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Modules over regular algebras of dimension 3. (English) Zbl 0763.14001
A 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.
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
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