Alphabeta MathCohomological Hall Algebras and Quiver Correspondences
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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
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- 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
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- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Solvability by Radicals and Kummer Theory
- Splitting Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Algebraic Geometry
- Affine Algebraic Sets and Coordinate Rings
- Affine Perverse Vanishing And Universal Hyperplanes
- Algebraic Differentials Separability and Smooth Local Presentations
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- Complex Algebraic Stratified Topology
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- Dimension Constructible Images and Dimensions of Fibres
- Higher-Dimensional Resolution of Singularities
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Nonaffine Algebraic Groups, Barsotti-Chevalley, and Abelian Varieties
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- Products Segre and Veronese Embeddings and Grassmannians
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Algebraic Topology
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- Fibrations Fiber Bundles and Homotopy Exact Sequences
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- Leray–Hirsch, the Thom Isomorphism, and Gysin Sequences
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Categorification and Geometric Representation Theory
Category Theory
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Commutative Algebra
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Complex Analysis
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Differential Geometry
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Differential Topology
Homological Algebra
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Linear Algebra
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Number Theory
Real Analysis
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- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Representation Theory of Groups
Scheme Theory
- Affine Group Schemes, Hopf Algebras, and Rational Representations
- Affine Schemes and the Structure Sheaf
- Algebraic Group Actions, Orbits, Stabilizers, and Controlled Quotients
- Algebraic Spaces, Stacks, and Derived Algebraic Geometry Foundations
- Analytic Derived Six Operations And Duality
- Blowups, Exceptional Divisors, and Strict Transforms
- Cartier and Weil Divisors Line Bundles and Picard Groups
- Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
- Compact Support Sheaf Calculus
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Flat Smooth and Etale Morphisms
- Group Schemes of Finite Type over a Field
- 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
- Filters and Ultrafilters
- 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
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Lemma
Affine quotients by products of general linear groups are algebraic stacks
Definition
Representation stacks and exact-sequence correspondences of a finite quiver
Lemma
The quiver exact-sequence correspondence is smooth and proper
Lemma
Algebraic Milnor neighborhoods admit controlled sector transport
Theorem
Normalized rational vanishing cycles have coherent correspondence operations
Remark
Source-reported preprojective CoHA actions and their type boundaries
Definition
Intrinsic Borel–Moore grading on quiver quotient stacks
Definition
Quiver potentials and realized monodromic coefficient data
Lemma
Parabolic and Levi Borel models for quiver extension and flag stacks
Theorem
Borel–Moore operations for quiver extension and flag quotient stacks
Definition
The untwisted equivariant Borel–Moore CoHA of a finite quiver
Lemma
A quiver potential trace is additive on exact sequences
Theorem
The untwisted quiver CoHA is associative and unital
Lemma
Standard rational quiver vanishing cycles realize the critical coefficient package
Definition
Critical quiver CoHA and its standard rational realization
Theorem
Critical quiver CoHA is associative and unital, including the standard realization
Sources
- The Stacks Project, Algebraic Stacks, §§94.10–94.12
- J. S. Milne, Algebraic Groups, corrected 2022 printing, §2.8
- Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, §2.1–2.2
- Dominic Joyce, Cohomological Hall Algebras, Week 2 notes, Geometry Advanced Class Trinity Term 2026, §§1–2
- Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, §2.2
- Dominic Joyce, Cohomological Hall Algebras, Week 2 notes, Geometry Advanced Class Trinity Term 2026, §2.1
- David Massey, The Sebastiani-Thom Isomorphism in the Derived Category
- David Massey, Natural commuting of vanishing cycles and the Verdier dual
- Yaping Yang and Gufang Zhao, The cohomological Hall algebra of a preprojective algebra, §§0.1–0.4, 3.2, 5.1–5.6, and 7–8
- Hiraku Nakajima, Introduction to quiver varieties—for ring and representation theorists, §§2–4
- Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
- Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, §§4.7 and 7.4–7.6
- Ben Davison, The critical CoHA of a quiver with potential, §§2.2–2.7 and 3.1–3.2
- Yan Soibelman, Remarks on Cohomological Hall algebras and their representations, §§2.1 and 2.4
- Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, §§2.1–2.3
- Yan Soibelman, Remarks on Cohomological Hall algebras and their representations, §2.2
- Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, §4.7
- Ben Davison, The critical CoHA of a quiver with potential, §3.1
- Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, §2.3
- Dominic Joyce, Cohomological Hall Algebras, Week 2 notes, Geometry Advanced Class Trinity Term 2026, §§2.1–2.2
- Ben Davison, The critical CoHA of a quiver with potential
- Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, §§2.3 and 7.6
- Ben Davison, The critical CoHA of a quiver with potential, §§2.5–2.7 and 3.2