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Lubin-Tate theory, character theory, and power operations. (English) Zbl 1476.55010

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 891-929 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: Chromatic homotopy theory decomposes the category of spectra at a prime \(p\) into a collection of categories according to certain periodicities. Lubin-Tate theory plays an important role in local arithmetic geometry and so it is not too surprising that other important objects from arithmetic geometry, such as the Drinfeld ring, that are closely related to the Lubin-Tate ring arise in the construction of the character map. An important ingredient in understanding the relationship between power operations and character theory is a result of Ando, Hopkins, and Strickland that gives an algebro-geometric interpretation of a special case of the power operation in terms of Lubin-Tate theory. The chapter considers Morava \(E\)-theory using the Landweber exact functor theorem, calculates the \(E\)-cohomology of finite abelian groups, describes the Goerss-Hopkins-Miller theorem, and also describes the resulting power operations on \(E\)-cohomology. It discusses the relationship between the power operations and the stabilizer group action.
For the entire collection see [Zbl 1468.55001].

MSC:

55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55S99 Operations and obstructions in algebraic topology
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology

Citations:

Zbl 1468.55001
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