THE COHOMOLOGY OF THE KOSZUL COMPLEXES ASSOCIATED TO THE TENSOR PRODUCT OF TWO FREE MODULES

Abstract

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

and its dual

Let H _𝒩(m, n, p) and H _ℳ (m, n, p) be the homology of the above complexes at

respectively. In this paper, we investigate the modules H _𝒩(m, n, p) and H _ℳ(m, n, p) when −e ≤ m − n ≤ g. We record the fact, already implicitly calculated by Bruns and Guerrieri, that H _𝒩(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 H _ℳ (m, n, p) appear in one of the split exact sequences

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 (T, R), where P is a polynomial ring in eg variables over R and T is the determinantal ring defined by the 2×2 minors of the corresponding e×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.

Key Words:

• Buchsbaum–Rim complex
• Class group
• Determinantal ring
• Duality
• Eagon–Northcott complex
• Finite free resolution
• Koszul complex
• Koszul homology
• Perfect module
• Universal resolution

Mathematics Subject Classification:

• 13D25
• 13C20
• 13C40

Acknowledgments

Notes

^#Communicated by W. Bruns.