Alphabeta MathGrothendieck Groups and Graded Cartan Pairings
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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
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- 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
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
- Tensor and Fusion Categories
- Universal Properties, Representables and the Yoneda Lemma
Combinatorics
Commutative Algebra
Group Theory
Homological Algebra
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
Grothendieck group of an essentially small abelian category
Definition
Split Grothendieck group of an additive category
Theorem
Universal properties and functoriality of G0 and split K0
Theorem
Simple classes freely generate the Grothendieck group of a length category
Theorem
Indecomposable projective classes form a basis of split K0
Definition
Graded Grothendieck groups, shift action, and Cartan map
Lemma
Graded Fitting decomposition for degree-zero endomorphisms
Theorem
Graded Krull–Schmidt for finite-dimensional graded modules
Lemma
Finite-dimensional graded algebras have graded projective covers
Theorem
Shift-orbit bases for graded simple and projective classes
Definition
Projective–module Hom pairing on class generators
Theorem
Projective Hom pairing descends and is graded sesquilinear
Theorem
Projective and simple classes are dual bases under splitting
Theorem
Exact adjoints induce adjoint operators on Grothendieck groups
Sources
- Charles Weibel, The K-book, Chapter II, Definition 6.1.1 and §6.1.3
- The Stacks Project, Homological Algebra, Definition 12.11.1
- Charles Weibel, The K-book, Chapter II, §1, §2 and Definition 5.1.2
- Charles Weibel, The K-book, Chapter II, §§1–2 and 5–6
- The Stacks Project, Homological Algebra, §12.11, tag 02MT
- Charles Weibel, The K-book, Chapter II, Exercise 6.3
- Alexander Kleshchev, Representation Theory of Symmetric Groups and Related Hecke Algebras, §2.2
- Tom Leinster, The bijection between projective indecomposable and simple modules, arXiv:1410.3671v1, §3, Lemma 3.1 and Corollary 3.2 (ungraded finite-dimensional Fitting lemma and indecomposable endomorphism criterion; the degree-zero graded adaptation is proved here)
- Alexander Kleshchev, Representation Theory of Symmetric Groups and Related Hecke Algebras, §2.2 (graded module conventions only)
- Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft), §11.1, Theorem 11.1.6 (ungraded Krull–Schmidt theorem for modules over a ring; the graded-category version is proved here)
- Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft), §7.3, Proposition 7.3.3(2) and Theorem 7.3.10 (ungraded projective-cover uniqueness and existence; the graded-category version is proved here)
- Alexander Kleshchev, Representation Theory of Symmetric Groups and Related Hecke Algebras, §2.2 (grading and internal-shift conventions only)
- Alexander Kleshchev, Representation Theory of Symmetric Groups and Related Hecke Algebras, §2.2 (shift and pairing conventions only)
- Alexander Kleshchev, Representation Theory of Symmetric Groups and Related Hecke Algebras, §2.2 (graded pairing convention only)