Localisations and completions of nilpotent G-spaces

Abstract We develop the theory of nilpotent G -spaces and their localisations, for G a compact Lie group, via reduction to the non-equivariant case using Bousfield localisation. One point of interest in the equivariant setting is that we can choose to localise or complete at different sets of primes at different fixed point spaces—and the theory works out just as well provided that you invert more primes at $$K \le G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> than at $$H \le G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> , whenever K is subconjugate to H in G . We also develop the theory in an unbased context, allowing us to extend the theory to G -spaces which are not G -connected.