Abelian Group Theory: Proceedings of the Conference held at by E. L. Lady (auth.), Rüdiger Göbel, Lee Lady, Adolf Mader

By E. L. Lady (auth.), Rüdiger Göbel, Lee Lady, Adolf Mader (eds.)

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Additional resources for Abelian Group Theory: Proceedings of the Conference held at the University of Hawaii, Honolulu, USA, December 28, 1982 – January 4, 1983

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QC be the category of quasi- As in Section 1, we do not require rank I < "'. In [13] Arnold duality was described by specifying that QAG = Hom(G,Q) and d(I ® AG} = HornI((I 0 G)/d(I 0 G) ,QI} ~ HomI(I 0 G,QI) ~ I A QAG. The fact that A is an exact functor and A2 ~ 1 follows easily. How- ever, to derive the deeper properties in this Section and Section S, a less obvious definition seems more convenient. 1. If CONSTRUCTION. G Let a:Q QI ~ be the canonical inclusion of rings. is a finite rank I-split W-module and essential I-submodule of Hom(G,QI)/QHom(G,I}, be the quotient map and let We call A(G) L e = the Arnold dual of Su*:Hom(G,Q} G.

2 we have ~:I' 01 Hom(G,QI) ~ Hom(G,QI'). e:. 8 I' 01 HomI(I l=- ~ QL' '" I' 01 QL and we can choose L' = I' 0 1 L. is a pure submodule of L' . Since ljJ is monic, (TjJ8)·I(L'). 6. PROPOSITION. 2. + X. 7. Let LEMMA. + 0 G + + ~ 0 ~ ~ is an exact sequence of finite AG is exact in QC, X AG Hom( ,Q) and m) H + 0 is exact, let + = + 0 ~(p)m P G' be the image of a finitely generated projective Then the map 1;:P OR M+HomR(HomR(P,R),M) is an isomorphism. Since the result is clearly true for Lemma 1. 30.

4. DEFINITION. If G and H are modules in e, then Exte(G,H) 1 is the subgroup of Extw(G,H) consisting of those elements represented by short exact sequences 0 + H + X + G + 0 such that X E lei. We further define QExte(G,H) = Q 0 Exte(G,H). 5. LEMMA. 1) Exte(G,H) is a divisible summand of Extw(G,H) containing its torsion submodule and QExte(G,H) is the quotient of Exte(G,H) modulo its torsion submodule. 1 2) Exte(~'_) is a subfunctor of Ext W(_'_)' 3) If B is an I-split module and 0 ~ H ~ X + G + 0 is an exact sequence in e, then there are induced long exact sequences as follows: a) 0 + Hom(B,H) + Hom(B,X) + Hom(B,G) + + Exte(B,H) + Exte(B,X) + Exte(B,G) + 0; b) 0 QHom(B,H) + QHom(B,X) + QHom(B,G) + QExte(B,H) + QExte(B,X) + QExte(B,G) + + c) 0 + 0; Hom(G,B) + Hom(X,B) + Hom(H,B) + Exte(G,B) + Exte(X,B) + Exte(H,B) + 0; d) 0 + QHom(G,B) + QHom(X,B) + QHom(H,B) + + QExte(G,B) + QExte(X,B) + QExte(H,B) + O.

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