Subspaces of non-commutative spaces

Smith, S.

doi:10.1090/S0002-9947-02-02963-X

Subspaces of non-commutative spaces
HTML articles powered by AMS MathViewer

by S. Paul Smith

Trans. Amer. Math. Soc. 354 (2002), 2131-2171

DOI: https://doi.org/10.1090/S0002-9947-02-02963-X

Published electronically: February 14, 2002

PDF | Request permission

Abstract:

This paper concerns the closed points, closed subspaces, open subspaces, weakly closed and weakly open subspaces, and effective divisors, on a non-commutative space.

References

M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 33–85. MR 1086882
M. Artin, J. Tate, and M. Van den Bergh, Modules over regular algebras of dimension $3$, Invent. Math. 106 (1991), no. 2, 335–388. MR 1128218, DOI 10.1007/BF01243916
M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228–287. MR 1304753, DOI 10.1006/aima.1994.1087
Maria Jesus Asensio, Michel Van den Bergh, and Freddy Van Oystaeyen, A new algebraic approach to microlocalization of filtered rings, Trans. Amer. Math. Soc. 316 (1989), no. 2, 537–553. MR 958890, DOI 10.1090/S0002-9947-1989-0958890-X
Gérard Cauchon, Les $T$-anneaux, la condition $(\textrm {H})$ de Gabriel et ses consequences, Comm. Algebra 4 (1976), no. 1, 11–50 (French). MR 393112, DOI 10.1080/00927877608822093
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821
K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, vol. 16, Cambridge University Press, Cambridge, 1989. MR 1020298
Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Peter Jørgensen, Intersection theory on non-commutative surfaces, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5817–5854. MR 1695026, DOI 10.1090/S0002-9947-00-02565-4
P. Jørgensen and S.P. Smith, Curves on non-commutative surfaces, in preparation.
P. Jørgensen and S.P. Smith, Intersection theory for the blow-up of a non-commutative surface, in preparation.
André Joyal and Ross Street, An introduction to Tannaka duality and quantum groups, Category theory (Como, 1990) Lecture Notes in Math., vol. 1488, Springer, Berlin, 1991, pp. 413–492. MR 1173027, DOI 10.1007/BFb0084235
A. Kapustin, A. Kuznetzov, and D. Orlov, Noncommutative Instantons and Twistor Transform, Comm. Math. Phys. 221 (2001), 385–432.
Lieven Le Bruyn and Michel Van den Bergh, On quantum spaces of Lie algebras, Proc. Amer. Math. Soc. 119 (1993), no. 2, 407–414. MR 1149975, DOI 10.1090/S0002-9939-1993-1149975-2
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
C. Pappacena, The injective spectrum of a non-commutative space, manuscript, February 2001, QA 0108175.
N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs, No. 3, Academic Press, London-New York, 1973. MR 0340375
Alexander L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, Mathematics and its Applications, vol. 330, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 1347919, DOI 10.1007/978-94-015-8430-2
Alexander L. Rosenberg, Noncommutative schemes, Compositio Math. 112 (1998), no. 1, 93–125. MR 1622759, DOI 10.1023/A:1000479824211
S. Paul Smith and James J. Zhang, Curves on quasi-schemes, Algebr. Represent. Theory 1 (1998), no. 4, 311–351. MR 1683617, DOI 10.1023/A:1009984608942
S.P. Smith, Integral non-commutative spaces, J. Algebra, to appear, QA 0104046.
S.P. Smith, Maps between non-commutative spaces, Preprint, University of Washington, 2000.
M. Van den Bergh, Blowing up on non-commutative smooth surfaces, Mem. Amer. Math. Soc., 154 (2001), No. 734.
Martine Van Gastel and Michel Van den Bergh, Graded modules of Gelfand-Kirillov dimension one over three-dimensional Artin-Schelter regular algebras, J. Algebra 196 (1997), no. 1, 251–282. MR 1474172, DOI 10.1006/jabr.1997.7072
A. B. Verëvkin, On a noncommutative analogue of the category of coherent sheaves on a projective scheme, Algebra and analysis (Tomsk, 1989) Amer. Math. Soc. Transl. Ser. 2, vol. 151, Amer. Math. Soc., Providence, RI, 1992, pp. 41–53. MR 1191171, DOI 10.1090/trans2/151/02