Let
be a profinite group with finite virtual cohomological dimension and let
be a discrete
–spectrum.
If and
are closed
subgroups of , with
, then, in general,
the –spectrum
is not known to be a
continuous –spectrum,
so that it is not known (in general) how to define the iterated homotopy fixed point
spectrum .
To address this situation, we define homotopy fixed points for delta-discrete
–spectra and show that the
setting of delta-discrete –spectra
gives a good framework within which to work. In particular, we show that by using delta-discrete
–spectra,
there is always an iterated homotopy fixed point spectrum, denoted
, and it is
just .
Additionally, we show that for any delta-discrete
–spectrum
, there is an
equivalence .
Furthermore, if
is an arbitrary profinite group, there is a delta-discrete
–spectrum
that is equivalent
to and, though
is not even known in general to
have a –action, there is always
an equivalence Therefore,
delta-discrete –spectra,
by letting
equal and
, give a way
of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete
–spectra.
Keywords
homotopy fixed point spectrum, discrete $G$–spectrum,
iterated homotopy fixed point spectrum
