On the resolutions of the powers of the Pfaffian ideals
Let \(R\) be a Noetherian ring and \(X\) an alternating \(n\times n\)-matrix of indeterminates \(X_{ij}\) over \(R\) for some odd integer \(n\). Then the ideal \(I\) in the polynomial ring \(S= R[X_{ij};\;1\leq i<j\leq n]\) generated by the maximal order pfaffians of \(X\) is a generically perfect Gorenstein ideal of grade 3. This is due to \textit{D. A. Buchsbaum} and \textit{D. Eisenbud} [Am. J. Math. 99, 447-485 (1977; Zbl 0373.13006)] who constructed an explicit minimal \(S\)-free resolution for \(S/I\) of length 3. The paper under review deals with the construction of minimal \(S\)-free resolutions for the powers \(I^ m\) of \(I\). In case \(m=2\), such a resolution has been described by the authors in a preprint some years before [see also the authors in Semin. Geom., Univ. Studi Bologna 1988- 1991, 27-35 (1991; Zbl 0743.13006)]. It seems remarkable that until that time even the projective dimension of \(S/I^ m\) was unknown except when \(m\leq 2\) [\textit{J. Herzog}, J. Reine Angew. Math. 318, 83-105 (1980; Zbl 0425.13005)]. In fact, these cases turn out to be the only ones in which \(I^ m\) is a perfect ideal for all \(n\). It has also to be noted that Kustin and Ulrich at the same time obtained a minimal \(S\)-free resolution for \(I^ m\), ``the approach and the techniques'' being ``different from'' the authors' ones [\textit{A. R. Kustin} and \textit{B. Ulrich}, Mem. Am. Math. Soc. 461 (1992; Zbl 0753.13005)].
scientific article; zbMATH DE number 130484
scientific article