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Delta-discrete $G$-spectra and iterated homotopy fixed points | | Daniel G. Davis
; | | Date: |
14 Jun 2010 | | Abstract: | Let G be a profinite group with finite virtual cohomological dimension and
let X be a discrete G-spectrum. If H and K are closed subgroups of G, with H
normal in K, then, in general, the K/H-spectrum X^{hH} is not known to be a
continuous K/H-spectrum, so that it is not known (in general) how to define the
iterated homotopy fixed point spectrum (X^{hH})^{hK/H}. To address this
situation, we define homotopy fixed points for delta-discrete G-spectra and
show that the setting of delta-discrete G-spectra gives a good framework within
which to work. In particular, we show that by using delta-discrete K/H-spectra,
there is always an iterated homotopy fixed point spectrum, denoted
(X^{hH})^{h_delta K/H}, and it is just X^{hK}. Additionally, we show that for
any delta-discrete G-spectrum Y, (Y^{h_delta H})^{h_delta K/H} simeq
Y^{h_delta K}. Furthermore, if G is an arbitrary profinite group, there is a
delta-discrete G-spectrum {X_delta} that is equivalent to X and, though X^{hH}
is not even known in general to have a K/H-action, there is always an
equivalence ((X_delta)^{h_delta H})^{h_delta K/H} simeq
(X_delta)^{h_delta K}. Therefore, delta-discrete L-spectra, by letting L
equal H, K, and K/H, give a way of resolving undesired deficiencies in our
understanding of homotopy fixed points for discrete G-spectra. | | Source: | arXiv, 1006.2762 | | Services: | Forum | Review | PDF | Favorites | |
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