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By J. A. Hillman

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2-1ink Thus in 26 with group free. Since this I-link is not a boundary link (see Chapter VI) the 2-1ink is nontrivial and so this gives a simple example of the phenomenon first observed by Poenaru [149]. ) Similarly the knot in Figure 6 is a slice of a 2-knot with group ~. Yanagawa [207] has shown that it is in fact a nontrivial slice of a trivial 2-knot. 1 Figure 6 He showed there also that a ribbon 2-knot with group Eg is trivial. In [193] Tristram shows that concordance of B-component l-links is generated as an equivalence relation by concordances of the form L ---+ L +b DR, where R : ~D 2 ----+ S 3 - imL is a ribbon map with image disjoint from that of L, and where +b denotes (iterated) band connected sum.

In the knot theoretic 47 case (~= ]) the results of part (iii) were first obtained by Crowell [39 ] (Note that a Al-mOdule is pseudozero if and only if it is finite). In [40 ] he showed that AI(L) annihilates G'/G" (under an unnecessary further hypothesis). The knot 946 [ 157 ;page 399] G'/G" = (Al/(t- 2)) (~) (Al/(2t-I)) (3,t + I) is not principal. has and so ~2(G'/G") = ( t - 2 , 2 t - I ) = The argument of our next theorem is related to that of Crowell in [39 ] Theorem 4 Let M be a finitely senerated A-module of rank r such that Er(M) is principal and suppose that C(Ar(M)) = • Then M is torsion free as an abelian group.

Is compatible with localization. // Corollary (~r+jM) If the coefficient ring is factorial, = (Ar+j_I(M)/Ar+j(M)) for each j ~ I. then Hence (Ar+j(M)) = (Aj(tM)) for each j ~ 0. ~ Proof theorem. Remarks By Lemma 3 (~r+jM) ~ = (~jtM) for each j ~ I. Now apply the // I. 10~ . 2. If M is a torsion module (over a factorial domain) which has a square presentation matrix then it follows easily from Cramer's rule that (Ao(M)/AI(M)) ~ Ann M. Ann M ffi (Ao(M)/AI(M)). The Corollary then implies that (Buchshaum and Eisenhud show that if R is any noetherian ring and M is an R-module with a square presentation matrix whose determinant is not a zero divisor, then Ann M = (Eo(M):EI(M)) [19]).

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