Koszul cohomology

In mathematics, the Koszul cohomology groups formula_1 are groups associated to a projective variety "X" with a line bundle "L". They were introduced by , and named after Jean-Louis Koszul as they are closely related to the Koszul complex.
 surveys early work on Koszul cohomology, gives an introduction to Koszul cohomology, and 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 formula_2 of "M" is the cohomology of the sequence 
If "L" is a line bundle over a projective variety "X", then the Koszul cohomology formula_1 is given by the Koszul cohomology formula_2 of the graded module formula_6, viewed as a module over the symmetric algebra of the vector space formula_7.