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THE COMPLEXITY OF THE EMBEDDABILITY RELATION BETWEEN TORSION-FREE ABELIAN GROUPS OF UNCOUNTABLE SIZE
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- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 2 / June 2018
- Published online by Cambridge University Press:
- 01 August 2018, pp. 703-716
- Print publication:
- June 2018
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- Article
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We prove that for every uncountable cardinal κ such that κ<κ = κ, the quasi-order of embeddability on the κ-space of κ-sized graphs Borel reduces to the embeddability on the κ-space of κ-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability relation on the κ-space of κ-sized R-modules, for every
$\mathbb{S}$-cotorsion-free ring R of cardinality less than the continuum. As a consequence we get that all the previous are complete 
$\Sigma _1^1$ quasi-orders.
COMMUTING PROPERTIES OF EXT
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 94 / Issue 2 / April 2013
- Published online by Cambridge University Press:
- 25 February 2013, pp. 276-288
- Print publication:
- April 2013
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- Article
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We characterize the abelian groups
$G$ for which the functors 
$\mathrm{Ext} (G, - )$ or 
$\mathrm{Ext} (- , G)$ commute with or invert certain direct sums or direct products.
