ScalingStacks

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Proof. This is clear by Proposition 2.88. ∎

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In this chapter

Chapter overview

1. Introduction
2. Reminders on ultrametric functional analysis
2.1. Seminormed vector spaces
2.1.1. Basic constructions
2.1.2. Orthogonal basis
2.2. Banach algebra
2.2.1. Basic constructions
2.2.2. Spectrum
2.2.3. Continuous map
2.2.4. Spectral seminorm
2.2.5. Banach module
2.3. Affinoid algebras
2.3.1. Basic constructions
2.3.2. Algebraic structures: Noetherianity
2.3.3. Topological structures: the spectral norm
2.3.4. Affinoid space as locally ringed space
2.4. Spectral calculus
2.4.1. Holomorphic envelop
2.4.2. Holomorphic functional calculus
2.5. Analytification of scheme of finite type
2.5.1. Local situation
2.5.2. Global situation
3. Normed section algebra
3.1. Basic setting
3.2. Algebraic properties of normed section algebra
3.3. Spectrum of normed section algebra
3.4. Fubini-Study metrics
3.5. Dual unit disc bundle
3.6. Comparison of algebra norms
4. Geometric approximation
4.1. Localization of spectrum by affinoid domain covering
4.2. Localization of Banach algebra homomorphism
5. Algebraic approximation
5.1. Algebra norm induced by Fubini-Study metric
5.1.1. Case for $(\mathbb{P}^{d},\mathscr{O}(1))$
5.1.2. Case for general $(X,L)$
5.2. Algebra norm induced by asymptotic Fubini-Study metric
References

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