Alphabeta MathKunneth Exactness and Splittings over Principal Ideal Domains
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- 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
Homological 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
Lemma
Under Choice, a submodule of an arbitrary-rank free module over a PID is free
Lemma
A free PID complex decomposes into two-term cycle-boundary pieces
Lemma
The cycle-boundary tensor sequence has the Kunneth kernel and cokernel
Theorem
The natural PID Kunneth sequence is exact
Theorem
The PID Kunneth sequence admits a section after choices
Sources
- Garrett, Finitely-generated modules, §6, Theorem 6.0.1 and proof, printed pp.177–178
- Goel, Commutative Algebra, Chapter 8, footnote 2, printed p.137 (PDF index 137)
- tom Dieck, Algebraic Topology, proof of Theorem 11.10.1, printed pp.298–299
- Friedman, Singular Intersection Homology, §6.4.5, (6.11), printed p.315 and Remark 6.4.18, p.318
- tom Dieck, Algebraic Topology, Theorem 11.10.1, kernel/cokernel proof, printed pp.298–299
- Friedman, Singular Intersection Homology, §6.4.5, (6.11)–(6.13), printed pp.315–317
- tom Dieck, Algebraic Topology, Theorem 11.10.1, printed pp.298–299
- Friedman, Singular Intersection Homology, §6.4.5, (6.12)–(6.13), printed pp.316–317
- tom Dieck, Algebraic Topology, final paragraph of proof of Theorem 11.10.1, printed p.299
- Friedman, Singular Intersection Homology, §6.4.5, Splitting, printed p.318