We describe vector bundles over a class of noncommutative curves, namely, over noncommutative nodal curves of string type and of almost string type. We also prove that, in other cases, the classification of vector bundles over a noncommutative curve is a wild problem.
Similar content being viewed by others
References
M. Atiyah, “Vector bundles over an elliptic curve,” Proc. London Math. Soc., 7, 414–452 (1957).
G. Birkhoff, “A theorem on matrices of analytic functions,” Math. Ann., 74, 122–133 (1913).
L. Bodnarchuk and Y. Drozd, “Stable vector bundles over cuspidal cubics,” Cent. Eur. J. Math., 1, 650–660 (2003).
V. M. Bondarenko, “Representations of bundles of semichain sets and their applications,” Algebra Anal., 3, No. 5, 38–61 (1991).
I. Burban and Y. Drozd, “Derived categories of nodal algebras,” J. Algebra, 272, 46–94 (2004).
I. Burban and Y. Drozd, “Coherent sheaves on rational curves with simple double points and transversal intersections,” Duke Math. J., 121, 189–229 (2004).
C. W. Curtis and I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Orders, Vol. 1, Wiley (1981).
Y. Drozd, “Reduction algorithm and representations of boxes and algebras,” C. R. Math. Acad. Sci. Canada, 23, 97–125 (2001).
Y. Drozd and G.-M. Greuel, “Tame and wild projective curves and classification of vector bundles,” J. Algebra, 246, 1–54 (2001).
W. Geigle and H. Lenzing, “A class of weighted projective curves arising in representation theory of finite-dimensional algebras,” Singularities, Representations, and Vector Bundles. Lect. Notes Math., 1273 (1987).
A. Grothendieck, “Sur la classification des fibrés holomorphes sur la sphère de Riemann,” Am. J. Math., 79, 121–138 (1956).
J. Milnor, Introduction to Algebraic K-Theory, Princeton Univ. Press, Princeton (1971).
J.-P. Serre, Cohomologie Galoisienne, Springer, Berlin (1963).
D. E. Voloshyn, “Structure of nodal algebras,” Ukr. Math. J., 63, No. 7, 880–888 (2011).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 2, pp. 185–199, February, 2012.
Rights and permissions
About this article
Cite this article
Drozd, Y.A., Voloshyn, D.E. Vector bundles over noncommutative nodal curves. Ukr Math J 64, 208–224 (2012). https://doi.org/10.1007/s11253-012-0639-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-012-0639-8