By E. L. Lady (auth.), Rüdiger Göbel, Lee Lady, Adolf Mader (eds.)
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Extra info for Abelian Group Theory: Proceedings of the Conference held at the University of Hawaii, Honolulu, USA, December 28, 1982 – January 4, 1983
C 0 In this section we show is equivalent to a full subcategory of the dual of the category of finitely generated modules over a hereditary noetherian ring A. f. curiosity. 5, which considerably simplifies the development in Section 4. The reader who is willing to accept on faith the equivalence of conditions 3) and 4) of this Lemma may wish on the first reading to skip directly to Section 4. We let I ~ W be a finite rank splitting ring and let C be the category of homomorphisms of finite rank I-split W-modules.
Integral extension, we may suppose that D is integrally closed in E of F admissible if sure of V in ring on F containing F F. D. 12, if V' D by a finite is galois over Call a valuation ring p-rank V < 1 contains Replacing F V Q and that on a subfield and the integral cloV' is any valuation is the integral closure of some on some subfield of F. Since D is integrally closed, it is the intersection of the valuation rings that contain it. 13, it suffices to see that each some of with ITW i W, with p-rank Wi < 1 there can be at most p-rank V f 0, rings we can write all 1.
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.