Koszul complexes and hyperdeterminants
The starting point is the paper ``Hyperdeterminants'' by \textit{I. M. Gelfand}, \textit{M. M. Kapranov} and \textit{A. V. Zelevinsky} [Adv. Math. 96, 226-263 (1992; Zbl 0774.15002)], where the concept of hyperdeterminant \(\text{Det}(A)\) and the Cayley-Koszul complexes \(C(m_1, \dots, m_r,A)\) are introduced, for any \(R\)-linear form \(A\) in the polynomial ring \(R=C[x_1^{(1)} \dots x_{d_1}^{(1)}; \cdots;x_1^{(r)} \dots x_{d_r}^{(r)}]\), \(r\geq 2\), \(d_i\geq 2\) \((i=1,\dots,r)\), with the condition \(d_{i_0}-1\leq \sum_{i\neq i_0}(d_i-1)\), for all \(i_0\in\{1, \dots,r\}\). In the present paper, the authors introduce a Koszul complex \({\mathcal K}_A\) on a free \(R\)-module \(M\) and prove that the Cayley-Koszul complexes are multigrade components of such a \({\mathcal K}_A\). Moreover, they introduce a double complex \({\mathcal D}_A\) of free \(R\)-modules and express \(\text{Det}(A)\) in terms of the determinant of a multigraded component of \({\mathcal D}_A\).
scientific article; zbMATH DE number 1523165