TWO VECTORS AND A RECTANGULAR MATRIX 3 where F and G are free R—modules with rankF = / g = rankG, t, X, and u are matrices of indeterminates, and R k[v,X,u] for some field k. Let K be the R—ideal Ij(X) + Ii(uX) in case 1, 4 7i(Xv) + If(X) + h{uX) in case 2, and p be the R—module homomorphism \[v 1 [l®X*(u) r ] : ( F ® F ) e G * - 4 F * in easel, X*(u) X*]: A2 ^* 8 (F* g F) 0 G* - F* in case 2. The integer 5 plays the role of the projective dimension of R/K as an R—module hence, { g in case 1, and g + / 1 in case 2. The integer m is defined by property (d) hence, g / 1 in case 1, and m : » g / in case 2. Then, in each case (1) and (2), there is a family of complexes {C^} which satisfy properties (a)—(g), provided (g) is modified to read (0.1) 0 -* cM(p) - C* ®R R/(ug) - &z-lHp)[-l] - 0. Case (1) is treated in [13] the present paper is devoted to finding the family of complexes {C^} in case (2). In fact, given the data of case (2), we produce two families of complexes. The complexes {I^z)} of section 2 are not minimal, but the maps are well understood. The complexes {M^} of section 4 are minimal, but the maps are very complicated, and less well understood. We begin by recording what is known about R/K in case (2). Theorem 0.2 has been established by De Concini and Strickland [10] using Hodge algebra techniques. Theorem 0.2. Let RQ be a commutative noetherian ring, 2 / g be integers, VfXi, Xgxf and Uixg be matrices of indeterminates, R be the polynomial ring Ro[v,X,u], and K be the R-ideal h(uX) + If{X) + h(Xv). (a) The ring R/K is reduced [respectively, Cohen-Macaulay, a domain, a normal domain) if and only if Ro satisfies the same property. (b) The ideal K is generically perfect of grade f + g 1. (c) The ring R/K satisfies Serre's condition (S{) if and only if RQ satisfies (Si). The proof and notation of Theorem 0.3 may be found in Bruns [4]. The form of the divisor class group of R/K, but not its generators, may also be found in Yoshino [20].
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