Relations on pfaffians. II: A counterexample (Q1191310)

Let \(R\) be a commutative ring with unity, and fix integers \(n\geq 2\), \(t\geq 1\) such that \(2t\leq n\). Let \(Pf_{2t}\) denote the ideal generated by the \(2t\)-order pfaffians of the generic \(n\) by \(n\) antisymmetric matrix in the polynomial ring over \(R\) in \(n(n-1)/2\) variables. In a previous paper [J. Math. Kyoto Univ. 31, No. 3, 713-731 (1991; see the preceding review)] the author has proved that the module of relations between the pfaffians is generated by relations of degree 1 when \(n\leq 2t+3\). In the present paper, he proves that, when \(n=8\), \(t=2\) and \(R\) is a field of characteristic 2, the module of relations between the pfaffians has a minimal generator which is a relation of degree 2. Since in characteristic 0 this module is generated by relations of degree 1, it follows that \(Pf_{2t}\) does not have, in general, generic minimal free resolutions.