Alphabeta MathFlat Smooth and Etale Morphisms
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Fields and Cyclotomic Extensions
- 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
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
Algebraic Geometry
- Affine Algebraic Sets and Coordinate Rings
- Algebraic Differentials Separability and Smooth Local Presentations
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Dimension Constructible Images and Dimensions of Fibres
- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Commutative Algebra
- Algebraic Zariski Main for Quasi-Finite Morphisms
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Dedekind Domains and Ideal Classes
- Depth and Cohen Macaulay Modules
- Flatness and Faithful Flatness
- Homogeneous Resultants and Projective Intersection Length
- Integral Extensions and Going Up
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Prime Spectra and Radicals
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Regular Local Rings and Homological Dimension
- Valuation Rings and Discrete Valuation Rings
- Zariski Topology on Prime Spectra
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Delta Functors and Universality
- Derived Functors
- Ext and Balanced Resolutions
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Tor Flatness and Global Dimension
- Universal Coefficients and Kunneth Theorems
- Yoneda Extensions and Homological Dimension
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Kahler Differentials Conormal Sequences and Infinitesimal Lifting
- Presheaves Sheaves Stalks and Sheafification
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
2 · Logical flowchart
3 · Items
Definition
Constructible subsets of a scheme
Definition
Flat morphism of schemes
Lemma
Affine-local flatness
Lemma
Flatness is stable under arbitrary base change
Lemma
Flatness is stable under composition
Definition
Faithfully flat scheme morphism
Lemma
A flat local map is faithfully flat
Theorem
Descent of vanishing along a faithfully flat morphism
Lemma
Generic freeness over a Noetherian domain
Theorem
Generic flatness for finite type morphisms over Noetherian integral bases
Lemma
Constructible images for finite-presentation affine maps
Lemma
Constructible subsets stable under generalisation are open in an affine spectrum
Definition
Open and universally open morphisms of schemes
Theorem
Flat finite-presentation morphisms are open
Lemma
Local fibre-dimension bound from polynomial quasi-finiteness
Lemma
Upper semicontinuity of proper fibre dimension
Definition
Smooth morphism of schemes
Definition
Relative dimension of a smooth morphism at a point
Theorem
Smoothness survives base change and composition
Theorem
Smooth morphisms are exactly the formally smooth locally finitely presented morphisms
Theorem
Relative Jacobian criterion with its presentation hypothesis
Theorem
Differentials of a smooth morphism
Lemma
Fibres of a smooth morphism are smooth
Definition
Étale morphism of schemes
Definition
Etale neighbourhoods and elementary etale neighbourhoods of a point
Theorem
Smooth maps have étale local affine-space form
Theorem
Étale equals flat and unramified in finite presentation
Theorem
Etale morphisms are the formally etale morphisms locally of finite presentation
Lemma
Étale stability
Theorem
Etale morphisms are universally open and quasi-finite at every point
Definition
Standard étale algebra
Theorem
Étale morphisms are locally standard étale
Theorem
Finite and finite type etale schemes over an algebraically closed field
Definition
The smooth locus of a morphism
Theorem
The smooth locus is open
Definition
The étale locus of a morphism
Theorem
The etale locus is open
Corollary
Classical and scheme smoothness over a perfect field
Lemma
Fibrewise injectivity lifts and leaves a flat cokernel over a Noetherian target
Lemma
A finite module with free fibre and flat base is free over a Noetherian target
Lemma
The flat locus of a finitely presented algebra is open
Lemma
Local flatness criterion for a module finite over a larger Noetherian local algebra
Lemma
Finite presentation data descend to Noetherian algebra and module stages
Lemma
