On a subcomplex of the Schur complex
Let \(R\) be a commutative ring and \(E\) any free \(R\)-module of finite rank. The authors generalize hook complexes considered by \textit{D. A. Buchsbaum} and \textit{D. Eisenbud} [Adv. Math. 18, 245-301 (1975; Zbl 0336.13007)] by associating to every partition \(\gamma\) a family of subcomplexes \(C^\bullet_{\gamma : q} E\) of a Schur complex [\textit{K. Akin}, \textit{D. A. Buchsbaum} and \textit{J. Weyman}, Adv. Math. 44, 207-278 (1982; Zbl 0497.15020)], \(q\) a positive integer such that \(\text{rk} (\gamma) \leq q \leq \text{length} (\gamma)\). When \(R\) contains a field \(k\), an easy sufficient condition for \(H^\bullet (C^\bullet_{\gamma : q} E)\) to be equal to \(H^0 (C^\bullet_{\gamma : q} E)\), and in fact to the Schur functor \(L_{\overline \gamma} E\) (regardless of \(q)\), is exhibited.
scientific article; zbMATH DE number 815870