Alexander Ideals of Links by J. A. Hillman

By J. A. Hillman

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All the (q-r) x (q-r) minors of Q. generated module. true that y(M) = p(M) If R = Q [ x , y , ~ for every finitely and M is the ideal y(M) = p(M) = the class of M, which is not principal, = R. a l s o pg~er ~(M) = 0(K ) = p(M ) A s i m p l e a r g u m e n t due t o S t e i n i t z It is not generally p(M) is It would he interesting (x,y,z) then but M = R and to have a simple general characterization of the column invariant of a module and to find other invariants isomorphism classes of (not necessarily projective) modules.

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 ] .

K(Aj (Rr) ~ ~ _ j t M ) [|4;pages 515-518] j ~ I. n-k+l and f(n-k+l) = I. ~k M = Ann(AkM) = pe(n-k+l) Therefore = (Ak_I(M)/Ak(M)). H ~ n}. Er+k(M) ~ Er+k_l(M)} when R is a discrete valuation ring, then we may extend this result as follows.

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