Alphabeta MathThe Serre Spectral Sequence and Applications
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.
✓ 22 results · all verified · 10 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 12 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Free Products and Amalgamation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- 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
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
Algebraic Topology
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Higher Homotopy Groups and Cofiber Sequences
- Homology Axioms Degree and Classical Applications
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Local Coefficients, Twisted Homology, and Duality
- Obstruction Theory, Postnikov Towers, and Classifying Spaces
- Relative Homology Excision and Mayer Vietoris
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
- Singular Cohomology and Coefficient Theorems
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
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Long Exact Sequences in Homology
- Projective and Injective Resolutions
- Spectral Sequences
- Tor Flatness and Global Dimension
- Universal Coefficients and Kunneth Theorems
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- 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
Topology
- Classification of Covering Spaces
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Lemma
Serre-fibration replacement preserves fiber homology transport
Definition
Fiber homology local system of a Serre fibration
Lemma
Fiber transport is functorial on the base fundamental groupoid
Definition
Serre filtration over the base skeleta
Lemma
Relative homology over one base cell is shifted fiber homology
Lemma
The first Serre differential is the cellular boundary with local coefficients
Theorem
Homological Serre spectral sequence
Theorem
Naturality of the homological Serre spectral sequence
Definition
Serre edge homomorphisms and transgression
Proposition
Serre edge maps come from projection and fiber inclusion
Proposition
Serre transgression agrees with the relative connecting construction
Theorem
Cohomological Serre spectral sequence
Lemma
Multiplicative filtered cochains induce products on every spectral-sequence page
Theorem
Multiplicative cohomological Serre spectral sequence
Proposition
Degree and parity criteria for Serre collapse
Theorem
Gysin sequence from a sphere-fiber Serre spectral sequence
Theorem
Wang sequence for a fibration over the circle
Definition
Serre classes, Serre rings, ideals, and modulo-C morphisms
Lemma
Serre classes are stable under finite filtrations
Theorem
First-quadrant spectral-sequence transfer modulo a Serre class
Theorem
Serre-class transfer through a simply connected fibration
Corollary
Finite-generation, torsion, and p-primary Serre transfer
Theorem
PID finite-generation transfer for simply connected base and fiber
Lemma
Circle and path-loop models for Eilenberg–Mac Lane induction
Theorem
Rational cohomology of Eilenberg–Mac Lane spaces in one generator
Sources
- Miller, MIT 18.906 notes, Lectures 24 and 28
- Hatcher, Algebraic Topology, Chapter 5, generalizations after Theorem 5.3
- Miller, MIT 18.906 notes, Lecture 24
- Hatcher, Algebraic Topology, proof of Theorem 5.3
- Hatcher, Algebraic Topology, Theorem 5.3
- Miller, MIT 18.906 notes, Lectures 23–24
- Hatcher, Algebraic Topology, naturality after Theorem 5.3
- Miller, MIT 18.906 notes, Lecture 26
- Hatcher, Algebraic Topology, Proposition 5.14
- Hatcher, Algebraic Topology, Theorem 5.15
- Miller, MIT 18.906 notes, Lecture 29
- Hatcher, Algebraic Topology, Chapter 5, Multiplicative Structure
- Hatcher, Algebraic Topology, multiplicative Serre spectral sequence
- Miller, MIT 18.906 notes, Product structure
- Hatcher, Algebraic Topology, Appendix, Theorem A.6
- Hatcher, Algebraic Topology, Serre spectral sequence examples
- Miller, MIT 18.906 notes, Euler class and integration along the fiber
- Hatcher, Algebraic Topology, Serre spectral sequence over the circle
- Miller, MIT 18.906 notes, Serre classes
- Miller, MIT 18.906 notes, finite filtrations modulo a Serre class
- Miller, MIT 18.906 notes, Serre classes in spectral sequences
- Miller, MIT 18.906 notes, Serre classes in the Serre spectral sequence
- Miller, MIT 18.906 notes, examples and transfer for Serre classes
- Miller, MIT 18.906 notes, Serre-class finite-generation method
- Rolf Schön, Fibrations Over a CWh-Base
- Hatcher, Algebraic Topology, path-space fibration
- Hatcher, Algebraic Topology, Proposition 5.21