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.
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