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Friday, May 1, 2020 | History

5 edition of Galois extensions of structured ring spectra found in the catalog.

Galois extensions of structured ring spectra

John Rognes

# Galois extensions of structured ring spectra

## by John Rognes

Written in

Subjects:
• Galois theory,
• Ring extensions (Algebra),
• Homology theory,
• Homotopy theory,
• Commutative algebra

• Edition Notes

Includes bibliographical references (p. 131-132) and index.

Classifications The Physical Object Statement John Rognes. Series Memoirs of the American Mathematical Society -- no. 898 LC Classifications QA211 .R64 2008 Pagination vii, 137 p. : Number of Pages 137 Open Library OL18500212M ISBN 10 9780821840634 LC Control Number 2007060583

This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of E ∞ –ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus by: The Hirzebruch genus of complex-oriented manifolds associated to Euler’s Γ-function lifts to a homomorphism of ring-spectra associated to a family of deformations of the Dirac operator, parametrized by the homogeneous space Sp/U. Thom Spectra and Bousfield localization (Piotr Pstragowski) Infinite loop space machines and K-theory (Alyson Bittner) (DG and simplicial) rings and structured ring spectra (Magdalena Kedziorek) Tuesday: Galois extensions of structured ring spectra (Bregje Pauweis) Picard and Brauer groups (Drew Heard).

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### Galois extensions of structured ring spectra by John Rognes Download PDF EPUB FB2

The author introduces the notion of a Galois extension of commutative $$S$$-algebras ($$E_\infty$$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg–Mac Lane spectra Galois extensions of structured ring spectra book commutative rings, real and complex topological $$K$$-theory, Lubin–Tate spectra and.

by Friedhelm Waldhausen, that structured ring spectra are an unavoidable gener-alization of discrete rings, with arithmetic properties captured by their algebraic K-theory, (2) the presumption that algebraic K-theory will satisfy an extended form of the ´etale- and Galois descent foreseen by.

Galois extensions of structured ring spectra Article in Memoirs of the American Mathematical Society () March with 16 Reads How we measure 'reads'Author: John Rognes.

Galois extensions of structured ring spectra 1. Introduction 2. Galois extensions in algebra 3. Closed categories of structured module spectra 4.

Galois extensions in topology 5. Examples of Galois extensions 6. Dualizability and alternate characterizations 7.

Galois theory I 8. Pro-Galois extensions and the Amitsur complex 9. Galois extensions of structured ring spectra -- 1. Introduction -- 2. Galois extensions in algebra -- 3. Closed categories of structured module spectra -- 4.

Galois extensions in topology -- 5. Examples of Galois extensions -- 6. Dualizability and alternate characterizations 7. Galois theory I -- 8. Pro-Galois extensions and the Amitsur complex. Galois theory of commutative ring spectra is a relatively new eld of mathe-matics, introduced in by Rognes in his article Galois extensions of struc-tured ring spectra [Rog08a].

It is inspired by the de nition of Galois extension on commutative rings by Auslander and Goldman [AG60], and motivated by.

Abstract: We introduce the notion of a Galois extension of commutative S-algebras (E_infty ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological K-theory, Lubin-Tate spectra and cochain by: G-Galois extension in the sense of Rognes.

This notion is due to J. Rognes (Rognes ).Let A be an E ∞-ring with an action of a finite group G and B = A hG its invariant subring. Then B → A (the map of B-algebras in E ∞-sense) is said to be a G-Galois extension if the natural map ⊗ → ∏ ∈ (which generalizes ⊗ ↦ (()) in the classical setup) is an equivalence.

In mathematics, a highly structured ring spectrum or ∞-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory.A commutative version of an ∞-ring is called an ∞ originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory.

We introduce the notion of a Galois extension of commutative S-algebras (E∞ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological K-theory, Lubin-Tate spectra and cochain S-algebras.

Structured Ring Spectra - TNG. 1st to 5th AugustHamburg, Germany. website. about structured ring spectra. This is the next in the informal series of conferences on Structured Ring Spectra in GlasgowBonn and Banff The conference will begin Monday morning and end Friday evening.

