Sheaves of noncommutative algebras and the Beilinson-Bernstein equivalence of categories
Authors: T. J. Hodges and S. P. Smith
Journal: Proc. Amer. Math. Soc. 93 (1985), 379-386
MSC: Primary 17B35; Secondary 14A20, 16A63, 22E46, 57S25
MathSciNet review: 773985
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Abstract: Let be an irreducible algebraic variety defined over a field , let be a sheaf of (noncommutative) noetherian -algebras on containing the sheaf of regular functions and let be the ring of global sections. We show that under quite reasonable abstract hypotheses (concerning the existence of a faithfully flat overring of obtained from the local sections of ) there is an equivalence between the category of -modules and the category of sheaves of -modules which are quasicoherent as -modules. This shows that the equivalence of categories established by Beilinson and Bernstein as the first step in their proof of the KazhdanLusztig conjectures (where is a primitive factor ring of the enveloping algebra of a complex semisimple Lie algebra, and is a sheaf of twisted differential operators on a generalised flag variety) is valid for more fundamental reasons than is apparent from their work.
- [BB] Alexandre Beĭlinson and Joseph Bernstein, Localisation de 𝑔-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
- [BG] J. N. Bernstein and S. I. Gel′fand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), no. 2, 245–285. MR 581584
- [G] Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 0232821
- [H] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- [HS] T. J. Hodges and S. P. Smith, Differential operators on the flag variety and the Conze embedding (preprint).
- [JS] A. Joseph and J. T. Stafford, Modules of 𝔨-finite vectors over semisimple Lie algebras, Proc. London Math. Soc. (3) 49 (1984), no. 2, 361–384. MR 748996, https://doi.org/10.1112/plms/s3-49.2.361
- [St] Bo Stenström, Rings of quotients, Springer-Verlag, New York-Heidelberg, 1975. Die Grundlehren der Mathematischen Wissenschaften, Band 217; An introduction to methods of ring theory. MR 0389953
- A. Beilinson and J. N. Bernstein, Localisation de -modules, C. R. Acad. Sci. Sér. A-B 292 (1981), 15-18. MR 610137 (82k:14015)
- J. N. Bernstein and S. I. Gelfand, Tensor products of finite and infinite dimensional representations of semi-simple Lie algebras, Compositio Math. 41 (1980), 245-285. MR 581584 (82c:17003)
- P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France 90 (1962), 323-448. MR 0232821 (38:1144)
- R. Hartshorne, Algebraic geometry, Graduate Texts in Math., no. 52, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
- T. J. Hodges and S. P. Smith, Differential operators on the flag variety and the Conze embedding (preprint).
- A. Joseph and J. T. Stafford, Modules of -finite vectors over semi-simple Lie algebras, Proc. London Math. Soc. (3) 49 (1984), 361-384. MR 748996 (86a:17004)
- B. Stenstrom, Rings of quotients, Springer-Verlag, New York, 1975. MR 0389953 (52:10782)