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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)

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
Full Text: DOI EuDML
[1] [AmSm] Amitsur, S.A., Small, L.W.: Prime ideals in P.I. rings. J. Algebra62, 358-383 (1980) · Zbl 0424.16009
[2] [ArSch] Artin, M., Schelter, W.: Graded algebras of global dimension 3. Adv. Math.66, 171-216 (1987) · Zbl 0633.16001
[3] [ATV] Artin, M., Tate, J., Van den Bergh, M.: Some algebras associated to automorphisms of elliptic curves. The Grothendieck Festschrift, vol. 1, pp. 33-85, Boston Basel Stuttgart: Birkhäuser 1990 · Zbl 0744.14024
[4] [AV] Artin, M., Van den Bergh, M.: Twisted homogeneous coordinate rings. J. Algebra133, 249-271 (1990) · Zbl 0717.14001
[5] [Bj] Björk, J.-E.: The Auslander condition on noetherian rings. Séminaire Dubreil-Malliavin 1987-8. Lect. Notes Math., vol. 1404, pp. 137-173, Berlin Heidelberg New York: Springer 1990
[6] [BLR] Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Berlin Heidelberg New York: Springer 1990
[7] [BS] Borevitch, Z.I., Shafarevitch, I.R.: Number theory. New York: Academic Press 1966
[8] [EG] Evans, E.G., Griffith, P.: Syzygies. Lond. Math. Soc. Lect. Note Ser. vol. 106, Cambridge: Cambridge University Press 1986
[9] [KL] Krause, G.R., Lenagan, T.H.: Growth of Algebras and Gelfand-Kirillov Dimension. Res. Notes Math. vol. 116, Boston: Pitman 1985 · Zbl 0564.16001
[10] [NV] Nastacescu, C., Van Oystaeyen, F.: Graded ring theory, p. 16. Amsterdam: North Holland 1982
[11] [OF] Odeskii, A.B., Feigin, B.L.: Sklyanin algebras associated to elliptic curves. (Manuscript)
[12] [Ra] Ramras, R.: Maximal orders over regular rings of dimension 2. Trans. Am. Math. Soc.142, 457-474 (1969) · Zbl 0186.07101
[13] [Re] Revoy, M.P.: Algèbres de Weyl en charactéristique p. C.R. Acad. Sci. Sér.A276, 225-227 (1973) · Zbl 0265.16007
[14] [Ro] Rowen, L.: Polynomial identities in ring theory. New York London: Academic Press 1980 · Zbl 0461.16001
[15] [Sn] Snider, R.L.: Noncommutative regular local rings of dimension 3. Proc. Am. Math. Soc.104, 49-50 (1988) · Zbl 0669.16013
[16] [SSW] Small, L.W., Stafford, J.T., Warfield, R.B.: Affine algebras of Gelfand-Kirillov dimension one are PI. Math. Proc. Camb. Philos. Soc.97, 407-414 (1985) · Zbl 0561.16005
[17] [Staf] Stafford, J.T.: Noetherian full quotient rings. Proc. Lond. Math. Soc.44, 385-404 (1982) · Zbl 0485.16009
[18] [Stan] Stanley, R.P.: Generating functions. In: Studies in Combinatorics. MAA Stud. Math., vol. 17, pp. 100-141, Washington: MAA, Inc. 1978
[19] [VdB] Van den Bergh, M.: Regular algebras of dimension 3. Séminaire Dubreil-Malliavin 1986. Lect. Notes Math., vol. 1296, pp. 228-234, Berlin Heidelberg New York: Springer 1987
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