Ideals Associated to Two Sequences and a Matrix

Let u1×n, Xn×n, and vn×1 be matrices of indeterminates, AdjX be the classical adjoint of X, and H(n) be the ideal I1(uX)+ I1(Xv)+ I1(vu−AdjX). Vasconcelos has conjectured that H(n) is a perfect Gorenstein ideal of grade 2n. In this paper, we obtain the minimal free resolution of H(n); and thereby establish Vasconcelos’ conjecture. Let u1×n, Xn×n, and vn×1 be matrices of indeterminates over a commutative noetherian ring R0, and let H(n) be the ideal I1(uX) + I1(Xv) + I1(vu− AdjX) of the polynomial ring R = R0[{ui, vi, xij | 1 ≤ i, j ≤ n}]. Vasconcelos has conjectured, in [10, Conjecture 3.3.1], that the ideal H(n) is a perfect Gorenstein ideal of grade 2n. In this paper, we obtain the minimal homogeneous resolution of R/H(n) by free R−modules; and thereby establish Vasconcelos’ conjecture. In fact, we produce two resolutions of R/H(n). The complex F of section 2 is never minimal, but it is relatively straightforward. The complex M of section 3 is a quotient of F. It is more complicated than F, but it is minimal. The exactness of F and M is established in section 4. In section 5 we consider the singular locus and linkage history of R/H(n); we also consider the algebra structure of Tor• (R/H(n), R0). At the end of the paper we record some open questions about the ring R/H(n). The arguments of sections 2, 3, and 4 are long, but routine. In section 2 we prove that F is a complex. In section 3 we split off a split exact subcomplex N of F. In section 4 we apply the acyclicity lemma and reduce the problem to one involving generic data with the parameter n replaced by n − 1. The most interesting part of the argument is the discovery of the complex F. This complex is obtained by merging four Koszul complexes: (*) F(1) ←→ F(2) l l F(3) ←→ F(4), where F(1) and F(4) are both Koszul complexes on the entries of [u v], F(2) is the Koszul complex on the entries of [uX v], and F(3) is the Koszul complex on the 1991 Mathematics Subject Classification. 13H10, 13D25.