Galois extensions of structured ring spectra. Stably dualizable groups
نویسندگان
چکیده
منابع مشابه
Realizability of Algebraic Galois Extensions by Strictly Commutative Ring Spectra
We discuss some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context ...
متن کاملRealisibility of Algebraic Galois Extensions by Strictly Commutative Ring Spectra
We describe some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups. We describe the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace or norm mapping constructed by Greenlees and May in the context of generalized Tate cohomology. ...
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Let p be an odd prime and F a field of characteristic different from p containing a primitive p\h root of unity. Assume that the Galois group G of the maximal p-extension of F has a finite normal series with abelian factor groups. Then the commutator subgroup of G is abelian. Moreover, G has a normal abelian subgroup with pro-cyclic factor group. If, in addition, F contains a primitive p2th roo...
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Theorem 1.1 (Kummer theory). Let m ∈ Z>0, and suppose that the subgroup μm(K) = {ζ ∈ K∗ : ζ = 1} of K∗ has order m. Write K∗1/m for the subgroup {x ∈ K̄∗ : x ∈ K∗} of K̄∗. Then K(K∗1/m) is the maximal abelian extension of exponent dividing m of K inside K̄, and there is an isomorphism Gal(K(K∗1/m)/K) ∼ −→ Hom(K∗, μm(K)) that sends σ to the map sending α to σ(β)/β, where β ∈ K∗1/m satisfies β = α.
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Let En be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E n whose coefficients are built from the coefficients of En and contain all roots of unity whose order is not divisible by p. For odd primes p we show that E n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that the...
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ژورنال
عنوان ژورنال: Memoirs of the American Mathematical Society
سال: 2008
ISSN: 0065-9266,1947-6221
DOI: 10.1090/memo/0898