Families of Berkovich spaces

Introduction
1. First step: flatness in the Berkovich setting
2. Loci of validity
3. About our proofs
Acknowledgements
Chapter 1. Background material
1.1. Prerequisites and basic conventions
1.2. Analytic geometry: basic definitions
1.3. Coherent sheaves, Zariski topology and closed immersions
1.4. Dimension theory
1.5. Irreducible components
Chapter 2. Algebraic properties in analytic geometry
2.1. Analytification of schemes, algebraic properties of analytic rings
2.2. A rather abstract categorical framework
2.3. Formalisation of algebraic properties
2.4. Validity in analytic geometry
2.5. Fibers of coherent sheaves
2.6. Ground field extension
2.7. Complements on analytifications
Chapter 3. Germs, Temkin's reduction and -strictness
3.1. -strictness
3.2. Analytic germs
3.3. Around graded Riemann-Zariski spaces
3.4. Temkin's construction
3.5. Temkin's reduction and -strictness
Chapter 4. Flatness in analytic geometry
4.1. Naive and non-naive flatness
4.2. Algebraic flatness versus analytic flatness
4.3. The flat, (locally) finite morphisms
4.4. Naive flatness is not preserved by base change
4.5. Analytic flatness has the expected properties
Chapter 5. Quasi-smooth morphisms
5.1. Reminders about the sheaf of relative differentials
5.2. Quasi-smoothness: definition and first properties
5.3. Quasi-smoothness, flatness and fiberwise geometric regularity
5.4. Links with étale and smooth morphisms
5.5. Transfer of algebraic properties
Chapter 6. Generic fibers in analytic geometry
6.1. Preliminary lemmas
6.2. Relative polydiscs inside relative smooth spaces
6.3. Local rings of generic fibers
Chapter 7. Images of morphisms: local results
7.1. Maps between Riemann-Zariski Spaces: the non-graded case
7.2. Maps between Riemann-Zariski spaces: the general case
7.3. Applications to analytic geometry
7.4. Complement: around quantifier elimination in the theory acvf
Chapter 8. Dévissages à la Raynaud-Gruson
8.1. Universal injectivity and flatness
8.2. Dévissages: definition and existence
8.3. Flatness can be checked naively in the inner case
8.4. The relative CM property
Chapter 9. Quasi-finite multisections and images of maps
9.1. Flat, quasi-finite multisections of flat maps
9.2. Images of maps
Chapter 10. Constructible loci
10.1. Constructibility in analytic spaces
10.2. The diagonal trick
10.3. The flat locus
10.4. The fiberwise closure of a locally constructible set
10.5. The fiberwise exactness locus
10.6. Regular sequences
10.7. The main theorem
Chapter 11. Algebraic properties: target, fibers and source
11.1. A weak ``dominant factorization theorem''
11.2. The axiomatic setting
11.3. The main theorem
Appendix A. Graded commutative algebra
A.1. Basic definitions
A.2. Graded linear algebra
A.3. Graded algebras and graded extensions
A.4. Graded valuations
Index of notation
Index
Bibliography