By J. A. Hillman
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Extra info for Alexander Ideals of Links
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.
Lemma I (Cochran [ 29]) Proof If ~(L) = 2, then H2(X;A) ~- A. Let u and v belong to H2(X) ~ A n . there are ~ and ~ in A such that ~u = By. no common factor. Then since rank Hi(X;A) = l, We may assume that ~ and B have Since fi is factorial v = ~w for some w in A n , which must actually be in H2(X;A) by the exactness of is torsion free. (2) and the fact that A n+l Therefore every 2-generator submodule of the finitely generated rank 1 A-module H2(X;A) is cyclic. The lemma follows easily. // Cochran's result extended to embeddings of arbitrary finite graphs and was published in [ 3 0 ] .
TM j Proof k ~ r and for each j ~ I. Let p be the maximal ideal of R. By the structure theorem for finitely generated modules over principal ideal domains, M~ R r ~ tM ~ R r ~ (I ~ i~ n (R/pC(i))) where 0 < e(i) ~ e(i+l) for 1 $ i ~ n.
Alexander Ideals of Links by J. A. Hillman