Alphabeta MathBounded Bimodule Complexes and Derived Tensor
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Homomorphisms and the Isomorphism Theorems
- 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
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Categories
- Derived Functors
- Ext and Balanced Resolutions
- Graded Bimodules and Tensor Functors
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Tor Flatness and Global Dimension
- Triangulated Categories
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
Bounded graded bimodule complexes and signed tensor totalization
Lemma
Bimodule tensor totalization respects differentials and homotopies
Theorem
Bounded bimodule tensor is associative, unital, and compatible with cones
Theorem
A bounded two-sided projective bimodule complex defines exact derived tensor functors
Proposition
Bimodule homotopy equivalences induce natural tensor-functor isomorphisms
Theorem
Supplied inverse bimodule complexes give derived tensor equivalences
Sources
- Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §2c
- Charles A. Weibel, An Introduction to Homological Algebra, §10.6
- Stacks Project, Differential Graded Algebra, §22.33, tag 09LP
- Khovanov and Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §2c and Proposition 2.4
- Stacks Project, More on Algebra, §15.60, tag 06XY
- Charles A. Weibel, An Introduction to Homological Algebra, §10.6, printed pp. 395–396
- Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §2c and Proposition 2.4
- Weibel, An Introduction to Homological Algebra, §10.6, printed pp. 395–396