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By J. A. Hillman

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For each k ~ 0 the k th elementary Ek(M) generated by the (q-k)• representing determinantal [43 ; page 101] . ) subdeterminants Q if k < q and by ! if k ~ q. Crowell and Fox Eisenbud (We use the terminology of [13 ; page 573 ] and the (k+l) st Fitting [19 ] . is well known, system S in R. of the matrix This ideal is called the k th Clearly Ek(M) ~ Ek+I(M) multiplicative ideal of M is the ideal That it depends only on M, not and is proven for instance by Crowell and Ek(Ms) = Ek(M) S for any 28 For each k ~ 0 let AkM be the k th exterior power of M El4 ; page 507 ] and let ~k M = Ann AkM.

The dimension of the vector space M / ~ M Let q be over the field R/~ . Then 2g ~k(M/~M) = 0 if k ~ q and eq+1 (M/~M) = R/~ , so Ok(M) ~ only if k ~ q. if and By Nakayama's le~m~a [ 4; page 21 2, M has a presentation with q generators. Since M / ~ M has dimension q, all the entries of the presentation matrix are in ~ , and hence Ek(M ) ~ and only if k < q, that is, if and only if ~k+|(M) ~ ~k(M) = n{~ prime I Ek(M) C ~ } = N{~ These results are well known. prime I ~k+l(M) ~ if In other words } = ~k+|(M).

2o Similarly, if G'/G" = 0 then ~ = ! ~ 0. d. d. ~ $ 2. d. 10. // Part (i) of this Theorem was first proven by Torres who used properties of the Wirtinger presentation [189]. If the commutator sub- group of a 2-component link is perfect, then AI(L) = I, so the linking number is • by the second Torres condition. the linking number is for G/G 3 of Chapter I. • (See Chapter Vll. That also follows from the Milnor presentation See also Chen [27]). 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).

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