Alphabeta MathDouble Complexes Exact Couples and Convergence
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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
Category Theory
Number Theory
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
Homological double complex
Remark
Commuting versus anticommuting double complex conventions
Definition
Direct sum total complex of a double complex
Lemma
The total differential squares to zero
Definition
Product total complex of a double complex
Proposition
Sum and product totalisations agree on finite diagonal double complexes
Lemma
Countable sequence groups and tail filtrations
Definition
Row and column filtrations of a first quadrant double complex
Theorem
The row filtration spectral sequence of a first quadrant double complex
Theorem
The column filtration spectral sequence of a first quadrant double complex
Proposition
The two double complex spectral sequences have the same abutment but not the same pages
Theorem
Acyclic assembly lemma for a first quadrant double complex
Definition
Exact couple
Definition
Differential associated to an exact couple
Lemma
The exact couple differential squares to zero
Definition
Derived exact couple
Lemma
The derived couple maps are well defined
Theorem
The derived couple is exact
Theorem
An exact couple generates a spectral sequence
Theorem
A filtered complex produces an exact couple
Proposition
The exact couple and subquotient constructions of the filtered complex spectral sequence agree
Definition
Regular spectral sequence
Definition
Weak convergence of a spectral sequence
Definition
Strong convergence of a spectral sequence
Theorem
A first quadrant filtered complex spectral sequence converges to filtered homology
Definition
Lim one obstruction to completeness
Lemma
Countable tower completion obstruction exact sequence
Lemma
Countable tower six term limit sequence
Lemma
Two by two Delta complex for a double tower
Lemma
Approximate cycle obstruction sequence for a complete filtered complex
Theorem
Complete exhaustive filtered complex convergence criterion
Proposition
Failure of separatedness or completeness can destroy the claimed abutment
Lemma
Finite and complete filtered isomorphism lifting
Theorem
Spectral sequence comparison theorem
Corollary
Quasi isomorphism criterion from a filtered map
Theorem
Five term exact sequence of a first quadrant cohomological spectral sequence
Theorem
Five term exact sequence of a first quadrant homological spectral sequence
Proposition
Collapse with projective associated graded pieces splits the finite filtration noncanonically
Proposition
A map of exact couples induces a map of spectral sequences
Proposition
Short exact sequences of filtered complexes give compatible exact couples
Counterexample
Sum and product totalisations can differ on infinite diagonals
False statement
The two spectral sequences of a double complex have identical e one pages
False statement
Direct sum and product totalisations are always isomorphic
False statement
Every exact couple is a long exact sequence with no extra grading data
False statement
First quadrant support alone identifies the abutment without a filtration
False statement
An isomorphism on e infinity automatically gives an isomorphism of unfiltered targets
False statement
Exhaustive filtration implies separated and complete filtration
Sources
- Stacks Project, Definition 12.18.1 (indices and signs translated)
- Weibel, Chapter 5, Section 5.6
- Stacks Project, Definitions 12.18.1 and 12.18.3
- Stacks Project, Definition 12.18.3 (anticommuting convention)
- Stacks Project, Definition 12.18.3; direct verification with translated signs
- Weibel, Chapter 5, totalisation conventions
- Stacks Project, Section 12.18, finite diagonal totalisation
- Weibel, Chapter 5, completion examples; explicit binary model supplied locally
- Stacks Project, Section 12.25, the two filtration formulas (homological translation)
- Stacks Project, Lemmas 12.25.1 and 12.25.3 (translated conventions; calculation supplied here)
- Stacks Project, Lemmas 12.25.1 and 12.25.3 (homological anticommuting convention)
- Stacks Project, Lemmas 12.25.1 and 12.25.3; explicit witnesses supplied locally
- Stacks Project, Lemma 12.25.4 (homological projection variant)
- Stacks Project, Definition 12.21.1 and Remark 12.21.5 (homological grading)
- Stacks Project, Lemma 12.21.2 and Remark 12.21.5
- Stacks Project, Lemma 12.21.2; indexed calculation supplied locally
- Stacks Project, Lemma 12.21.2; full descent argument supplied here
- Stacks Project, Lemma 12.21.2 (omitted chase supplied in full here)
- Stacks Project, Definition 12.21.3 and Lemma 12.21.4 (full local proof supplied)
- Weibel, Section 5.9, filtered-complex exact couple
- Weibel, Section 5.9, filtered-complex comparison
- Weibel, An Introduction to Homological Algebra, Chapter 5
- The Stacks Project, Homological Algebra
- Boardman, Conditionally Convergent Spectral Sequences, section 1
- Weibel, Chapter 5, Proposition 5.5.9 and the interchange issue
- Weibel, Chapter 5, Corollary 5.5.8 and Proposition 5.5.9
- Weibel, Chapter 5; explicit infinite-diagonal binary witness