A killed first Tor obstruction yields flatness after local Noetherian base change
Lemma
Flatness over a local filtered colimit appears at a Noetherian stage
Lemma
Noetherian fibrewise flatness for a module finite over the target
Lemma
Flatness over a polynomial chart from base and fibre flatness
Lemma
Dense relative-dimension strata in flat finitely presented fibres
Lemma
Lower semicontinuity of flat finitely presented fibre dimension
Theorem
Fibre dimension of proper flat finitely presented families
Remark
Flatness does not force isomorphic or smooth fibres
Lemma
Coprime polynomial factorisations lift after an etale localisation
Example
Open immersions are etale
Lemma
Finite neighbourhood of an isolated fibre point after elementary etale change
Lemma
Finite decomposition around isolated fibre points after an elementary étale change
Lemma
Structure sheaf of a quasi-compact quasi-separated morphism is affine-local
Lemma
Integral closure commutes with étale base change
Lemma
Integral quasi-coherent algebras over qcqs bases are unions of finite subalgebras
Lemma
An étale universally injective morphism is an open immersion
Lemma
Relative normalization and its finite-stage reduction
Lemma
Scheme Zariski Main factorization for separated quasi-finite morphisms
Theorem
A proper quasi-finite morphism is finite
Lemma
Fibrewise exact finite flat complexes lift over a Noetherian target
Lemma
A sharp dimension bound makes equations regular on an equidimensional Cohen-Macaulay fibre
Lemma
A regular sequence lowers dimension exactly in a Cohen-Macaulay local ring
Lemma
Local fibre dimension equals local ring dimension plus residue transcendence degree
Lemma
Fibrewise regular sequences persist openly in equidimensional Cohen-Macaulay fibres
Lemma
The highest positive homology of a depth-bounded finite complex has positive depth
Lemma
Reduction of an exact free complex by a nonzerodivisor stays exact above degree one
Lemma
A unit differential entry splits a contractible two-term summand
Lemma
Positive-degree exact free complexes split over a depth-zero local ring
Lemma
Associated-prime localizations detect elements and have depth zero
Lemma
Exact free complexes have the expected ranks and determinantal regular sequences
Lemma
Expected ranks and determinantal regular sequences force a free complex to be exact
Theorem
Buchsbaum-Eisenbud rank and grade criterion for exact free complexes
Lemma
Fibrewise exactness of a finite free complex is open in a flat Cohen-Macaulay family
Lemma
Syzygies of a base-flat module over a flat algebra stay base-flat and fibrewise exact
Sources
- The Stacks Project, Topology, Section 5.15 (tags 0059, 04ZC) and Morphisms of Schemes, Section 29.23
- The Stacks Project, Commutative Algebra, Section 10.29
- The Stacks Project, Morphisms of Schemes, Definition 29.25.1
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Definition 25.1.1
- The Stacks Project, Morphisms of Schemes, Section 29.25
- The Stacks Project, Commutative Algebra, Section 10.39 and Lemma 10.18.7
- The Stacks Project, Morphisms of Schemes, Sections 29.25-29.26
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapters 25-26
- The Stacks Project, Morphisms of Schemes, Sections 29.25 and 29.34-29.36
- The Stacks Project, Commutative Algebra, Lemma 10.18.7 and Section 10.39 (faithfully flat modules)
- The Stacks Project, Morphisms of Schemes, Section 29.26 (flat morphisms, descent)
- The Stacks Project, Commutative Algebra, Section 10.40 (faithfully flat descent)
- The Stacks Project, Commutative Algebra, Section 10.108 (generic flatness)
- The Stacks Project, Commutative Algebra, Lemma 10.118.1 (tag 051R)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Theorem 25.5.12
- The Stacks Project, Commutative Algebra, Section 10.29 (Theorem 10.29.10, tags 00F5-00F7)
- The Stacks Project, Morphisms of Schemes, Section 29.23
- The Stacks Project, Topology, Lemma 5.23.6 (tag 0903) and Section 5.19 (tags 0062, 0065)
- The Stacks Project, Commutative Algebra, Section 10.41 (tags 00HY, 00I0, 00I1)
- The Stacks Project, Morphisms of Schemes, Definition 29.24.1 (tag 01U0) and Section 29.24
- The Stacks Project, Morphisms of Schemes, Section 29.26 (flat morphisms)
