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Koszul cohomology

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In mathematics, the Koszul cohomology groups K p , q ( X , L ) {\displaystyle K_{p,q}(X,L)} 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.

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 ) {\displaystyle K_{p,q}(M,V)} of M is the cohomology of the sequence

⋀ p + 1 M q − 1 → ⋀ p M q → ⋀ p − 1 M q + 1 {\displaystyle \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 ) {\displaystyle K_{p,q}(X,L)} is given by the Koszul cohomology K p , q ( M , V ) {\displaystyle K_{p,q}(M,V)} of the graded module M = ⨁ q H 0 ( L q ) {\displaystyle M=\bigoplus _{q}H^{0}(L^{q})} , viewed as a module over the symmetric algebra of the vector space V = H 0 ( L ) {\displaystyle V=H^{0}(L)} .

References

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