By J. P. Levine
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E A, Since Z ~ Conversely, corresponding k (~, ~) R as a subring, suppose A, AT field rather than only be the localization at Q(A) with d e n o m i n a t i o n s or A Clearly ~, A a) A b) A ÷ A ~A c) A0 we c o r r e s p o n d i n g l y not divisible by field ~. A /~A ks = 0 The following are equivalent: has ~-only torsion. A T is injective. (R = A/(~)) n e A, the subring of the quotient has n-only torsion if is R - t o r s i o n free generated by to work with a general ~/~. 3: ~. ms = 0. , Choose w h i c h annihilates Given an irreducible element discrete v a l u a t i o n ring with residue F-primary A-module m.
R-torsion ~ A I ~S is trivial, S = A/(~d), A") 0 ~ ( A ' ) 0 ~ ( A " ) are e l e m e n t a r y , G i v e n any ideal be our " b u i l d i n g free is D e d e k i n d , of h o m o g e n e o u s A 0 ~ M. the free S - m o d u l e , Furthermore, will and any of R, it s u f f i c e s of R, there (we r e g a r d since we can c h o o s e of the same r a n k as 0, direct M ~ M'~ exists sums of I, where M' to prove: an ideal R = S/nS), and M. T o__ff S, ~T = T~ ~S. is 34 other h a n d Ak+ 1 = 0 since nk+iA = 0.
An i n j e c t i o n There B k = 0 = Bk. Ak ÷ B0 via the i < k A to to A k a n d is a s t r i n g This composite the T - p r i m a r y isomor- need some and B. sequence additional = 0 for B; its relation to the of e p i m o r p h i s m s injection epimorphism Thus -- the a d d i t i o n a l B k-I -- the of information. i > k. 1 A k + Ak_ 1 = Bk~ 1 because are sequences we w i l l for from precisely BZ+ 1 + A£+ 1 try to d e s c r i b e Of c o u r s e B. and readily. of d e r i v e d is l o s t w h e n w e p a s s sequences and fact recovery turn from those Ak the B~ ÷ Az together B 0 + B k-I ~-primary B0 ÷ B1 + latter A k ~ B k-I information ...
Algebraic Structure of Knot Modules by J. P. Levine