- The Stacks Project, Morphisms of Schemes, Lemma 29.26.10 (tag 01UA) and Section 29.24 (tags 01U0, 01U1)
- The Stacks Project, Commutative Algebra, Lemmas 10.125.1-10.125.6 (tags 00QD-00QH)
- The Stacks Project, Morphisms of Schemes, Section 29.29 (tag 02FW)
- The Stacks Project, More on Morphisms, Section 37.30 (tags 05F6-0D4J)
- The Stacks Project, Morphisms of Schemes, Sections 29.28-29.30
- The Stacks Project, Morphisms of Schemes, Section 29.29 (morphisms and dimensions of fibres)
- The Stacks Project, Morphisms of Schemes, Section 29.34 (smooth morphisms, tags 01V4-01V9) and Section 29.24
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Sections 25.3 and 26.1
- The Stacks Project, Morphisms of Schemes, Section 29.35 (smooth morphisms: formal smoothness criterion)
- EGA IV, 17.5.2 (smooth iff formally smooth and locally of finite presentation)
- The Stacks Project, Morphisms of Schemes, Section 29.35 (smooth morphisms)
- H. Matsumura, Commutative Algebra, Ch. 6 (formal smoothness and the Jacobian criterion)
- The Stacks Project, Morphisms of Schemes, Section 29.32 (differentials, tags 01UN-02H4)
- The Stacks Project, Morphisms of Schemes, Sections 29.34-29.36 (smooth morphisms)
- The Stacks Project, Morphisms of Schemes, Sections 29.25-29.37 (fibres of smooth morphisms)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapter 25
- The Stacks Project, Morphisms of Schemes, Sections 29.25-29.37
- The Stacks Project, More on Morphisms, Definition 37.35.1 (tag 02LE) and Section 37.35
- The Stacks Project, Morphisms of Schemes, Section 29.37 (etale morphisms)
- The Stacks Project, Morphisms of Schemes, Section 29.34 (smooth morphisms are locally affine space over the base, tags 01V4-01V9)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Section 25.3
- The Stacks Project, Morphisms of Schemes, Sections 29.34-29.36 (etale morphisms, tag 02G4)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapter 26
- The Stacks Project, Morphisms of Schemes, Lemma 29.36.5 and Section 29.35 (tags 02G4, 02H9)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapter 26 (etale = formally etale + locally finitely presented)
- The Stacks Project, Morphisms of Schemes, Sections 29.25-29.37 (etale morphisms)
- The Stacks Project, Morphisms of Schemes, Section 29.36 (etale morphisms are open and quasi-finite, tag 02G4)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapter 26 (etale maps are open and quasi-finite)
- The Stacks Project, Algebra, Section 10.144 (tag 00UE), Definition 10.144.1 and Proposition 10.144.4
- The Stacks Project, Algebra, Section 10.144, Proposition 10.144.4 (standard étale local form)
- The Stacks Project, Morphisms of Schemes, Section 29.36 (etale morphisms, tag 02G4) and Algebra, Section 10.144
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapter 26 (etale morphisms) and Exercise 26.1.B
- The Stacks Project, Morphisms of Schemes, Section 29.34 (smooth morphisms, tags 01V4-01V9)
- The Stacks Project, Morphisms of Schemes, Lemma 29.36.6 and Section 29.36 (tag 02G4)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapter 26 (the etale locus is open)
- The Stacks Project, Morphisms of Schemes, Sections 29.25-29.37; Varieties, Section 33.12
- The Stacks Project, Algebra, Lemma 10.99.1 (tag 00ME), fibre-injective maps
- The Stacks Project, Algebra, Lemma 10.99.4 (tag 00MH), free fibre and flatness
- The Stacks Project, Commutative Algebra, Sections 10.99, 10.127-10.130 and Morphisms of Schemes 29.26
- The Stacks Project, Algebra, Theorem 10.129.4 (tag 00RC), flat locus of a finitely presented module
- The Stacks Project, Algebra, Lemma 10.129.3 (tag 00RB), openness of fibrewise exactness
- The Stacks Project, Algebra, Lemma 10.99.10 (tag 00ML), variant of the local criterion
- The Stacks Project, Algebra, Lemma 10.99.7 (tag 00MK), local criterion for flatness
- The Stacks Project, Algebra, Lemma 10.127.18 (tag 00R1), finite-presentation approximation
- The Stacks Project, Algebra, Lemma 10.127.13 (tag 00QX), local approximation with modules
- The Stacks Project, Algebra, Lemma 10.99.14 (tag 00MO), Tor-killing base-change criterion
- The Stacks Project, Algebra, Lemma 10.99.12 (tag 00MM), first Tor comparison
