Alphabeta MathKahler Differentials Conormal Sequences and Infinitesimal Lifting
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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
- 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
Commutative Algebra
Homological Algebra
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
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
Derivation of an algebra
Definition
Universal Kähler differential module
Theorem
Existence and generators of Kähler differentials
Corollary
Derivations are maps out of Ω
Lemma
Polynomial differentials are free
Theorem
Conormal exact sequence for an algebra quotient
Corollary
Jacobian presentation of Ω
Theorem
Transitivity sequence for differential modules
Lemma
Kähler differentials commute with localization
Lemma
Kähler differentials commute with scalar base change
Definition
Sheaf of relative Kähler differentials
Theorem
Universal property of relative differential sheaves
Lemma
The sheaf attached to a module on an affine scheme
Lemma
Affine charts recover the algebraic module of differentials
Theorem
Conormal sequence for a closed immersion
Theorem
Transitivity sequence for schemes
Lemma
Relative differentials commute with scheme base change
Definition
Relative cotangent and tangent spaces
Theorem
Cotangent space at a rational point
Theorem
Tangent vectors as dual-number points
Lemma
Differential of an S-morphism
Definition
Formally unramified morphism
Definition
Formally smooth morphism
Definition
Formally etale morphism
Lemma
The diagonal ideal modulo its square is Omega
Theorem
Formal unramifiedness iff Omega vanishes
Definition
Unramified morphism
Theorem
An unramified morphism has an open diagonal
Lemma
Finite-type field extensions with zero Ω
Lemma
Unramified residue extensions are finite separable
Definition
Relative differential-rank condition
Remark
The conormal sequence is only right exact
Remark
Differential rank alone does not prove smoothness
Sources
- Stacks Algebra 10.131.1
- Vakil §22.2.17, p.582
- Stacks Algebra 10.131.2–3
- Vakil §22.2.2, p.575
- Stacks Algebra 10.131.3
- Stacks Algebra 10.131.14
- Vakil 22.2.3, p.575
- Stacks Algebra 10.131.9
- Vakil 22.2.12, pp.579–580
- Stacks Algebra 10.131.9, 14–15
- Vakil §22.2.3 and §22.2.12
- Stacks Algebra 10.131.7
- Vakil 22.2.9–11, pp.578–579
- Stacks Algebra 10.131.8
- Vakil §22.2.L, pp.583–584
- Stacks Algebra 10.131.12
- Vakil §22.2.K, p.583
- Stacks Modules 17.28.4 and 17.28.10, tags 08TD and 08RT
- Stacks Morphisms 29.33.1 and 29.33.5, tags 01UQ and 01UT
- Vakil §22.2.20, pp.584–585
- Stacks Modules, Lemma 17.28.4 and Definition 17.28.10, tags 08TD and 08RT
- Stacks Morphisms, Lemma 29.33.2, tag 01UR
- Stacks Schemes, Lemma 26.7.1 (tag 01I7)
- Stacks Modules, Definition 17.10.1 (tag 01BE), Lemma 17.10.5 (tag 01BH) and Definition 17.10.6 (tag 01BI)
- Stacks Morphisms, Lemma 29.33.5, tag 01UT
- Stacks Morphisms, Lemma 29.33.3, tag 01US
- Vakil 22.2.20, pp.584–585
- Stacks Morphisms, Lemma 29.33.15 (tag 01UT)
- Vakil 22.2.12 and 22.2.15, pp.579-581
- Stacks Morphisms, Lemma 29.33.9 (tag 01UX)
- Vakil 22.2.9-11, pp.578-579
- Stacks Morphisms, Lemma 29.33.10 (tag 01UY)
- Vakil 22.2.K, p.583
- Stacks Morphisms, Sections 29.33 and 29.36
- Vakil 22.2.18, pp.582-583
- Stacks Algebra, Lemma 10.131.10 (tag 00RW)
- Stacks Morphisms, Section 29.33 and Stacks Properties of Schemes, Section 28.16
- Stacks Morphisms, Lemma 29.33.8 (tag 01UW)
- Stacks Algebra, Definition 10.148.1 (tag 00UM) and Stacks More on Morphisms, Section 37.6 (tag 02G3)
- Stacks Algebra, Definition 10.138.1 (tag 00TH) and Stacks More on Morphisms, Section 37.11
- Stacks Algebra, Definition 10.150.1 (tag 00U7) and Stacks More on Morphisms, Section 37.7
- Stacks Algebra, Lemma 10.131.13 (tag 00RV)
- Vakil 22.2.20, p.584
- Stacks Algebra, Lemma 10.148.3 (tag 00UO) and Stacks More on Morphisms, Lemma 37.6.7
- Stacks Morphisms, Definition 29.36.1 (tag 02G4) and Lemma 29.36.2
- Stacks Morphisms, Lemma 29.36.13 (tag 02GE) and Stacks Algebra, Lemma 10.151.4
- Stacks Algebra, Lemma 10.158.1 (tag 090W) and Lemma 10.151.5 (tag 00UW)
- Stacks Algebra, Lemma 10.151.5 (tag 00UW) and Stacks Morphisms, Lemma 29.36.12 (tag 02G8)
- Stacks Morphisms 29.35.12-13 and the 29.35.14 warning
- Vakil 22.2.12-13, pp.579-580
- Stacks Morphisms 29.35.13 (tag 02G2) and the warning following it
- Stacks Morphisms, Lemma 29.36.2: unramifiedness and vanishing differentials
- Stacks Algebra 10.137.1, smoothness of a ring map via finite presentation and the naive cotangent complex