Alphabeta MathHochschild Homology and Diagonal Koszul Resolutions
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- 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 Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- 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
- 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
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
Enveloping algebra and the bimodule–module dictionary
Definition
The augmented two-sided bar complex
Lemma
The bar boundary squares to zero and is augmented
Theorem
The two-sided bar complex is a projective Aᵉ-resolution
Definition
Hochschild chains and Hochschild homology with coefficients
Lemma
Hochschild chains are bar tensor chains
Theorem
Hochschild homology is Tor over the enveloping algebra
Theorem
Functoriality and coefficient long exact sequences for Hochschild homology
Proposition
Degree-zero Hochschild homology is bimodule coinvariants
Definition
The polynomial diagonal Koszul bimodule complex
Lemma
Polynomial diagonal differences form a regular sequence
Theorem
The diagonal Koszul complex is a finite free resolution of R
Theorem
Polynomial Hochschild homology from the diagonal Koszul complex
Corollary
Diagonal Hochschild homology of a polynomial ring
Sources
- Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9: Hochschild and Cyclic Homology, §9.1.1–9.1.3
- Mikhail Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, Hochschild homology section
- Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9: Hochschild and Cyclic Homology, §9.1.3–9.1.5
- Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9: Hochschild and Cyclic Homology, §9.1.3
- Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9: Hochschild and Cyclic Homology, §9.1.1
- Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9, §§9.1.3–9.1.5
- Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9, §9.1.2, Exercise 9.1.2
- Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9: Hochschild and Cyclic Homology, Exercise 9.1.3
- The Stacks Project, Section 15.31: Koszul regular sequences, Lemma 15.31.2
- Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9, §9.1.3 and Exercise 9.1.3
- Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9, §9.1.1 and Exercise 9.1.3