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

Bavula, V.; van Oystaeyen, F.

doi:10.1090/S0002-9947-01-02701-5

Simple holonomic modules over rings of differential operators with regular coefficients of Krull dimension 2
HTML articles powered by AMS MathViewer

by V. Bavula and F. van Oystaeyen

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

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

Published electronically: January 29, 2001

PDF | Request permission

Abstract:

Let $K$ be an algebraically closed field of characteristic zero. Let $\Lambda$ be the ring of ($K$-linear) differential operators with coefficients from a regular commutative affine domain of Krull dimension $2$ which is the tensor product of two regular commutative affine domains of Krull dimension $1$. Simple holonomic $\Lambda$-modules are described. Let a $K$-algebra $D$ be a regular affine commutative domain of Krull dimension $1$ and $\mathcal {D} (D)$ be the ring of differential operators with coefficients from $D$. We classify (up to irreducible elements of a certain Euclidean domain) simple $\mathcal {D}(D)$-modules (the field $K$ is not necessarily algebraically closed).

References

V. V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz 4 (1992), no. 1, 75–97 (Russian); English transl., St. Petersburg Math. J. 4 (1993), no. 1, 71–92. MR 1171955
Vladimir Bavula, Generalized Weyl algebras, kernel and tensor-simple algebras, their simple modules, Proceedings of the Sixth International Conference on Representations of Algebras (Ottawa, ON, 1992) Carleton-Ottawa Math. Lecture Note Ser., vol. 14, Carleton Univ., Ottawa, ON, 1992, pp. 23. MR 1206933
V. Bavula, The simple modules of the Ore extensions with coefficients from a Dedekind ring, Comm. Algebra 27 (1999), no. 6, 2665–2699. MR 1687317, DOI 10.1080/00927879908826587
V. V. Bavula, Identification of the Hilbert function and Poincaré series, and the dimension of modules over filtered rings, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 2, 19–39 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 44 (1995), no. 2, 225–246. MR 1275900, DOI 10.1070/IM1995v044n02ABEH001595
Vladimir Bavula, Each Schurian algebra is tensor-simple, Comm. Algebra 23 (1995), no. 4, 1363–1367. MR 1317405, DOI 10.1080/00927879508825284
Joseph Bernstein and Valery Lunts, On nonholonomic irreducible $D$-modules, Invent. Math. 94 (1988), no. 2, 223–243. MR 958832, DOI 10.1007/BF01394325
J.-E. Björk, Rings of differential operators, North-Holland Mathematical Library, vol. 21, North-Holland Publishing Co., Amsterdam-New York, 1979. MR 549189
Richard E. Block, Classification of the irreducible representations of ${\mathfrak {s}}{\mathfrak {l}}(2,\,\textbf {C})$, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 1, 247–250. MR 513751, DOI 10.1090/S0273-0979-1979-14573-7
Richard E. Block, The irreducible representations of the Weyl algebra $A_{1}$, Séminaire d’Algèbre Paul Dubreil 31ème année (Paris, 1977–1978) Lecture Notes in Math., vol. 740, Springer, Berlin, 1979, pp. 69–79. MR 563495
Richard E. Block, The irreducible representations of the Lie algebra ${\mathfrak {s}}{\mathfrak {l}}(2)$ and of the Weyl algebra, Adv. in Math. 39 (1981), no. 1, 69–110. MR 605353, DOI 10.1016/0001-8708(81)90058-X
A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic $D$-modules, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Boston, MA, 1987. MR 882000
V. Bavula and F. van Oystaeyen, The simple modules of certain generalized crossed products, J. Algebra 194 (1997), no. 2, 521–566. MR 1467166, DOI 10.1006/jabr.1997.7038
V. V. Bavula and F. Van Oystaeyen, Simple holonomic modules over the second Weyl algebra $A_2$, Adv. in Math., 150 (2000), 80–116.
S. C. Coutinho, $d$-simple rings and simple ${\scr D}$-modules, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 3, 405–415. MR 1656789, DOI 10.1017/S0305004198002965
Albert Eagle, Series for all the roots of a trinomial equation, Amer. Math. Monthly 46 (1939), 422–425. MR 5, DOI 10.2307/2303036
Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
Huishi Li and Freddy van Oystaeyen, Zariskian filtrations, $K$-Monographs in Mathematics, vol. 2, Kluwer Academic Publishers, Dordrecht, 1996. MR 1420862, DOI 10.1007/978-94-015-8759-4
Valery Lunts, Algebraic varieties preserved by generic flows, Duke Math. J. 58 (1989), no. 3, 531–554. MR 1016433, DOI 10.1215/S0012-7094-89-05824-9
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
J. T. Stafford, Nonholonomic modules over Weyl algebras and enveloping algebras, Invent. Math. 79 (1985), no. 3, 619–638. MR 782240, DOI 10.1007/BF01388528