Alphabeta MathFlat Smooth and Etale Morphisms — Examples
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 9 results · all verified · 2 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 7 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
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
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
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Dedekind Domains and Ideal Classes
- Depth and Cohen Macaulay Modules
- Flatness and Faithful Flatness
- 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
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Flat Smooth and Etale 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
Example
Polynomial rings are flat and smooth
Example
The square-root standard etale chart
Counterexample
A flat family with a nodal special fibre is not smooth at the node
Counterexample
The affine line is smooth but not etale
Counterexample
Unramified of finite presentation does not imply flat or etale
Counterexample
Flat and finite type is not open without finite presentation
Example
Finite field extensions and etaleness
Counterexample
Frobenius on the affine line is finite flat but not smooth
Example
The family xy=t
Sources
- The Stacks Project, Morphisms of Schemes, Sections 29.25 and 29.34-29.35
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapters 25-26
- The Stacks Project, Algebra, Section 10.143 and Morphisms of Schemes, Section 29.36 (standard etale and the square-root chart)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapter 26 (the T^2-a square-root chart)
- The Stacks Project, Morphisms of Schemes, Section 29.25 (flatness) and Section 29.34 (smoothness)
- J. S. Milne, Algebraic Geometry v6.10, §4b Definition 4.9 and Example 4.10
- The Stacks Project, Morphisms of Schemes, Section 29.36 (etale versus smooth of relative dimension zero, tag 02G4)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapter 26 (the affine line is smooth but not etale)
- The Stacks Project, Morphisms of Schemes, Section 29.36 and Lemma 29.36.7 (unramified does not imply flat)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapter 26 (closed immersions are unramified but usually not flat)
- The Stacks Project, Morphisms of Schemes, Section 29.26 (flat morphisms)
- The Stacks Project, Commutative Algebra, Tag 00R2 (Lemma 10.6.3: finite presentation and kernels of surjections)
- The Stacks Project, Commutative Algebra, Section 10.21 (idempotents and connected components)
- The Stacks Project, Morphisms of Schemes, Section 29.36 (tag 02G4) and Algebra, Section 10.143 (standard etale)
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapter 26 (finite etale k-schemes and separable extensions)
- The Stacks Project, Morphisms of Schemes, Sections 29.25 and 29.34-29.36
- Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Chapters 25-26 (relative Frobenius)
- The Stacks Project, Morphisms of Schemes, Sections 29.25 and 29.34-29.36 (flatness, smoothness, Jacobian criterion)