Download PDF by P. C. Eklof, A. H. Mekler: Almost Free Modules: Set-theoretic Methods

By P. C. Eklof, A. H. Mekler

ISBN-10: 0444504923

ISBN-13: 9780444504920

This e-book offers a complete exposition of using set-theoretic equipment in abelian crew concept, module concept, and homological algebra, together with functions to Whitehead's challenge, the constitution of Ext and the lifestyles of almost-free modules over non-perfect jewelry. This moment version is totally revised and udated to incorporate significant advancements within the decade because the first variation. between those are purposes to cotorsion theories and covers, together with an evidence of the Flat hide Conjecture, in addition to using Shelah's pcf idea to constuct nearly loose teams. As with the 1st version, the e-book is essentially self-contained, and designed to be obtainable to either graduate scholars and researchers in either algebra and good judgment. they are going to locate there an advent to robust strategies which they could locate worthy of their personal paintings.

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5 L e m m a . PROOF. Let S - {X c_ n. sup(X) < n}. By hypothesis S has cardinality a, so we can identify it with a and use this identification to define functions from a to S. In fact we define a function fA" n --4 S for every A C_ a: let fA (a) -- A M c~. Now if A ~- B, then there exists /3 E (A \ B ) U (B \ A); for all c~ >/3, fA(c~) # fB(~). [:] Let a be a cardinal such that 2 <~ - n. Let D be a uniform ultrafilter on a set I of cardinality n. Let Mi be a module of cardinality n for all i E I.

Then there is a decomposition of S, S - Hze~Sz, into ~ pairwise disjoint Sz that all \ SZ r F. PROOF. For each a < a, choose a surjection g~" ~ --+ c~. ) For all v E ~, ~ E ~, let - e s. - We claim that there is a , such that {S~" /3 e ~, a \ S ~ r F} has cardinality ~. First fix /3 and let YZ - U{S~" - e ~}; then YZ - {a E g ' a > /3} M S, since the ga's are surjective. Since F is t~-complete and contains the cofinite filter, the hypothesis implies that {a C n" a > /3} C F; so a \ Y Z does not belong t o F .

By hypothesis on ~, F is contained in a A-complete ultrafilter D. Form the u l t r a p r o d u c t I-Iye~(YI/D which we will denote by M*. 8, M* is free. 4 Clubs and stationary sets 35 and let ~(a) - a D . Now a E Y for some Y E I, so g(Z) - a for all Z E Uy. Therefore it is clear that ~a is an embedding. , it follows t h a t M is free. 11 C o r o l l a r y . If ~ is an Lw~o~-compact cardinal, then every ~-free abelian group (of arbitrary cardinality) is free. 3. 4 Clubs and s t a t i o n a r y sets Our theme in this section will be the description of "relatively large" subsets of a limit ordinal 7; we will have analogs of "sets of measure 1" (the closed u n b o u n d e d sets, or clubs) and of "sets of non-zero measure" (the stationary sets).

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Almost Free Modules: Set-theoretic Methods by P. C. Eklof, A. H. Mekler

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