Posted at Febru PM UTC. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. and see also the answers to the MO question Is there a Galois correspondence for ring extensions. Ferrero; A. Paques. Galois Theory of Commutative Rings Revisited.

Contributions to Algebra and Geometry, Galois extensions of structured ring spectra Actually, KU is a commutative KO-algebra spectrum. Complex conjugation gives rise to a C 2-action on KU with homotopy xed points KO.

In a suitable sense KU is unrami ed over KO: KU ^ KO KU ’KU KU. Rognes ’ KU is a C 2-Galois extension of KO. De nition (Rognes ’08) (up to co brancy issues. In the setting of structured ring spectra, however, there are additional morphisms of nonconnective ring spectra that have formal properties similar to those of etale morphisms, though they are not etale on homotopy groups.

The faithful Galois extensions of Rognes [Rog08] are key examples here. [GH04b] P. Goerss and M. Hopkins. Moduli spaces of commutative ring spectra. In Structured ring spectra, volume of London Math. Soc. Lecture Note Ser., pages – Cambridge Univ.

Press, Cambridge, [Gre92] Cornelius Greither. Cyclic Galois extensions of commutative rings, volume of Lecture Notes in Mathematics. This article introduces a theory of homotopic Hopf-Galois extensions in a monoidal category with compatible model category structure that generalizes the case of structured ring spectra.

We prove analogs of faithfully flat descent and Galois descent for categories of modules over $$E_{\infty }$$-ring spectra using the $$\infty$$-categorical Barr-Beck theorem proved by particular, faithful G-Galois extensions are shown to be of effective descent for this we study the category of ER(n)-modules, where ER(n) is the $$\mathbb {Z}/2$$-fixed points under Cited by: 2.

For separable extensions of ring spectra we would like to prove the same statement as was proved for Galois extensions in the previous section.

From the ideological point of view this is the expected statement because as in algebra one should expect that any commutative separable extension embeds into a $$G$$ -Galois one, for some $$G$$.Author: Stanislaw Betley. This book aims to make the general theory of field extensions accessible to any reader with a modest background in groups, rings and vector spaces.

Galois theory is regarded amongst the central and most beautiful parts of algebra and its creation marked the culmination of Cited by: BACKGROUND ON GALOIS EXTENSIONS GALOIS EXTENSIONS IN HOMOTOPY THEORY DEFINITION (ROGNES) A map ˚: R!S of commutative ring spectra is G-Galois if G acts on S and ˚induces an equivalence R ’ShG, and the natural map S ^ R S!F(G +;S) is an equivalence.

GALOIS CORRESPONDENCE [ROGNES] Suppose R!S is a faithful G-Galois extension of commutative ring. Moduli Problems for Structured Ring Spectra, ≥ Galois extensions of structured ring spectra, John Rognes: The Davis-Mahowald spectral sequence, slides, An untitled book project about symmetric spectra, Stefan Schwede: Lectures on equivariant stable homotopy theory.

Structured ring spectra, London Math. Soc. Lecture Note Ser.– [Ro] J. Rognes, Galois extensions of structured ring spectra, preprint (ArXiv.

THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS MARK BEHRENS1 AND DANIEL G. DAVIS2 Abstract.

Let E be a k-local pro nite G-Galois extension of an E1-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful. The assertion is Lemma in Rognes's "Galois extensions of structured ring spectra" available on the fact, the $K(n)$ spectra do not even admit \$E_2.

Advancing research. Creating by: The Mathematical Sciences Research Institute (MSRI), founded inis an independent nonprofit mathematical research institution whose funding sources include the National Science Foundation, foundations, corporations, and more than 90 universities and institutions.

The Institute is located at 17 Gauss Way, on the University of California, Berkeley campus, close to Grizzly Peak, on the. At first, there are many things that are unclear, but it is similar to being immersed in a language.

Please direct questions, comments or concerns to feedback inspirehep. Jacob Lurie, Derived Algebraic Geometry. Galois extensions of structured ring spectra. Part 1, Proc. In particular, the construction of irreducible polynomials and the normal basis of finite fields are included.

