Alphabeta MathClassical and Derived Deformation Data
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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 over a Principal Ideal Domain and the Canonical Forms
- 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
- 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 Categories
- Derived Functors
- Differential Graded Algebras and Homological Foundations
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Grothendieck Spectral Sequences and Computations
- Hochschild Homology and Diagonal Koszul Resolutions
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Spectral Sequences
- Tor Flatness and Global Dimension
- Triangulated Categories
- 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
Quantum Field Theory Mathematics
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Suprema and Infima
- Topology of ℝ
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Algebraic Spaces, Stacks, and Derived Algebraic Geometry Foundations
- Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
- Deformation Theory of Schemes and Obstruction Spaces
- Diagonals Separated Morphisms and Valuative Uniqueness
- 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
- Proj Projective Schemes Twisting Sheaves and Ampleness
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Cohomology Cech Cohomology and Comparison
- 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
String Theory
2 · Logical flowchart
3 · Items
Lemma
Marked algebra and sheaf extensions from polynomial presentations
Lemma
Scheme effectivity and flatness for marked Artin lifts
Definition
Strict Calabi–Yau variety in the Morimura source convention
Definition
Small dg categories, perfect modules, and Morita deformation
Lemma
Projective volume trace duality for perfect complexes under derived base change
Lemma
A finite sum of ample powers generates perfect complexes
Lemma
Small-extension comparison for nilpotent Maurer–Cartan gauge classes
Theorem
Affine square-zero lifting obstruction, torsor, and automorphisms
Definition
Geometric orientation and dg-category Calabi–Yau structure
Definition
Derived moduli prestack of perfect complexes
Theorem
Global cotangent-complex obstruction and lifting torsor
Definition
Relative closed shifted symplectic form
Sources
- Stacks Project, Deformation Theory
- Stacks Project, The Cotangent Complex
- Stacks Project, Deformations of ringed spaces and the naive cotangent complex
- Stacks Project, More on Morphisms
- Hayato Morimura, Versal dg deformation of Calabi–Yau manifolds, arXiv:2111.11778v3
- The Stacks Project, Varieties
- Christopher Brav and Tobias Dyckerhoff, Relative Calabi-Yau Structures, arXiv:1606.00619v1
- Pinned mathematical supplier, dualizing line bundle and trace datum
- Pinned mathematical supplier, Serre duality for locally free sheaves
- The Stacks Project, Derived Categories of Schemes, An example generator
- Bondal and Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry
- Marco Manetti, Deformation theory via differential graded Lie algebras
- The Stacks Project Authors, The Cotangent Complex, Lemma 92.16.1/tag 08SP
- The Stacks Project Authors, Deformation Theory, Chapter 91
- Robin Hartshorne, Lectures on Deformation Theory, Math 274 draft
- Pantev, Toën, Vaquié, Vezzosi, Shifted Symplectic Structures
- Christopher Brav and Tobias Dyckerhoff, Relative Calabi-Yau Structures
- Matthew Huynh, Degenerations of Shifted Symplectic Moduli Stacks of Sheaves
- Toën and Vezzosi, Homotopical Algebraic Geometry II: Geometric Stacks and Applications
- The Stacks Project Authors, The Cotangent Complex, Lemma 92.21.1/tag 08UZ
- Damien Calaque and Stefano Ronchi, Shifted Symplectic Geometry by Examples