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By J. P. Levine

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Then define for i > k. and, therefore, The O} 28 epimorphisms obvious Ci + projection course, this projection To and For C i+l C k-I map must onto Cok define i > k, we With be we zero. ¢i we ¢ ÷ A map when ¢i ~ Bi ÷ i = k - i, and, Ci and as Ci Ck = C = Ck For C = Ck i < k, proceed B i ÷ Cok have take i < k - i. = C~Cok Ci ÷ Ci the projection i = k, for and we zero when induce take definitions, commutativity ¢ of the of the wish to p r o v e that are, in the + bottom the row + as q-primary above.

Can be chosen Yi(1) of type is a principal of n e c e s s i t y , of type = i, are the r a t i o n a l of the elements m. ,ym } ~ A = It is easy it s u f f i c e s pA = O, since, Ym(1) completely satisfying choose and yi/Yi+l to them as the free m o d u l e s then free m o d u l e This divisor K, F ) - i, Yi(1) A = (k e A : kA = 0}. ,ym free m o d u l e Y1 is not a unit determined The f o l l o w i n g Z-torsion for where 6i~)J will be the d e s i r e d element. = ±i. 20 (See [L, Cor. =l free module of type §8.

The p r o c e d u r e p r o c e e d i n g i n d u c t i v e l y on k. 24 We w i l l assume and a d j o i n block" to c r e a t e falls i) into We ~k+iA and the of A Constructing Inductive first A. the look at let Kk(A) "building B B from Define Ci+ I + C i Ai+l, maps (i). that We C- = A. 1 1 of " b u i l d i n g implementation between of this the q - p r i m a r y = Ker ~ k , Let a q-primary A be Let show k+iA where sequences first for = 0. A-module such that {0 + Ai+ 1 ÷ A i ÷ A i ÷ A i+l + 0} be how the r-primary to d i r e c t l y are obviously and C 1• + and i < k, C.

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