Koszul cohomology

In mathematics, the Koszul cohomology groups ${\displaystyle K_{p,q}(X,L)}$ 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 ${\displaystyle K_{p,q}(M,V)}$ of M is the cohomology of the sequence

${\displaystyle \bigwedge ^{p+1}M_{q-1}\rightarrow \bigwedge ^{p}M_{q}\rightarrow \bigwedge ^{p-1}M_{q+1}}$

If L is a line bundle over a projective variety X, then the Koszul cohomology ${\displaystyle K_{p,q}(X,L)}$ is given by the Koszul cohomology ${\displaystyle K_{p,q}(M,V)}$ of the graded module ${\displaystyle M=\bigoplus _{q}H^{0}(L^{q})}$, viewed as a module over the symmetric algebra of the vector space ${\displaystyle V=H^{0}(L)}$.

References

• Aprodu, Marian; Nagel, Jan (2010), Koszul cohomology and algebraic geometry, University Lecture Series, 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, 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