In mathematics, the Koszul cohomology groups are groups associated to a projective variety X with a line bundle L. They were introduced by Mark Green (1984, 1984b), and named after Jean-Louis Koszul as they are closely related to the Koszul complex.

Green (1989) surveys early work on Koszul cohomology, Eisenbud (2005) gives an introduction to Koszul cohomology, and Aprodu & Nagel (2010) gives a more advanced survey.

Definitions

If M is a graded module over the symmetric algebra of a vector space V, then the Koszul cohomology of M is the cohomology of the sequence

If L is a line bundle over a projective variety X, then the Koszul cohomology is given by the Koszul cohomology of the graded module , viewed as a module over the symmetric algebra of the vector space .

References

  • Aprodu, Marian; Nagel, Jan (2010), Koszul cohomology and algebraic geometry, University Lecture Series, vol. 52, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4964-4, MR 2573635
  • Eisenbud, David (2005), The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Berlin, New York: Springer-Verlag, doi:10.1007/b137572, ISBN 978-0-387-22215-8, MR 2103875
  • Green, Mark L. (1984), "Koszul cohomology and the geometry of projective varieties", Journal of Differential Geometry, 19 (1): 125–171, ISSN 0022-040X, MR 0739785
  • Green, Mark L. (1984), "Koszul cohomology and the geometry of projective varieties. II", Journal of Differential Geometry, 20 (1): 279–289, ISSN 0022-040X, MR 0772134
  • Green, Mark L. (1989), "Koszul cohomology and geometry", in Cornalba, Maurizio; Gómez-Mont, X.; Verjovsky, A. (eds.), Lectures on Riemann surfaces, Proceedings of the First College on Riemann Surfaces held in Trieste, November 9–December 18, 1987, World Sci. Publ., Teaneck, NJ, pp. 177–200, ISBN 9789971509026, MR 1082354
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.