Koszul cohomology

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

[math]\displaystyle{ \bigwedge^{p+1}M_{q-1}\rightarrow \bigwedge^{p}M_{q} \rightarrow \bigwedge^{p-1}M_{q+1} }[/math]

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

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, https://books.google.com/books?id=Pxw9_38LlnYC 
• 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 
• Green, Mark L. (1984), "Koszul cohomology and the geometry of projective varieties", Journal of Differential Geometry 19 (1): 125–171, ISSN 0022-040X, http://projecteuclid.org/getRecord?id=euclid.jdg/1214438426 
• Green, Mark L. (1984), "Koszul cohomology and the geometry of projective varieties. II", Journal of Differential Geometry 20 (1): 279–289, ISSN 0022-040X, http://projecteuclid.org/getRecord?id=euclid.jdg/1214439000 
• Green, Mark L. (1989), "Koszul cohomology and geometry", in Cornalba, Maurizio; Gómez-Mont, X.; Verjovsky, A., 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