The minimal free resolutions of the Huneke-Ulrich deviation two Gorenstein ideals (Q1103678)

The deviation of the ideal I is the minimal number of generators of I minus the grade of I. A deviation zero ideal is a complete intersection. \textit{E. Kunz} [J. Algebra 28, 111-115 (1974; Zbl 0275.13025)] proved that there are no deviation one Gorenstein ideals. A hypersurface section of a grade \(g-1\) deviation two Gorenstein ideal is a grade g deviation two Gorenstein ideal. We consider this to be a trivial example. Very few non- trivial examples of deviation two Gorenstein ideals are known. The only known non-trivial deviation two Gorenstein ideals of grade at least four are the ideals of \textit{C. Huneke} and \textit{B. Ulrich} [Am. J. Math. 107, 1265-1303 (1985; Zbl 0587.13006)], which are the ideals studied in this paper. For each integer s, let \(\underline y^{(s)}\) be a \(1\times s\) matrix of indeterminates, \(X^{(s)}\) be an \(s\times s\) alternating matrix of indeterminates, \(\underline g^{(s)}\) be the product \(\underline y^{(s)}X^{(s)}\), and \(\underline c^{(s)}\) be the sequence \(g_ 1^{(s)},...,g^{(s)}_{s-1}\). Let \(A_ s\) be the ideal of the ring \({\mathbb{Z}}[X^{(s)},\underline y^{(s)}]\) generated by the entries of \b{g}\({}^{(s)}\) together with the pfaffian of \(X^{(s)}.\) In this paper we produce the minimal free resolution of the algebra defined by \(A_{2n}\). Once and for all we fix an integer \(n\geq 2\), X a 2n\(\times 2n\) alternating matrix of indeterminates, and \b{y} a \(1\times 2n\) matrix of indeterminates. Let \(R=R_ n\) be the ring \({\mathbb{Z}}[X,\underline y]\) and let \(A=A_{2n}\) be the ideal of R generated by the entries of the product \(\underline g = \underline yX\) together with the pfaffian of X. The ideal A is prime and Gorenstein. It has grade \(2n-1\) and has \(2n+1\) generators. We will produce two resolutions \({\mathbb{F}}={\mathbb{F}}^{(n)}\) and \({\mathbb{M}}={\mathbb{M}}^{(n)}\) of \(R/A\) by free R-modules. The resolution \({\mathbb{M}}\) will be minimal; however, \({\mathbb{F}}\) will be easier to manipulate.