Alphabeta MathGraded Quiver Algebras and Derived Tensor Functors
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- 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 over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
Category Theory
- Abelian Categories
- 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
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Categories
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Graded Bimodules and Tensor Functors
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Spectral Sequences
- Tor Flatness and Global Dimension
- Triangulated Categories
Linear Algebra
Real Analysis
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
Integral path ring of a finite quiver
Definition
Khovanov–Seidel type A algebra
Lemma
The 4m+1 path basis
Definition
Finite graded Aₘ-modules, internal shifts and the vertex projectives
Definition
The vertex modules Sᵢ and their prime quotients
Lemma
The Khovanov-Seidel grid resolutions of the vertex modules
Lemma
The graded horseshoe lemma for finite graded projective resolutions
Theorem
Finite homological dimension of the finite graded Khovanov-Seidel module category
Lemma
The bounded projective comparison for the derived category
Definition
The bounded projective homotopy category Cₘ and the two shifts
Definition
The two-sided projective bimodules Uᵢ and their tensor functors
Theorem
Corner computations: the Uᵢ satisfy the Temperley-Lieb relations
Definition
The Khovanov–Seidel bimodule maps βᵢ and γᵢ
Definition
Signed totalization of graded Aₘ-bimodule actions
Lemma
Bounded two-sided projective bimodule complexes act on Cₘ
Definition
The twist complexes Rᵢ and Rᵢ⁻¹
Definition
The triangulated K₀ of the Khovanov–Seidel projective category
Lemma
Homological and internal shifts on K₀(Cₘ)
Sources
- Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §1b, printed pp. 3-4
- Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §1b and §§2a-2c, printed pp. 3-4 and 9-11
- Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §2a, printed pp. 9-11
- Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §2a Proposition 2.1 and its proof, printed pp. 9-10
- Charles Weibel, An Introduction to Homological Algebra, ch. 2 §2.2, pp. 36-38
- Charles Weibel, An Introduction to Homological Algebra, ch. 10 §10.4, pp. 387-390
- Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §§2a-2c, printed pp. 9-11
- Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §2b, printed pp. 11-12
- Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §2d, printed pp. 11-12
- Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §2c, printed pp. 10-11
- The Stacks Project, Derived Categories, section 28, K-groups (tag 0FCM), Definition 13.28.1