Ideals generated by Pfaffians

Abstract

Influenced by the structure theorem of Buchsbaum and Eisenbud for Gorenstein ideals of depth 3, [4], we study in this paper ideals generated by Pfaffians of given order of an alternating matrix. Let ,x7 be an n by n alternating matrix (i.e., xi9 = --xji for i < j and xii = 0) with entries in a commutative ring R with identity. One can associate with X an element Pf(-k-) of R called Pfaffian of X (see [I] or [2] for the definition). For n odd det S ~-= 0 and Pf(X) = 0; for n even det X is a square in R and Pf(X)z = det X. For a basis-free account and basic properties ofpfaffians we send interested readers to Chapter 2 of [4] or to [IO]. For a sequence i1 ,..., &, 1 < i r < n, the matrix obtained from X by omitting rows and columns with indices i1 ,..., ik is again alternating; we write Pf il*....ik(X) for its Pfaffian and call it the (n k)-order Pfaffian of X. We are interested in ideals Pf,,(X) generated by all the 2p-order Pfaffians of X, 0 ,( 2p < n. After some preliminaries in Section 1 we prove in Section 2 that the height and the depth of E-“&X) are bounded by the number