- The Stacks Project, Algebra, Lemma 10.99.13 (tag 00MN), second Tor comparison
- The Stacks Project, Algebra, Lemma 10.128.3 (tag 00R6), eventual flatness
- The Stacks Project, Algebra, Lemma 10.127.13 (tag 00QX), local approximation
- The Stacks Project, Algebra, Lemma 10.99.15 (tag 00MP), Noetherian fibrewise flatness criterion
- The Stacks Project, Algebra, Lemma 10.128.8 (tag 00R7), critère de platitude par fibres
- The Stacks Project, Algebra, Lemma 10.99.15 (tag 00MP), Noetherian case of the critère de platitude par fibres
- The Stacks Project, More on Morphisms, Section 37.22 (tags 045U, 054T): the Cohen-Macaulay locus of a flat, locally finitely presented morphism
- The Stacks Project, Commutative Algebra, Section 10.130 (tags 00RE, 00RH, 00RI, 00RL): openness of Cohen-Macaulay loci
- The Stacks Project, Commutative Algebra, Sections 10.99 and 10.128 (tags 00MI, 00R4): flatness criteria and miracle flatness
- The Stacks Project, More on Morphisms, Section 37.30, Lemma 37.30.5
- The Stacks Project, More on Morphisms, Section 37.30 (dimension of fibres)
- The Stacks Project, Commutative Algebra, Tag 00R2 (Lemma 10.6.3)
- The Stacks Project, Commutative Algebra, Lemma 10.143.13 and Section 10.143 (etale local factorization of a polynomial)
- The Stacks Project, Morphisms of Schemes, Section 29.36 (standard etale and the Jacobian criterion)
- The Stacks Project, Morphisms of Schemes, Lemma 29.36.1 (open immersions are etale, tag 02GJ)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapter 26 (open immersions are etale)
- The Stacks Project, Algebra, Lemma 10.145.2 (tag 00UJ): etale local structure of quasi-finite ring maps
- The Stacks Project, Algebra, Lemma 10.122.2 (tag 00PK): isolated points in fibres and quasi-finiteness
- The Stacks Project, Algebra, Lemma 10.145.1 (tag 00UI) and Definition 10.143.1 (tag 00U1)
- The Stacks Project, More on Morphisms, Section 37.41 (etale neighbourhoods) and Morphisms, Section 29.21
- The Stacks Project, More on Morphisms, Section 37.41 (étale neighbourhoods)
- The Stacks Project, Morphisms of Schemes, Section 29.11 (quasi-coherent sheaves and pushforwards)
- The Stacks Project, Morphisms of Schemes, Section 29.25 (flat morphisms)
- The Stacks Project, Commutative Algebra, Section 10.147 (étale ring maps and integral closure)
- The Stacks Project, Properties of Schemes, Lemma 28.23.13 (tag 0817) and Section 28.23
- The Stacks Project, Étale Morphisms of Schemes, Section 41.14 (tag 025F), Theorem 41.14.1 (tag 025G)
- The Stacks Project, More on Morphisms, Sections 37.41–37.43 (étale neighbourhoods and Zariski Main)
- The Stacks Project, More on Morphisms, Section 37.43 (Zariski Main Theorem)
- The Stacks Project, Morphisms of Schemes, Section 29.44 (finite morphisms)
- The Stacks Project, Algebra, Lemma 10.99.5 (tag 00MI), exact complexes and flat cokernels
- The Stacks Project, Algebra, Lemma 10.129.1 (tag 00R9), dimension bound and regular sequences
- The Stacks Project, Algebra, Section 10.129 (tag 00R8), dimension drop in Cohen-Macaulay fibres
- The Stacks Project, Algebra, Lemma 10.116.3 (tag 00P1), dimension at an affine point
- The Stacks Project, Algebra, Lemma 10.129.2 (tag 00RA), openness of regular sequences in fibres
- The Stacks Project, Algebra, Lemma 10.102.8 (tag 00N0), acyclicity lemma
- The Stacks Project, Algebra, Lemma 10.102.7 (tag 00MZ), reduction of exact free complexes
- The Stacks Project, Algebra, Lemma 10.102.2 (tag 00MT), unit-entry splitting
- The Stacks Project, Algebra, Lemma 10.102.3 (tag 00MY), depth-zero splitting
- The Stacks Project, Algebra, Lemma 10.63.19 (tag 0311), detection at associated localizations
- The Stacks Project, Algebra, Section 10.102 (tag 00MR), associated-prime step in the acyclicity criterion
- The Stacks Project, Algebra, Proposition 10.102.9 (tag 00N1), necessity direction
- The Stacks Project, Algebra, Proposition 10.102.9 (tag 00N1), sufficiency direction
- The Stacks Project, Algebra, Proposition 10.102.9 (tag 00N1), exactness criterion
- The Stacks Project, Algebra, Theorem 10.129.4 (tag 00RC), flat syzygy and fibre reduction step
- The Stacks Project, Algebra, Lemma 10.39.13 (tag 00HM), flatness in exact sequences