If M is a graded module over the symmetric algebra of a vector space V, then the Koszul cohomology ${\displaystyle K_{p,q}(M,V)}$  of M is the cohomology of the sequence

${\displaystyle \bigwedge ^{p+1}M_{q-1}\rightarrow \bigwedge ^{p}M_{q}\rightarrow \bigwedge ^{p-1}M_{q+1}}$ 

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

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