When is Ext(A,B) torsion-free?, and related problems. (English) Zbl 0653.20055

Goeters, H. Pat

Commun. Algebra 16, No. 8, 1605-1619 (1988).

This paper considers the problem: Given torsion-free abelian groups A and B, when is Ext(A,B) torsion-free? The main result is the following: Theorem. Let A be countable, let B be finite rank with outer type of B not equal to the type of Q, the rational numbers, and let X be a subgroup of Q of type equal to the inner type of B. Then Ext(A,B) is torsion-free if and only if \(\cap \{Ker f|\) f: \(A\to X\}\) is divisible by all primes p such that pX\(\neq X.\)
This theorem extends an earlier result of Huber and Warfield for the case when B is rank one. The author also shows that under the hypotheses of the theorem, if Ext(A,B) is torsion-free, then there is a projective- injective relationship between A and B. Specifically, (1) B is injective with respect to any pure embedding \(0\to C\to A\) where C is finite rank, and (2) A if projective with respect to any epimorphism \(B\to C\to 0\) where C is torsion-free.

Reviewer: C.Vinsonhaler

MSC:

20K15	Torsion-free groups, finite rank
20K35	Extensions of abelian groups
20K40	Homological and categorical methods for abelian groups

Keywords:

torsion-free abelian groups; type; Ext; injective; pure embedding; projective

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