Alphabeta MathKahler Differentials Conormal Sequences and Infinitesimal Lifting — 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.
1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Modules, Exact Sequences, Projective and Injective Modules
- 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 Finite Abelian Groups
Algebraic Geometry
Category Theory
Commutative Algebra
Linear Algebra
Number Theory
Real Analysis
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Fibre Products Base Change and Scheme Theoretic Fibres
- 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
Differentials of k[x,y]
Example
Differentials of a plane hypersurface
Example
Differentials of dual numbers in both characteristics
Example
Finite separable extensions have zero Omega
Counterexample
A purely inseparable field has nonzero Omega
Counterexample
A conormal left map with nonzero kernel
Example
Dual-number vectors in affine space
Example
A closed point immersion is unramified
Counterexample
Zero Frobenius tangent map does not imply formal etaleness
Sources
- Stacks Algebra 10.131.14
- Vakil 22.2.3, p.575
- Stacks Algebra 10.131.9
- Vakil 22.2.12, p.579
- Vakil 22.2.7, pp.577-578
- Stacks Algebra 10.158.1
- Vakil 22.2.F, p.577
- Stacks Algebra 10.131.9 and 10.158.1
- Vakil 22.2.12-13, pp.579-580
- Stacks Morphisms 29.33; Vakil 22.2.18
- Stacks Morphisms 29.36.8
- Stacks Morphisms 29.35.13 warning and 29.33