Koszul cohomology

In mathematics, the Koszul cohomology groups $$K_{p,q}(X,L)$$ 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 $$K_{p,q}(M,V)$$ of M is the cohomology of the sequence
 * $$\bigwedge^{p+1}M_{q-1}\rightarrow \bigwedge^{p}M_{q} \rightarrow \bigwedge^{p-1}M_{q+1}$$

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