Alphabeta MathAlgebraic Differentials Separability and Smooth Local Presentations
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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 Finite Abelian Groups
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
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
Universal algebraic differentials and A-derivations
Lemma
Universal property of algebraic differentials
Lemma
Differentials of a polynomial quotient and the Jacobian cokernel
Lemma
Localization, base change and functoriality of differentials
Lemma
Transitivity sequence for differentials
Definition
Separating transcendence basis and separably generated extensions
Theorem
Finitely generated extensions of a perfect field are separably generated
Theorem
Differentials of a separably generated field extension
Lemma
Separable residue and the cotangent sequence of a local algebra
Theorem
Extension of scalars of a scheme along a field extension
Definition
Geometrically regular algebras and geometrically regular fibres
Definition
Standard smooth presentations and locally standard smooth maps
Lemma
Base change of standard smooth presentations
Lemma
Invertible Jacobian minor gives regular parameters in a polynomial fibre
Lemma
Local flatness criterion by regular parameters
Lemma
Standard smooth algebras are finitely presented and flat
Lemma
Fibres of standard smooth algebras are regular of relative dimension
Lemma
Regularity ascends and descends along a flat local homomorphism
Lemma
Separable generation after finite purely inseparable extensions
Lemma
Field tests for geometric regularity
Theorem
Regular algebras over a perfect field are geometrically regular
Theorem
Jacobian criterion and openness of the regular locus over a perfect field
Lemma
Flat maps with geometrically regular fibres have standard smooth local presentations
Theorem
Locally standard smooth iff flat with geometrically regular fibres
Theorem
Base change and composition of standard smooth presentations
Theorem
Submersion criterion for locally standard smooth morphisms
Example
Base change of an inseparable field extension is a thickening
Sources
- Stacks Algebra 10.131.1–2
- Vakil §22.2.2, p.575
- Stacks Algebra 10.131.3
- Vakil §22.2.17, pp.582–583
- Stacks Algebra 10.131.9, 14–15
- Vakil §§22.2.3, 22.2.12, pp.575, 579
- Stacks Algebra 10.131.8, 12
- Vakil §§22.2.K–L, pp.583–584
- Stacks Algebra 10.131.7
- Vakil §§22.2.9–11, pp.577–579
- Stacks Algebra 10.42.1, 3
- Stacks Algebra 10.44.1–2 and 10.45.2
- Stacks Algebra 10.131.7–8 and 10.44.1–2
- Vakil §22.2.M, p.584
- Stacks Algebra 10.131.9–10 and 10.140.4–5
- Vakil §§22.2.18 and 22.3.9, pp.582–583, 590
- Stacks common principal neighbourhoods, Lemma 26.11.5
- Stacks Algebra 10.131.12 and standard affine tensor base change
- Stacks Algebra 10.166.1–2
- Stacks Algebra 10.137.5–7
- Stacks Algebra 10.137.5–6
- Stacks Algebra 10.137.4 and 10.106.3
- Stacks Algebra 10.99.6–7
- Vakil §25.6.2–3, pp.678–679
- Stacks Algebra 10.99.7, 10.106.3, 10.136.12 and 10.137.5–7
- Vakil §25.6.2–3 and §26.2, pp.678–679, 689–690
- Stacks Algebra 10.137.5–6 (tags 00T6, 00T7)
- Stacks Algebra 10.110.3, 10.133.4–5 (tags 00OF, 00OD, 00OE)
- Vakil §26.2, pp.689–690
- Stacks Algebra 10.42.4 (tag 04KM)
- Stacks Algebra 10.43.3, 10.45.3 and 10.45.4
- Stacks Algebra 10.45.3–4
- Stacks Algebra 10.131.9 and 10.140.3 (Jacobian criterion, smooth locus)
- Vakil §21.2.9 and §25.6, pp.600–601, 678–679
- Stacks Algebra 10.137.15–16 (tags 00TE, 00TF) and 10.136.15 (tag 00SY)
- Vakil §26.2.4, pp.691–693
- Stacks Algebra 10.137.16 (tag 00TF), 10.137.5-6 (tags 00T6, 00T7) and 10.137.12 (tag 00TC)
- Vakil §26.2.2 and proof of §26.2.4, pp.690–693
- Stacks Algebra 10.137.6 (tag 00T7) and 10.137.8 (tag 00T9)
- Vakil §26.2.2 and the Jacobian-block argument of §26.2.4, pp.690–693
- Vakil §26.2.F and the proof of §26.2.4, pp.690–693
- Stacks Algebra 10.140.5 (tag 00TV), 10.137.16 (tag 00TF) and 10.137.5–6 (tags 00T6, 00T7)
- Stacks Algebra 10.166.1–2 (tags 0381, 0382)
- Vakil §22.2.10 and §26.2.4, pp.578, 690–693