Gamma-invariant in the theory of Abelian groups

$ \Gamma $- invariant

An invariant associated to an uncountable Abelian group and taking values in a Boolean algebra. Two groups with different invariants are non-isomorphic, but the converse fails in general: groups with the same invariant need not be isomorphic. The invariant is most commonly defined for almost-free groups (groups such that every subgroup of strictly smaller cardinality is a free Abelian group). By a theorem of S. Shelah (see [a7]), such a group is free if it is of singular cardinality, so the invariant is defined for groups of regular cardinality (see Cardinal number). If $ A $ is an Abelian group of regular uncountable cardinality $ \kappa $, $ A $ is said to be $ \kappa $- free if and only if every subgroup of $ A $ of cardinality $ < \kappa $ is free. In that case $ A $ can be written as the union of a continuous chain (called a $ \kappa $- filtration) of free subgroups of cardinality $ < \kappa $: $ A = \cup _ {\nu < \kappa } A _ \nu $, where the continuity condition means that for every limit ordinal $ \nu < \kappa $, $ A _ \nu = \cup _ {\mu < \nu } A _ \mu $. The $ \Gamma $- invariant of $ A $, denoted by $ \Gamma _ \kappa ( A ) $ or just $ \Gamma ( A ) $, is defined to be the equivalence class, $ {\widetilde{S} } $, of

$$ S = \left \{ {\nu < \kappa } : {\textrm{ for some } \tau > \nu, A _ \tau /A _ \nu \textrm{ is not free } } \right \} ; $$

$ {\widetilde{S} } $ is defined to be the set of all subsets $ T $ of $ \kappa $ for which $ S \cap C = T \cap C $ for some closed unbounded subset $ C $ of $ \kappa $. (See Suslin hypothesis for the definitions of closed unbounded and stationary.) The equivalence class depends only on the isomorphism type of $ A $ and not on the choice of $ \kappa $- filtration, because any two $ \kappa $- filtrations agree on a closed unbounded subset of $ \kappa $; the equivalence classes of subsets of $ \kappa $ form a Boolean algebra, $ D ( \kappa ) $, under the partial order induced by inclusion. The least element of this Boolean algebra, $ 0 $, is the class of all non-stationary subsets of $ \kappa $. It can be proved that $ \Gamma ( A ) = 0 $ if and only if $ A $ is free (see [a1]). For $ \kappa = \aleph _ {n + 1 } $( $ n \in \omega $), every one of the $ 2 ^ \kappa $ members of the Boolean algebra $ D ( \kappa ) $ is the $ \Gamma $- invariant of some $ \kappa $- free group of cardinality $ \kappa $( see [a6]). Assuming Gödel's axiom of constructibility, $ V = L $( see Gödel constructive set), the same holds for all regular $ \kappa $ which are not too large (e.g., less than the first inaccessible cardinal, or even the first Mahlo cardinal); in fact, a complete characterization, for any regular $ \kappa $, of the range of $ \Gamma $ can be given, assuming $ V = L $( see [a6] and [a5]).

Another $ \Gamma $- invariant can be defined for use in connection with the Whitehead problem in Abelian group theory, and its generalizations. In this case, for any Abelian groups $ A $ and $ M $, $ \Gamma _ {M} ( A ) $ is defined to be the equivalence class of

$$ \left \{ {\nu < \kappa } : {\textrm{ for some } \tau > \nu, { \mathop{\rm Ext} } ( A _ \tau /A _ \nu , M ) \neq 0 } \right \} $$

when $ A $ is $ \kappa $- free of cardinality $ \kappa $ and $ A $ is written as the union, $ \cup _ {\nu < \kappa } A _ \nu $, of a $ \kappa $- filtration. Then $ \Gamma _ {M} ( A ) = 0 $ implies $ { \mathop{\rm Ext} } ( A, M ) = 0 $; the converse holds for $ M $ of cardinality at most $ \kappa $, assuming $ V = L $( see [a2]).

Useful references for additional information are [a3] and [a4].

References

[a1]	P.C. Eklof, "Methods of logic in abelian group theory" , Abelian Group Theory , Lecture Notes in Mathematics , 616 , Springer (1977) pp. 251–269
[a2]	P.C. Eklof, "Homological algebra and set theory" Trans. Amer. Math. Soc. , 227 (1977) pp. 207–225
[a3]	P.C. Eklof, "Set-theoretic methods: the uses of gamma invariants" , Abelian Groups , Lecture Notes in Pure and Appl. Math. , 146 , M. Dekker (1993) pp. 143–153
[a4]	P.C. Eklof, A.H. Mekler, "Almost free modules" , North-Holland (1990)
[a5]	P.C. Eklof, A.H. Mekler, S. Shelah, "Almost disjoint abelian groups" Israel J. Math. , 49 (1984) pp. 34–54
[a6]	A.H. Mekler, "How to construct almost free groups" Canad. J. Math. , 32 (1980) pp. 1206–1228
[a7]	S. Shelah, "A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals" Israel J. Math. , 21 (1975) pp. 319–349