A family of functors defined on generalized primary groups

Ray Mines, III

Vol. 26 (1968), No. 2, 349–360

DOI: 10.2140/pjm.1968.26.349

Abstract

Let G denote an abelian group; G is called a generalized p-primary group if qG = G for all primes q≠p. Let α be an ordinal, and let δ : G → E_α satisfy the following four conditions: (1) E_α is p^α Ext-injective, (2) p^αE_α = 0, (3) δ′G is p^α-pure in E_α, (4) ker δ = p^αG. Define p^α∗G to be that subgroup of E_α such that p^α(E_α∕δ(G)) = p^α∗G∕δ(G). If α is a limit ordinal, let L_α(G) = _β<αG∕p^βG. Let

Then we have the following p^α-pure containments: G∕p^αG≅ δ(G) ⊆ U_α(G) ⊆ p^α∗(G) ⊆ L_αU_α(G), whenever α is a countable limit of lesser hereditary ordinals. We have p^α∗G = U_α(G) for all groups G if and only if p^α Ext is hereditary. From this we obtain a new proof of the fact that p^α Ext is hereditary when α is a countable limit of lesser hereditary ordinals. We also obtain an example of a cotorsion group G such that G∕p^αG is not equal to L_α(G), thus refuting a conjecture of Harrison. A group G is called generally complete if L_α(G)∕δ(G) is reduced for all limit ordinals α. A generalized p-primary group G is generally complete if and only if it is cotorsion.

Mathematical Subject Classification

Primary: 20.50

Milestones

Received: 1 August 1966

Published: 1 August 1968