Koszul resolutions of linear subschemes in projective space (Q1814971)

Let \(R\) be a commutative Noetherian ring with multiplicative unit, \(K\) be the Koszul complex over the polynomial algebra \(R[e_1,\dots, e_d]\) on a sequence of \(R\)-linear forms \(L_1,\dots,L_n\). The author proves the assertion of \textit{W. Bruns} and \textit{U. Vetter} [``Determinantal rings'', Lect. Notes Math. 1327 (1988; Zbl 0673.13006)] that, in the case \(n\geq d\), the fulfilment of conditions on the Fitting ideals for the corresponding matrix \(A\) of these forms [\textit{L. L. Avramov}, J. Algebra 73, 248-263 (1981; Zbl 0516.13001)] is equivalent to the acyclicity of the homogeneous components \(K^\nu\) of the complex \(K\) for \(\nu\geq n-d+1\).