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].