The essentials of Galois rings are also presented. This invaluable book has been written in a friendly style, so that lecturers can easily use it as a text and students can use it for by: Idea. The terminology (commutative) ring spectrum refers either to a (commutative) monoid in the stable homotopy category regarded as monoidal category via the smash product of spectra, or to the richer structure of a monoid in a model structure for spectra equipped with a symmetric smash product of spectra.

In the first case a ring spectrum is a spectrum equipped with a unit and product. 1-ring spectra and the X(n) were only known to be E 2.

It seemed like a shot in the dark, but rather than try to recover everything Rognes had done, but for E n-ring spectra, I went to Google and typed in \Hopf-Galois extensions of associative ring spectra." To my indescribable delight, I found that Fridolin Roth, a student of Birgit Richter.

tensions. In fact, every Galois extension with non-abelian Galois group has at least two Hopf Galois structures, one by the group ring of the Galois group, the other by the Hopf algebra H = L[], where acts on Lvia the Galois action and on by conjugation (a non-trivial ac-tion if is non-abelian).

They showed that for nite Galois extensions. In this thesis we will consider Galois closures for monogenic degree-4 ring extensions. We will start by giving the de nition of a G-closure for a degree-n ring extension as in O.

Biesel’s PhD thesis [1], where G S n. This de nition generalizes classical nite Galois Theory, with the property of having a G-closure corresponding to having the. Galois extensions of structured ring spectra.

Stably dualizable groups - John Rognes: MEMO/ Limit theorems of polynomial approximation with exponential weights - Michael I. Ganzburg: MEMO/ The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra - Michael Kapovich, Bernhard Leeb and John J.

Abstract: Let E be a k-local profinite G-Galois extension of an E_infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite by: 1.

Galois extensions for separable group rings 3 Galois Extensions Let R be a ring with 1, G a group, and RG the group ring of G over R with center RG is an Azumaya shall show that RG is not a Galois extension of (RG)G with Galois group G where G is the innerautomorphism group of RG induced by the elements of er, let K be a proper subgroup of G with an invertible Author: George Szeto, Lianyong Xue.

Galois extensions of structured ring spectra J. Rognes, in Memoirs of the American Mathematical Society, vol.no. (March ) Addendum: A Galois extension that is not faithful Correction: A not-necessarily commutative map Stably dualizable groups.

lois extensions \from the bottom" { for a xed eld F, Proposition tells us how to \produce" all nite Galois extensions of the form K=F. Below we will obtain a simple characterization of nite Galois extensions \from the top", called Artin’s lemma.

Lemma Let K=F be a separable extension, ans suppose that there. The workshop consists of four series of lectures on Structured Ring Spectra: Bill Dwyer (Notre Dame University): "Morita theory for ring spectra" Gerd Laures (Universität Bochum): "K(1)-local topological modular forms" John Rognes (Oslo University): "Galois theory for S-algebras".

ring spectra and applications Akhil Mathew Applications These ideas have several applications to the understanding of certain invariants of structured ring spectra.

1 A new point of view on Rognes’s Galois extensions [8] and a formulation in terms of axiomatic Galois theory (as well as several computations of Galois groups) in [5]. Proof: Suppose F 3 is the splitting eld of f2F 1[x].This means that F 3 is the smallest eld containing F 1 and all the roots of f.

Then f2F 2[x] since F 1 F 2 and therefore F 3 is the smallest eld containing F 2 and all the roots of f.:foorP De nition For elds F Ede ne aut(EjF) to be the set of all automorphisms. John Rognes, Galois extensions of structured ring spectra.

J.F. Jardine, Diagrams and torsors. January Submissions ; Daniel Dugger, Spectral enrichments of model categories. Michael Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity.We use a characterization of one-sided Galois extension for bialgebroids in terms of the corresponding one-sided D2 property of extensions in [10, Theorem ] or Theorem in this paper, which then yields endomorphism ring theorems for D2 extensions and Ga- lois extensions: viz., a right D2 (respectively Galois) extension has left D2 (Galois Cited by: Nonclassical Hopf-Galois structures can provide a variety of viewpoints on a given nite separable extension of elds L=K.

We will focus on Galois extensions of p-adic elds. De nition If H is a K-Hopf algebra then we say that H gives a Hopf-Galois Structure on the extension L=K, or that L is an H-Galois extension of K, if: L is an H-module algebra.