In mathematics, the Koszul cohomology groups Kp,q(X, L) are groups associated to a projective variety X with a line bundle L. They were introduced by 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 Kp,q(M,V) of M is the cohomology of the sequence
If L is a line bundle over a projective variety X, then the Koszul cohomology Kp,q(X,L) is given by the Koszul cohomology Kp,q(M,V) of the graded module M = ⊕qH0(Lq), as a module over the symmetric algebra of the vector space V=H0(L).