Vol. 101, No. 1, 1982

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The determinantal ideals of link modules. I

Lorenzo Traldi

Vol. 101 (1982), No. 1, 215–222
Abstract

Let m 1, and let H be the multiplicative free abelian group of rank m, with integral group ring ZH and augmentation ideal IH. Suppose 0 B A IH 0 is a short exact sequence of ZH-modules, and the module A is finitely generated. Then A and B are both finitely presented, and for any k Z the determinantal ideals Ek(A) and Ek1(B) satisfy the inclusions

                                        m−1
Ek−1(B)⋅(IH )m −1 ⊆ Ek (A) and Ek(A )⋅(IH )( 2 ) ⊆ Ek−1(B),

where (m−21) is the binomial coefficient (and in particular (m −21) = 0 if m 2), and (IH)0 = ZH. In particular, if m = 1 then Ek(A) = Ek1(B) for any k Z. A consequence of these inclusions is the fact that the greatest common divisors Δk(A) = Δk1(B) for any k Z.

Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 13C12
Milestones
Received: 24 April 1981
Published: 1 July 1982
Authors
Lorenzo Traldi