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 Template:Harvs, and named after Jean-Louis Koszul as they are closely related to the Koszul complex.

Template:Harvtxt surveys early work on Koszul cohomology, Template:Harvtxt gives an introduction to Koszul cohomology, and Template:Harvtxt 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

${\bigwedge }^{p+1}{M}_{q-1}\to {\bigwedge }^p{M}_q\to {\bigwedge }^{p-1}{M}_{q+1}$

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 $M=\underset{q}{⨁}{H}^0(L^q)$, viewed as a module over the symmetric algebra of the vector space $V=H^0(L)$.

References

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