By J. A. Hillman

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**Example text**

O. <(~k(Aj (Rr) ~ ~ _ j t M ) [|4;pages 515-518] j ~ I.

D. ll. Link module sequences his work on the A-modules A(L) and G'/G" Crowell defined a link In module sequence as an exact sequence 0 ---+ B ---+ A ---+ I ---+ 0 of A-modules (4) such that A has a presentation with n + I generators n relations (for some n) and where I is the augmentation ideal of A. arguments of Theorem 3 apply in this slightly more general showing for instance (i) that El(A) = (AI(A))I. The first elementary In [40] and setting, Crowell showed that ideal E l of A annihilates B.

D. A(L) The second of the spectral F(V)'/F(~)" o__r lemma. d. d. d. G'/G" sequence duality Note also that the resolution sequence Proof Moreover are also iu~nediate consequences of Chapter of IRV). exact = ZZ q follows from the long exact to the Crowell and the fact that e q-I I = e q ~ (2) and exact for q > 0. // sequence in the 45 An immediate consequence of this lemma is that if AI(L ) # 0 then G'/G" has a square presentation matrix if and only if ~ ~ 3. d. d. A(L) ~ p - I . Since projective free, a torsion module has a short projective resolution A-modules are if and only if it has a square presentation matrix.