We give a natural construction and a direct proof of the Adams isomorphism
for equivariant orthogonal spectra. More precisely, for any finite group
, any normal
subgroup
of
, and any orthogonal
–spectrum
, we construct a natural
map
of orthogonal
–spectra from the
homotopy
–orbits
of
to the derived
–fixed points of
, and we show that
is a stable weak
equivalence if
is
cofibrant and
–free.
This recovers a theorem of Lewis, May and Steinberger in the equivariant
stable homotopy category, which in the case of suspension spectra was
originally proved by Adams. We emphasize that our Adams map
is
natural even before passing to the homotopy category. One of the tools we develop is a
replacement-by-–spectra
construction with good functorial properties, which we believe is of independent
interest.
Keywords
Adams isomorphism, equivariant stable homotopy theory
