbb {P}^2$
where
$\lambda =1+\sqrt {3}$
. By Theorem 4.11, if
$p \in E[2]$
or
$p \notin E[6]$
, then
$N(E,\sigma _{p})=M(E,\sigma _{p})=Z(E,\sigma _{p})$
.
Let
$E \subset \mathbb {P}^2$
be a projective variety. For
$\tau \in \mathrm {Aut}_{k}\,E$
, we define
$$ \begin{align*}||\tau||:=\inf\{ i \in \mathbb{N}^{+} \mid \tau^i \in \mathrm{Aut}_{k}(E \uparrow \mathbb{P}^{2}) \}.\end{align*} $$
Corollary 4.13 For every three-dimensional nonexceptional geometric AS-regular algebra B, there exists a three-dimensional standard AS-regular algebra S such that
$\mathrm {Twist}(B)=\mathrm {Twist}_{\mathrm {alg}}(S)$
.
Proof By Lemma 2.10, there exists a three-dimensional nonexceptional standard AS-regular algebra
$A=\mathcal {A}(E,\sigma )$
such that
$\mathrm {GrMod}\,B \cong \mathrm {GrMod}\,A$
. If
$\sigma ^6 \neq \mathrm {id}$
, then
$\mathrm {Twist}(B)=\mathrm {Twist}(A)=\mathrm {Twist}_{\mathrm {alg}}(A)$
by Corollary 4.12, so we assume that
$\sigma ^6=\mathrm {id}$
. Set
$\tau :=\sigma ^2 \in \mathrm {Aut}_{k}\,E$
and
$S:=\mathcal {A}(E,\sigma ^3)$
. Since
$||\tau ||=||\sigma ^2||=|\sigma ^6|=1$
by [Reference Itaba and Mori9, Theorem 3.4],
$\tau \in \mathrm {Aut}_{k}(E \uparrow \mathbb {P}^{2})$
. Since
$\tau ^{i+1}\sigma =\sigma ^{2i+3}=\sigma ^3\tau ^i$
for every
$i \in \mathbb {Z}$
,
$\mathrm {GrMod}\,A \cong \mathrm {GrMod}\,S$
by Theorem 2.6. Since
$(\sigma ^{3})^2=\mathrm {id}$
,
$\mathrm {Twist}(B)=\mathrm {Twist}(A)=\mathrm {Twist}(S)=\mathrm {Twist}_{\mathrm {alg}}(S)$
by Corollary 4.12.
Example 4.14 shows that even if
$B \cong S^{\theta }$
for some three-dimensional quadratic Calabi–Yau AS-regular algebra S, there may be no
$\phi \in \mathrm {GrAut}_{k}\,S$
such that
$B \cong S^{\phi }$
. We need to carefully choose S in order that
$B \cong S^{\phi }$
for some
$\phi \in \mathrm {GrAut}_{k}\,S$
.
Example 4.14 Let
$E \subset \mathbb {P}^2$
be an elliptic curve. Assume that
$j(E) \neq 0,12^3$
. We set three geometric algebras of Type EC;
$B:=\mathcal {A}(E,\tau _{E}\sigma _{p})$
,
$A:=\mathcal {A}(E,\sigma _{p})$
, and
$S:=\mathcal {A}(E,\sigma _{3p})$
, where
$p \in E[6] \setminus (E[2] \cup E[3])$
. By [Reference Itaba and Matsuno8, Theorem 4.3], these algebras are three-dimensional quadratic AS-regular algebras. Moreover, A and S are standard. By [Reference Itaba and Matsuno7, Theorem 4.20],
$\mathrm {GrMod}\,B \cong \mathrm {GrMod}\,A \cong \mathrm {GrMod}\,S$
. Since
$|\sigma _{p}|=6$
and
$|\sigma _{3p}|=2$
,
$M(E,\sigma _{3p})=Z(E,\sigma _{3p}) \neq Z(E,\sigma _{p})$
by Table 4, so
$\mathrm {Twist}(B)=\mathrm {Twist}_{\mathrm {alg}}(S) \neq \mathrm {Twist}_{\mathrm {alg}}(A)$
.
Acknowledgment
The author is grateful to Professor Izuru Mori for his support and helpful discussions. The author would like to thank the referee for careful reading. The author also appreciates Yu Saito for careful reading.
