Alphabeta MathFibre Products Base Change and Scheme Theoretic Fibres
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1 · Prerequisites
- Abelian Categories
- Affine Algebraic Sets and Coordinate Rings
- Affine Schemes and the Structure Sheaf
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Artinian Rings and Length
- Binary Operations, Monoids, Groups and Subgroups
- Cardinal Arithmetic, Cofinality and the Alephs
- Categories, Functors and Natural Transformations
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Compactness
- Compactness in Metric Spaces
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exactness and the Member Calculus
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Limits and Colimits
- Linear Independence, Bases and Dimension
- Localisation of Modules and Support
- Metric Spaces
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Polynomial Rings, the Division Algorithm and Roots
- Preadditive and Additive Categories and Biproducts
- Presheaves Sheaves Stalks and Sheafification
- Prime Spectra and Radicals
- Products Segre and Veronese Embeddings and Grassmannians
- Projective Algebraic Sets Projective Morphisms and Cones
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Roots, Rational Powers, and Classical Inequalities
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tensor Products of Modules
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Field of Fractions and Localisation
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Zariski Topology on Prime Spectra
2 · Logical flowchart
3 · Items
Definition
Fibre product of schemes
Lemma
Uniqueness of the fibre product
Theorem
Affine fibre products are spectra of tensor products
Lemma
Projections on primes, stalks and residue fields
Lemma
Restricting fibre products to open subschemes
Lemma
Gluing fibre products along open covers
Theorem
Existence of all scheme fibre products
Corollary
Products and initial and terminal S-schemes
Lemma
Symmetry, associativity and units
Definition
Base change of objects, morphisms and properties
Lemma
Iterated base change
Lemma
Presentations and localization under base extension
Lemma
Field-valued points and local-ring points
Definition
Scheme-theoretic fibre
Theorem
Coordinate ring of an affine fibre
Lemma
Points and topology of a fibre
Lemma
Stalks of the scheme-theoretic fibre
Lemma
Points of a fibre product via residue-field tensors
Lemma
Fibres after base change
Lemma
Affine charts after extension of the ground field
Definition
Geometric fibres and geometric points
Lemma
Independence of the chosen algebraic closure
Definition
Geometric properties of fibres
Lemma
Base change of immersions
Definition
Scheme-theoretic inverse images of subschemes
Lemma
Intersections of subschemes
Lemma
Local finiteness conditions under base change
Lemma
Quasi-compactness is local on the target and survives base change
Corollary
Finite type under base change and products over a field
Definition
Affine morphisms
Lemma
Affineness from a finite principal cover
Lemma
Affineness is local on the target
Lemma
Base change and composition of affine morphisms
Definition
The diagonal morphism
Lemma
The diagonal commutes with base change
Definition
The graph morphism over a base
Lemma
The graph is a pullback of the diagonal
Theorem
Classical products and scheme products
Lemma
Surjectivity survives arbitrary base change
Lemma
Immersions and affine localizations are monomorphisms
Remark
Why geometric properties differ from ordinary ones
Remark
What the underlying fibre set forgets
Sources
- Stacks 26.17.1
- Vakil 10.1.3 and 10.1.5
- Vakil 10.1.B; Stacks 26.17.2
- Stacks 26.17.2 and 26.17.5
- Stacks 26.17.3; Vakil proof 10.1.1 Step 1
- Vakil proof 10.1.1 Steps 2–5
- Vakil Theorem 10.1.1, complete proof; Stacks 26.17.4
- Vakil 10.1.1 and 10.1.A (empty gluing specialization)
- Vakil 10.1.3; proof 10.1.1 Step 1
- Stacks 26.18.1 and 26.18.3
- Vakil proof 10.1.1 Step 1; 10.3.C
- Vakil 10.2.A, B, F
- Stacks 26.13, paragraphs preceding 26.13.3 and its field-valued special case
- Stacks 26.18.4; Vakil 10.3.2
- Vakil 10.3.2; Stacks 26.18.4–5
- Stacks 26.18.5; Vakil 10.3.B
- Stacks 26.18.6
- Stacks 26.17.5
- Vakil 10.3.C
- Vakil 10.2.3
- Vakil 10.4.3 and table (Rc), (Ic), (Cc)
- Vakil 10.4.3 and scalar extension 10.2.3
- Vakil 10.4.3 and table preceding 10.4.K
- Stacks 26.17.6 and 26.18.2
- Stacks 26.17.7
- Vakil 10.2.C and H
- Vakil 10.4.B(e,g); Stacks 29.15.4 and 29.22.4
- Stacks 26.19.2–3
- Vakil 10.2.D and 10.4.B(f)
- Stacks 29.11.1
- Stacks 28.28.3, with elementary finite-equalizer proof
- Stacks 29.11.3(1) iff (2), Remark 29.11.4; 26.11.5–6
- Stacks 29.11.8–10
- Stacks 26.21 introductory definition
- Vakil proof 11.1.10, pp.230–231
- Vakil 11.1.17, p.232
- Vakil proof 11.1.18, diagram (11.1.18.1), p.232
- Vakil 10.1.3 and 10.4.E; local published affine product theorem
- Vakil 10.4.D
- Vakil 10.2.G
- Vakil 10.4.1–2 and 10.4.G
- Vakil 10.1.2 and 10.3.3; Stacks 26.18.5–6