Perfection and representation of GL\((N)\) (Q1335092)

Let \(F_ 0\) and \(G_ 0\) be locally free modules of rank \(m\) and \(n\), \(m \geq n\), over a scheme. Put \(R = \text{sym} (F_ 0 \otimes G_ 0)\), \(X = \text{Spec} (R)\), \(F = F_ 0 \otimes R\) and \(G = G_ 0 \otimes R\). Let \(\varphi : F \to G^*\) be the generic map over \(X\). For any partition \(\lambda = (\lambda_ 1, \dots, \lambda_ l)\) we have an induced map \(\varphi_ \lambda : L_ \lambda F \to L_ \lambda G^*\) between the corresponding Schur functors \(L_ \lambda\). The aim of this paper is to study the perfectness of the \({\mathcal O}_ X\)-module \(M_ \lambda : = \text{coker} (\varphi_ \lambda)\). By giving an explicit description of a minimal free resolution of \(M_ \lambda\), when \({\mathcal O}_ X\) contains the field of rational integers, the author proves that \(hd_ RM_ \lambda = D (\lambda) (m - n) + 1\), where \(D (\lambda)\) denotes the Durfee square of \(\lambda\). From this it follows that \(M_ \lambda\) is perfect if and only if \(\lambda\) is a hook, i.e., \(D(\lambda) = 1\). It should be noted that the sufficient part of this statement was already proved by \textit{D. A. Buchsbaum} and \textit{D. Eisenbud} [Adv. Math. 18, 245-301 (1975; Zbl 0336.13007)], and that the case \(\lambda\) is a rectangle has been studied earlier by \textit{K. Akin}, \textit{D. A. Buchsbaum} and \textit{J. Weyman} [Adv. Math. 44, 207-278 (1982; Zbl 0497.15020)] and recently by the author [J. Algebra 167, No. 2, 233-257 (1994; Zbl 0809.13007)].