Modules over regular algebras of dimension 3.

*(English)*Zbl 0763.14001A finitely generated graded algebra \(A=k+A_ 1+A_ 2+\cdots\), where \(k\) is a field, is said to be regular if \(A\) has finite global dimension and polynomial growth and is Gorenstein. In an earlier work [in The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 33-85 (1990; Zbl 0744.14024)] the authors had classified regular algebras of global dimension 3 which are standard (i.e. generated in degree 1) by establishing a correspondence between such algebras and “regular” triples \((E,\sigma,L)\), with \(\sigma\) an automorphism of the scheme \(E\), of one of the following four types:

(1a) \(E\) is a cubic divisor in \(\mathbb{P}^ 2\) and \(L={\mathcal O}_ E(1)\),

(1b) \(E\) is a divisor of bidegree (2,2) in \(\mathbb{P}^ 1\times\mathbb{P}^ 1\) and \(L=pr^*_ 1({\mathcal O}_{\mathbb{P}^ 1}(1))\),

(2a) \(E=\mathbb{P}^ 2\) and \(L\cong{\mathcal O}_ E(1)\).

(2b) \(E=\mathbb{P}^ 1\times\mathbb{P}^ 1\) and \(L\cong pr^*_ 1({\mathcal O}_{\mathbb{P}^ 1}(1))\).

The regularity of the triple means that \(L^{(\sigma-1)(\sigma^ j- 1)}\cong{\mathcal O}_ E\) with \(j=1\) in case (a) and \(j=2\) in case (b).

One of the main results proved in the present paper is theorem II: The regular algebra \(A\) is a finite module over its centre if and only if the automorphism \(\sigma\) is of finite order.

Given the triple \((E,\sigma,L)\) the corresponding regular algebra \(A\) is obtained via an intermediary graded algebra, namely \(B=\sum_{n\geq 0}H^ 0(E,L\otimes L^ \sigma\otimes\cdots\otimes L^{\sigma^{n- 1}})\). — The algebras \(A\) and \(B\) are related by \(B\cong A/gA\) with \(g\) homogeneous. Since \(B\) has an explicit description, theorem II is easy to prove for \(B\). In order to deduce the corresponding result for \(A\), the authors introduce the \(\mathbb{Z}\)-graded ring \(\Lambda=A[g^{-1}]\). In analogy with the commutative case, one can think of the non-commutative affine scheme \(\text{Spec} \Lambda_ 0\) as the “open complement” of the non-commutative \(\text{Proj}(B)\) in \(\text{Proj}(A)\). The structures of \(A\) and \(\Lambda_ 0\) are closely related. On the structure of \(\Lambda_ 0\) the authors have the following result (which is used in proving theorem II):

Theorem I. Let \(s\) be the order of the \(\sigma\)-orbit of the class of \(L\) in \(\text{Pic}(E)\). If \(s<\infty\) then \(\Lambda_ 0\) is an Azumaya algebra of rank \(s^ 2\) over its centre, while if \(s=\infty\) then \(\Lambda_ 0\) is a simple ring.

A point module over \(A\) is a graded right \(A\)-module \(N\) such that \(N_ 0=k\), \(N_ 0\) generates \(N\) and \(\dim_ kN_ i=1\) for all \(i\geq 0\). It is shown that under the correspondence \(A\leftrightarrow(E,\sigma,L)\) the points of \(E\) parametrize the point modules over \(A\) and that the structure of these modules is related nicely to the geometry of \((E,\sigma,L)\). It is the study of this relationship that yields a proof of theorems I and II. The authors also describe a process of twisting a graded algebra by an automorphism to obtain a new algebra of the same dimension, and they use this to determine those regular algebras which correspond to non-reduced divisors \(E\) by showing that they are twists of a few special types. The following theorem is also proved: A regular noetherian algebra of global dimension at most 4 is a domain.

(1a) \(E\) is a cubic divisor in \(\mathbb{P}^ 2\) and \(L={\mathcal O}_ E(1)\),

(1b) \(E\) is a divisor of bidegree (2,2) in \(\mathbb{P}^ 1\times\mathbb{P}^ 1\) and \(L=pr^*_ 1({\mathcal O}_{\mathbb{P}^ 1}(1))\),

(2a) \(E=\mathbb{P}^ 2\) and \(L\cong{\mathcal O}_ E(1)\).

(2b) \(E=\mathbb{P}^ 1\times\mathbb{P}^ 1\) and \(L\cong pr^*_ 1({\mathcal O}_{\mathbb{P}^ 1}(1))\).

The regularity of the triple means that \(L^{(\sigma-1)(\sigma^ j- 1)}\cong{\mathcal O}_ E\) with \(j=1\) in case (a) and \(j=2\) in case (b).

One of the main results proved in the present paper is theorem II: The regular algebra \(A\) is a finite module over its centre if and only if the automorphism \(\sigma\) is of finite order.

Given the triple \((E,\sigma,L)\) the corresponding regular algebra \(A\) is obtained via an intermediary graded algebra, namely \(B=\sum_{n\geq 0}H^ 0(E,L\otimes L^ \sigma\otimes\cdots\otimes L^{\sigma^{n- 1}})\). — The algebras \(A\) and \(B\) are related by \(B\cong A/gA\) with \(g\) homogeneous. Since \(B\) has an explicit description, theorem II is easy to prove for \(B\). In order to deduce the corresponding result for \(A\), the authors introduce the \(\mathbb{Z}\)-graded ring \(\Lambda=A[g^{-1}]\). In analogy with the commutative case, one can think of the non-commutative affine scheme \(\text{Spec} \Lambda_ 0\) as the “open complement” of the non-commutative \(\text{Proj}(B)\) in \(\text{Proj}(A)\). The structures of \(A\) and \(\Lambda_ 0\) are closely related. On the structure of \(\Lambda_ 0\) the authors have the following result (which is used in proving theorem II):

Theorem I. Let \(s\) be the order of the \(\sigma\)-orbit of the class of \(L\) in \(\text{Pic}(E)\). If \(s<\infty\) then \(\Lambda_ 0\) is an Azumaya algebra of rank \(s^ 2\) over its centre, while if \(s=\infty\) then \(\Lambda_ 0\) is a simple ring.

A point module over \(A\) is a graded right \(A\)-module \(N\) such that \(N_ 0=k\), \(N_ 0\) generates \(N\) and \(\dim_ kN_ i=1\) for all \(i\geq 0\). It is shown that under the correspondence \(A\leftrightarrow(E,\sigma,L)\) the points of \(E\) parametrize the point modules over \(A\) and that the structure of these modules is related nicely to the geometry of \((E,\sigma,L)\). It is the study of this relationship that yields a proof of theorems I and II. The authors also describe a process of twisting a graded algebra by an automorphism to obtain a new algebra of the same dimension, and they use this to determine those regular algebras which correspond to non-reduced divisors \(E\) by showing that they are twists of a few special types. The following theorem is also proved: A regular noetherian algebra of global dimension at most 4 is a domain.

Reviewer: B.Singh (Bombay)

##### MSC:

14A22 | Noncommutative algebraic geometry |

11R54 | Other algebras and orders, and their zeta and \(L\)-functions |

16W50 | Graded rings and modules (associative rings and algebras) |

14J30 | \(3\)-folds |

16E10 | Homological dimension in associative algebras |

##### Keywords:

regular algebras of global dimension 3; non-commutative affine scheme; point modules; twisting a graded algebra
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\textit{M. Artin} et al., Invent. Math. 106, No. 2, 335--388 (1991; Zbl 0763.14001)

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