Alphabeta MathFormal Moduli, Derived Stacks and Shifted Symplectic Geometry
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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
- 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
- Tensor Products of Modules
- The Field of Fractions and Localisation
Algebraic Geometry
Algebraic Topology
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
Homological Algebra
- A∞, L∞, and Deformation Complexes
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Categories
- Derived Functors
- Differential Graded Algebras and Homological Foundations
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Tor Flatness and Global Dimension
- Triangulated Categories
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
- Sequences and Limits
- 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
- Kahler Differentials Conormal Sequences and Infinitesimal Lifting
- Presheaves Sheaves Stalks and Sheafification
- 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
2 · Logical flowchart
3 · Items
Definition
Formal moduli problem on small derived bases
Definition
Cohomological differential graded Lie algebra
Definition
Derived Artin stack and cotangent complex
Definition
Closed n-shifted symplectic form
Theorem
Derived Artin stacks are closed under homotopy fiber products
Lemma
Shuffle comparison and symmetric powers of simplicial modules
Lemma
Graded PBW, the CE resolution, and the free Lie comparison
Lemma
Characteristic-zero algebra models and the CE deformation context
Theorem
Transgression to shifted symplectic mapping stacks
Theorem
Derived intersection of Lagrangians is shifted symplectic
Sources
- Jacob Lurie, Derived Algebraic Geometry X: Formal Moduli Problems
- Joost Nuiten, Lecture 8: Koszul duality for formal moduli problems
- J. P. Pridham, Unifying derived deformation theories
- Bertrand Toën and Gabriele Vezzosi, Homotopical Algebraic Geometry II: Geometric Stacks and Applications
- Prestige Math Library, Derived schemes and cotangent complex
- Pantev, Toën, Vaquié, Vezzosi, Shifted Symplectic Structures
- Damien Calaque and Stefano Ronchi, Shifted Symplectic Geometry by Examples
- Toën and Vezzosi, Homotopical Algebraic Geometry II
- Vladimir Hinich, Homological Algebra of Homotopy Algebras
- Vladimir Hinich, Erratum to Homological Algebra of Homotopy Algebras