Properties and examples of 𝐴-Landweber exact spectra

It is classically known that Landweber exact homology theories (complex oriented theories which are completely determined by complex cobordism) admit no nontrivial phantom maps. Herein we propose a definition of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Landweber exact spectra, for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a compact abelian Lie group, and show that an analogous result on phantom maps holds. Also, we show that a conjecture of May on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K upper U Subscript upper G"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">KU_G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is false. We do not prove an equivariant Landweber exact functor theorem, and therefore our result on phantom maps only applies to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M upper U Subscript upper A"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">MU_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K upper U Subscript upper A"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">KU_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, their <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-localizations, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B upper P Subscript upper A"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">BP_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which are shown to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Landweber exact by ad-hoc methods.