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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 [[#References|[a7]]]), such a group is free if it is of [[singular cardinal]]ity, so the invariant is defined for groups of [[regular cardinal]]ity (see [[Cardinal number|Cardinal number]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100503.png" /> is an Abelian group of regular uncountable cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100505.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100507.png" />-free if and only if every subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100508.png" /> of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100509.png" /> is free. In that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005010.png" /> can be written as the union of a continuous chain (called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005012.png" />-filtration) of free subgroups of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005013.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005014.png" />, where the continuity condition means that for every [[limit ordinal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005016.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005017.png" />-invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005018.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005019.png" /> or just <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005020.png" />, is defined to be the equivalence class, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005021.png" />, of
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005022.png" /></td> </tr></table>
+
'' $  \Gamma $-
 +
invariant''
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005023.png" /> is defined to be the set of all subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005025.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005026.png" /> for some closed unbounded subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005028.png" />. (See [[Suslin hypothesis|Suslin hypothesis]] for the definitions of closed unbounded and stationary.) The equivalence class depends only on the isomorphism type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005029.png" /> and not on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005030.png" />-filtration, because any two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005031.png" />-filtrations agree on a closed unbounded subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005032.png" />; the equivalence classes of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005033.png" /> form a [[Boolean algebra|Boolean algebra]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005034.png" />, under the partial order induced by inclusion. The least element of this Boolean algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005035.png" />, is the class of all non-stationary subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005036.png" />. It can be proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005037.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005038.png" /> is free (see [[#References|[a1]]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005039.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005040.png" />), every one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005041.png" /> members of the Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005042.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005043.png" />-invariant of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005044.png" />-free group of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005045.png" /> (see [[#References|[a6]]]). Assuming Gödel's [[axiom of constructibility]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005046.png" /> (see [[Gödel constructive set|Gödel constructive set]]), the same holds for all regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005047.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005048.png" />, of the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005049.png" /> can be given, assuming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005050.png" /> (see [[#References|[a6]]] and [[#References|[a5]]]).
+
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 [[#References|[a7]]]), such a group is free if it is of [[singular cardinal]]ity, so the invariant is defined for groups of [[regular cardinal]]ity (see [[Cardinal number|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
  
Another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005051.png" />-invariant can be defined for use in connection with the [[Whitehead problem|Whitehead problem]] in Abelian group theory, and its generalizations. In this case, for any Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005054.png" /> is defined to be the equivalence class of
+
$$
 +
S = \left \{ {\nu < \kappa } : {\textrm{ for some  }  \tau > \nu, A _  \tau  /A _  \nu  \textrm{ is not  free  } } \right \} ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005055.png" /></td> </tr></table>
+
$  {\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|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|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 [[#References|[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 [[#References|[a6]]]). Assuming Gödel's [[axiom of constructibility]],  $  V = L $(
 +
see [[Gödel constructive set|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 [[#References|[a6]]] and [[#References|[a5]]]).
  
when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005056.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005057.png" />-free of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005059.png" /> is written as the union, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005060.png" />, of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005061.png" />-filtration. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005062.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005063.png" />; the converse holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005064.png" /> of cardinality at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005065.png" />, assuming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005066.png" /> (see [[#References|[a2]]]).
+
Another  $  \Gamma $-
 +
invariant can be defined for use in connection with the [[Whitehead problem|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 [[#References|[a2]]]).
  
 
Useful references for additional information are [[#References|[a3]]] and [[#References|[a4]]].
 
Useful references for additional information are [[#References|[a3]]] and [[#References|[a4]]].

Latest revision as of 19:41, 5 June 2020


$ \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
How to Cite This Entry:
Gamma-invariant in the theory of Abelian groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gamma-invariant_in_the_theory_of_Abelian_groups&oldid=47039
This article was adapted from an original article by P.C. Eklof (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article