부산대학교 도서관

This article presents several structural improvements in the tool set used with the Morava change of rings theorem to attempt computations of stable homotopy groups of finite spectra. \par As usual, let $E_n$ denote the Lubin-Tate spectrum obtained via the Landweber exact functor theorem and let $G_n$ denote the profinite group of automorphisms of $E_n$ as a ring spectrum. So-called chromatic techniques imply that understanding the continuous $G_n$-action on $E_n \wedge X$ is an important step in trying to compute the stable homotopy of the finite spectrum $X$. To this end, in [Topology {\bf 43} (2004), no.~1, 1--47; MR2030586 (2004i:55012)] E. S. Devinatz\ and M. J. Hopkins constructed a homotopy fixed point spectrum $E_n^{hG}$ for $E_n$ with respect to a closed subgroup $G$ of $G_n$, which comes with a ``descent'' spectral sequence from certain continuous cohomology groups of $G$ to the stable homotopy of $E_n^{hG} \wedge X$. However, their construction seems imperfect in that it has no clear connection to any total right derived functor of a fixed points functor and the $G_n$-action on $E_n$ is not shown to be continuous. \par After a compilation of useful facts for this subject matter, the article begins by considering an appropriate choice of Quillen model category structure on ${\rm Spt}_G$, the category of $G$-discrete spectra for a profinite group $G$. Despite its origins in the language of sheaves of spectra [see J. F. Jardine, in {\it Axiomatic, enriched and motivic homotopy theory}, 29--68, Kluwer Acad. Publ., Dordrecht, 2004; MR2061851 (2005f:55004); S. A. Mitchell, in {\it Algebraic $K$-theory (Toronto, ON, 1996)}, 221--277, Amer. Math. Soc., Providence, RI, 1997; MR1466977 (99f:19002); P. G. Goerss, in {\it The Čech centennial (Boston, MA, 1993)}, 187--224, Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995; MR1320993 (96a:55008)], this model structure is shown to inherit the definition of cofibrations and weak equivalences from ${\rm Spt}_{\{e\}}={\rm Spt}$, the Bousfield-Friedlander category of spectra. As a consequence, the fixed points functor is seen to have a total right derived functor, which is the chosen definition of homotopy fixed points. This definition is shown to be consistent with several previously settled cases, including an instance of Thomason's hypercohomology spectra [as in R. W. Thomason, Ann. Sci. École Norm. Sup. (4) {\bf 22} (1989), no.~4, 675--677; MR1026753 (91j:14013)]. \par The definition of a continuous $G$-spectrum is given as the inverse limit of a tower of discrete $G$-spectra. Extending the results above to towers produces a total right derived functor, ${\rm R} \lim_i (?)_i^G\colon {\rm Ho}({\rm tow}({\rm Spt}_G)) \to {\rm Ho}({\rm Spt})$, used to define the homotopy fixed points for a continuous $G$-spectrum. After showing that $E_n$ is a continuous $G_n$-spectrum in the sense above, which relies on the machinery of generalized Moore spaces in Brown-Peterson homology as well as the construction of Devinatz-Hopkins, the homotopy fixed points with respect to closed subgroups $G < G_n$ are shown to be functorial in the argument $G$ in the sense that the construction produces a presheaf of spectra on the orbit category of $G_n$. Assuming appropriate Mittag-Leffler conditions, two descent spectral sequences are constructed for $L_{K(n)}(E_n \wedge X)^{hG}$, and they are shown to be similar to those of Devinatz-Hopkins if $X$ is a finite spectrum. Determining the exact relationship between these three descent spectral sequences is a topic of later work by this author.