Vanishing of the homology modules of a Koszul complex
The author gives an upper bound for the degree of non-vanishing homology modules of the Koszul complex \(K_\mu (\varphi)\), for \(\varphi\) a homomorphism of finite dimensional free \(R\)-modules. More precisely, let \(R\) be a commutative Noetherian ring with a unit. Let \(\varphi : F \to E\) be an \(R\)-homomorphism of finite dimensional free \(R\)-modules of rank \(n\) and \(d\) respectively. Denote by \(S(E)\) the symmetric algebra of \(E\), and let \(F_i = F \otimes S_i (E)\). The Koszul complex splits: \(K (\varphi) = \bigoplus K_\mu (\varphi)\), for \(K_\mu (\varphi)\) the complex \(0 \to \bigwedge^\mu F_0 @>\partial>> \bigwedge^{\mu - 1} F_1 @>\partial>> \cdots @>\partial>> \bigwedge^0 F_\mu\), where \(\partial (x_1 \wedge\cdots \wedge x_k) = \sum_j (-1)^{j + 1} \varphi^* (x_j) x_1 \wedge \cdots \widehat x_j \cdots \wedge x_k\). This paper gives a lower bound \(\ell\), obtained from the grades of ideals generated by minors of \(\varphi\), such that \(H_k (K_\mu (\varphi)) = 0\) whenever \(\mu \geq n - d + 1\) and \(k > \ell\).
scientific article; zbMATH DE number 810224
scientific article