Foundations of the theory of phi Gamma modules over the Robba ring 1st edition by Maria Heuß, Prof Annette, Huber Klawitte

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Author:      Maria Heuß, Prof. Dr. Annette, Huber-Klawitte

Foundations of the Theory of φvarphiφ-Gamma Modules over the Robba Ring (1st Edition) by Maria Heuß, Prof. Annette Huber, and Klawitte is a scholarly work that focuses on the development of a specialized area within p-adic analysis and algebraic geometry. Specifically, this book investigates the theory of φvarphiφ-Gamma modules, which play a crucial role in the study of p-adic Hodge theory and the structure of the Robba ring.

The Robba ring, which is fundamental in the study of p-adic differential equations, is examined in this work in conjunction with φvarphiφ-modules, which are modules equipped with a Frobenius endomorphism. These structures arise naturally in the context of crystalline cohomology and play a significant role in understanding p-adic representations.

Book Description:

“Foundations of the Theory of φvarphiφ-Gamma Modules over the Robba Ring” introduces the foundational concepts required to study and understand the theory of φvarphiφ-modules and ΓGammaΓ-modules, especially in relation to the Robba ring. The Robba ring itself is a tool used in the study of p-adic differential equations and p-adic analytic functions, while φvarphiφ-modules and ΓGammaΓ-modules provide a framework for analyzing various structures in p-adic geometry.

This book is divided into several key parts:

• Introduction to the Robba Ring: The authors provide a thorough background on the Robba ring, its properties, and its significance in the context of p-adic analysis.

• The Theory of φvarphiφ-Modules: A detailed exploration of φvarphiφ-modules, including their construction, properties, and importance in modern p-adic geometry. This section includes foundational results and key examples of φvarphiφ-modules over the Robba ring.

• φvarphiφ-Gamma Modules: The book delves into the interaction between φvarphiφ-modules and ΓGammaΓ-modules, exploring how these structures work together to provide insight into the theory of p-adic representations. The authors also investigate the role these modules play in crystalline cohomology.

• Applications and Advanced Topics: Building on the basic theory, the book looks at applications of φvarphiφ-Gamma modules in higher-dimensional p-adic Hodge theory, as well as their connection to Fontaine’s theory of (φ,Γ)( varphi, Gamma)(φ,Γ)-modules.

• Mathematical Results and Techniques: Detailed proofs of fundamental theorems, as well as advanced methods in the theory of φvarphiφ-modules, are provided. The authors use rigorous mathematical tools to establish the key results.

Target Audience:

This book is intended for advanced graduate students, researchers, and academics working in the fields of algebraic geometry, p-adic analysis, and number theory. It is particularly useful for those interested in p-adic Hodge theory, crystalline cohomology, and the study of p-adic representations. The book assumes familiarity with basic algebraic geometry, p-adic analysis, and module theory.

Foundations of the theory of phi Gamma modules over the Robba ring 1st Table of contents:

1. Background on p-adic Analysis
1.1 Norms and valuations
1.2 Fréchet spaces
1.3 Definition of the p-adic numbers
1.4 The completion of the algebraic closure of Qpmathbb{Q}_pQp​
1.5 Laurent series of p-adic numbers
1.6 The composition of Laurent series
1.7 The cyclotomic character

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2. φvarphiφ-modules
2.1 Modules over a ring AAA
2.2 Definition of a φvarphiφ-module
2.3 The category of φvarphiφ-modules
2.4 The tensor product of φvarphiφ-modules
2.5 Étale φvarphiφ-modules
2.6 The functor α∗:MA1→MA2alpha^* : mathcal{M}_{A_1} to mathcal{M}_{A_2}α∗:MA1​​→MA2​​
2.7 Fontaine’s (φ,Γ)(varphi, Gamma)(φ,Γ)-modules

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3. The Robba Ring RRR
3.1 Convergent Laurent series in one variable
3.2 The Fréchet topology on L[0,r]L[0, r]L[0,r]
3.3 Definition of the Robba ring

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4. Properties of the Robba Ring
4.1 The Frobenius operator φvarphiφ and the action of ΓGammaΓ on RRR
4.2 The decomposition of elements of the Robba ring
4.3 The logarithm function on x+x^+x+ and xxx
4.4 The differential operator $\partial :R\to R$
4.5 The residue of an element of RRR

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5. (φ,Γ)(varphi, Gamma)(φ,Γ)-modules over RRR
5.1 Rank 1 (φ,Γ)(varphi, Gamma)(φ,Γ)-modules
5.2 Cohomology for (φ,Γ)(varphi, Gamma)(φ,Γ)-modules over RRR
5.3 The isomorphism Ext1(R,M)≅H1(M)text{Ext}^1(R, M) cong H^1(M)Ext1(R,M)≅H1(M)
5.4 The cohomology group H0(R(Γ))H^0(R(Gamma))H0(R(Γ))
5.5 The characterization of all (φ,Γ)(varphi, Gamma)(φ,Γ)-modules of rank 1 over RRR

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