Homological algebra and set theory

Abstract

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E x t Subscript normal upper Lamda Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>Ext</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">{\operatorname {Ext}}_\Lambda ^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for rings <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of global dimension 1. Using Martin’s Axiom, the necessity of these conditions is shown not to be a theorem of ZFC. Applications are given to abelian group theory, including a partial solution (assuming <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V equals upper L"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>V</mml:mtext> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>L</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {V}} = {\text {L}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) to a problem of Baer on the splitting of abelian groups. Some independence results in abelian group theory are also proved.