By J. A. Hillman
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D. d. G'/G" >. p - I . 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 .
Then M is finitely generated, There is a natural map 6:~ = det~fi(mj~, r(M*) + (~rM)* which is an isomorphism The definition of ~ is compatible with localization and with passage to a quotient with respect to an ideal. Therefore by Nakayama's le~m~a ~ is an epimorphism for each prime ideal p of R. P Hence 6 is an epimorphism and therefore 0(M ) = p(M) . 39 The column ideal class of M = coker Q is the row ideal class of coker Qtr, the cokernel of the transpose of Q. ) Q ~ Rp § coker Qtr § O and similarly (2) 0 ~ (coker Qtr)* ~ R p Q ~ R q ~ M ~ O.
In 1970 Murasugi proved that this condition is equivalent to "each of the subquotients G"Gn/G"Gn+ 1 is isomorphic to the corresponding subquotient of F(2) " and to "the longitudes of L are in G(~) = N (G"G n) ". n~2 Cochran showed in his 1970 Dartmouth thesis that "AI(L) = 0" implied that H2(X;A ) = A, and constructed a family of unsplittable 2-component links with first Alexander polynomial 0 [29, 30]. In this chapter is given a counter-example to the conjecture of Smythe, as an illustration of a new criterion for a ribbon link to be an homology boundary llnk.
Alexander Ideals of Links by J. A. Hillman