Let E and G be free modules of rank e and g, respectively, over a commutative noetherian ring R. The identity map on E* ⊗ G induces the Koszul complex ...→ SmE* ⊗ SnG ⊗ [This symbol cannot be presented in ASCII format] P (E* ⊗ G) → Sm+1E* ⊗ Sn+1 G ⊗ [This symbol cannot be presented in ASCII format] P-1 (E* ⊗ G) → ... and its dual ... → Dm+1 E ⊗ D n+1 G* ⊗ [This symbol cannot be presented in ASCII format]P-1 (E⊗G*) → DmE ⊗ DnG* ⊗ [This symbol cannot be presented in ASCII format] (E⊗G*) → ... Let HN(m,n,p) and HM(m,n,p) be the homology of the above complexes at SmE* ⊗ SnG ⊗ [This symbol cannot be presented in ASCII format] P (E* ⊗ G) and DmE ⊗ Dn G* ⊗ [This symbol cannot be presented in ASCII format] P (E ⊗G*), respectively. In this paper, we investigate the modules HN(m,n,p) and HM(m,n,p) when -e ≤ m - n ≤ g. We record the fact, already implicitly calculated by Bruns and Guerrieri, that HN(m,n,p) ≅ H M (m′,n′,p′), provided m + m′ = g - 1, n + n′ = e-1, p+p′ = (e-1)(g-1), and 1-e ≤ m - n ≤ g - 1. If m - n is equal to either g or -e, then we prove that the only nonzero modules of the form H N(m,n,p) and HM(m,n,p) appear in one of the split exact sequences. 0 → H M(g,0,p′) → [This symbol cannot be presented in ASCII format] g+p′ (E ⊗ G*) → H N(0,e,p) → 0, or 0 → H M(0,e,p′) → [This symbol cannot be presented in ASCII format] e+p′ (E ⊗ G*) → H N(g,0,p) → 0, where p + p′ = (e - 1)(g - 1) -1. The modules that we study are not always free modules. Indeed, if m=n, then the module H N(m,n,p) is equal to a homogeneous summand of the graded module Tor[This symbol cannot be presented in ASCII format](T,R), where P is a polynomial ring in eg variables over R and T is the determinantal ring defined by the 2 x 2 minors of the corresponding e x g matrix of indeterminates. Hashimoto's work shows that if e and g are both at least five, then H N(2,2,3) is not a free module when R is Z, and when R is a field, the rank of this module depends on the characteristic of R. When the modules H M (m,n,p) are free, they are summands of the resolution of the universal ring for finite length modules of projective dimension two. [ABSTRACT FROM AUTHOR]