Simple holonomic modules over rings of differential operators with regular coefficients of Krull dimension 2

Authors:
V. Bavula and F. van Oystaeyen

Journal:
Trans. Amer. Math. Soc. **353** (2001), 2193-2214

MSC (2000):
Primary 16S32, 32C38, 13N10

DOI:
https://doi.org/10.1090/S0002-9947-01-02701-5

Published electronically:
January 29, 2001

MathSciNet review:
1814067

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be an algebraically closed field of characteristic zero. Let be the ring of (-linear) differential operators with coefficients from a regular commutative affine domain of Krull dimension which is the tensor product of two regular commutative affine domains of Krull dimension . Simple holonomic -modules are described. Let a -algebra be a regular affine commutative domain of Krull dimension and be the ring of differential operators with coefficients from . We classify (up to irreducible elements of a certain Euclidean domain) simple -modules (the field is not necessarily algebraically closed).

**[Bav1]**V. V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz, 4 (1992), no. 1, 75-97; English transl. in St. Petersburg Math. J.**4**(1993), no. 1, 71-92. MR**93h:16043****[Bav2]**V. V. Bavula, Generalized Weyl algebras, kernel and tensor-simple algebras, their simple modules, Representations of algebras. Sixth International Conference, August 19-22, 1992. CMS Conference proceedings (V.Dlab and H.Lenzing, Eds.), v. 14, 83-106. MR**94i:16016****[Bav3]**V. V. Bavula, Simple modules of the Ore extensions with Dedekind coefficients, Comm. in Algebra,**27**(1999), no. 6, 2665-2699. MR**2000i:16050****[Bav4]**V. V. Bavula, Identification of the Hilbert function and Poincaré series, and the dimension of modules over filtered rings, Russian Acad. Sci. Izv. Math.,**44**(1995), no. 2, 225-246. MR**95j:13013****[Bav5]**V. V. Bavula, Each Schurian algebra is tensor simple, Comm. in Algebra**23**(1995), no. 4, 1363-1367. MR**96i:16008****[BeLu]**I. Bernstein and V. Lunts, On non-holonomic irreducible -modules, Inv. Math.,**94**(1988), 223-243. MR**90b:58247****[Bj]**J.-E. Björk, ``*Rings of differential operators*", North Holland, Amsterdam, 1979. MR**82g:32013****[Bl1]**R. E. Block, Classification of the irreducible representations of , Bull. Amer. Math. Soc.,**1**(1979), 247-250. MR**80i:17008****[Bl2]**R. E. Block, The irreducible representations of the Weyl algebra , in ``Séminaire d'Algébre Paul Dubreil (Proceedings, Paris 1977-1978)" (M. P. Malliavin, Ed.), Lecture Notes in Mathematics no. 740, pp. 69-79, Springer-Verlag, Berlin, New York, 1979. MR**81e:17010****[Bl3]**R. E. Block, The irreducible representations of the Lie algebra and of the Weyl algebra, Adv. in Math.**39**(1981), 69-110. MR**83c:17010****[Bor]**A. Borel et al., ``*Algebraic*-*modules*", Perspectives in Mathematics (J. Coates and S. Helgason, Eds.), vol. 2, Acad. Press, 1987. MR**89g:32014****[BVO1]**V. V. Bavula and F. Van Oystaeyen, The simple modules of certain generalized crossed products, J. of Algebra**194**(1997), 521-566. MR**98e:16006****[BVO2]**V. V. Bavula and F. Van Oystaeyen, Simple holonomic modules over the second Weyl algebra , Adv. in Math.,**150**(2000), 80-116. CMP**2000:09****[Co]**S. C. Coutinho, -simple rings and simple -modules, Math. Proc. Camb. Phil. Soc.**125**(1999). 405-415. MR**99j:16013****[Jac]**N. Jacobson, ``*The Theory of Rings*", Amer. Math. Soc., Providence, R.I., 1943. MR**5:31f****[KL]**G. R. Krause and T. H. Lenagan, ``*Growth of algebras and Gelfand-Kirillov dimension*", Revised edition, Graduate Studies in Math.,**22**, Amer. Math. Soc., Providence, R.I. MR**2000j:16035****[LVO]**Li Huishi and F. Van Oystaeyen, ``*Zariskian Filtratons*", Kluwer, 1996. MR**97m:16083****[Lu]**V. Lunts, Algebraic varieties preserved by generic flows, Duke Math. J.,**58**(1989), no. 3, 531-554. MR**91a:32015****[MR]**J. C. McConnell and J. C. Robson, ``*Noncommutative Noetherian rings*", Wiley, 1987. MR**89j:16023****[St]**J. T. Stafford, Non-holonomic modules over Weyl algebras and enveloping algebras, Invent. Math.,**79**(1985), 619-638. MR**86h:17009**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
16S32,
32C38,
13N10

Retrieve articles in all journals with MSC (2000): 16S32, 32C38, 13N10

Additional Information

**V. Bavula**

Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK

Email:
vbavula@sheffield.ac.uk, bavula@uia.ua.ac.be

**F. van Oystaeyen**

Affiliation:
Department of Mathematics and Computer Science, University of Antwerp (U.I.A), Universiteitsplein, 1, B-2610, Wilrijk, Belgium

Email:
francin@uia.ua.ac.be

DOI:
https://doi.org/10.1090/S0002-9947-01-02701-5

Received by editor(s):
September 15, 1998

Received by editor(s) in revised form:
March 23, 2000

Published electronically:
January 29, 2001

Additional Notes:
The first author was supported by a grant of the University of Antwerp as a research fellow at U.I.A

Article copyright:
© Copyright 2001
American Mathematical Society