Alphabeta MathGrothendieck Spectral Sequences and Computations
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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 Actions, Orbits, Stabilisers and Cayley's Theorem
- 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
- Semidirect Products, Automorphism Groups and Split Extensions
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- 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 Categories
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Group Cohomology as a Derived Functor
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Spectral Sequences
- Tor Flatness and Global Dimension
- Triangulated Categories
- Universal Coefficients and Kunneth Theorems
Linear Algebra
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
Cartan-Eilenberg injective resolution of a bounded-below complex
Lemma
Finite biproducts of injective objects are injective
Lemma
A Cartan-Eilenberg resolution totalizes to an injective replacement
Theorem
Cartan-Eilenberg injective resolutions exist
Definition
Right hyperderived functor of a complex
Lemma
Hyperderived functors are independent of the supplied resolution
Lemma
Cartan–Eilenberg comparisons preserve both filtrations
Lemma
Finite-diagonal cohomological double-complex spectral sequences
Theorem
First hypercohomology spectral sequence
Theorem
Second hypercohomology spectral sequence
Proposition
Hypercohomology edge maps are canonical
Definition
G-acyclic object for a left-exact functor
Lemma
Applying F gives a termwise G-acyclic complex
Lemma
The total Cartan-Eilenberg complex computes the derived composite
Lemma
The two filtrations identify E2 and the composite edge
Theorem
Grothendieck spectral sequence
Theorem
Naturality of the Grothendieck spectral sequence
Theorem
Five-term exact sequence of the Grothendieck spectral sequence
Corollary
Grothendieck collapse when one functor is exact
Corollary
Derived composition isomorphisms under total acyclicity
Remark
Dual left-derived Grothendieck spectral sequence
Theorem
Universal coefficient spectral sequence
Theorem
Kunneth Tor spectral sequence
Corollary
PID Kunneth is a two-column collapse
Proposition
UCT and Kunneth collapse retains an extension problem
Theorem
Hyper-Tor spectral sequence
Theorem
Hyper-Ext spectral sequence
Lemma
Invariants for a group extension compose
Lemma
N-invariants send injective G-modules to Q-acyclics
Lemma
Restriction of injective group modules is injective
Theorem
Lyndon-Hochschild-Serre spectral sequence
Theorem
Five-term exact sequence from LHS
Proposition
LHS collapse for a cohomologically trivial normal subgroup
Definition
Spectral-sequence computation record
Proposition
An E2 page alone does not determine the abutment
Proposition
Collapsed vector-space spectral sequences split noncanonically
False statement
Grothendieck needs only left exactness
False statement
RᵖG(R^qF) is the E₁ page
False statement
Cartan-Eilenberg only resolves terms
False statement
UCT collapse gives a natural splitting
False statement
LHS defines low-degree group cohomology
False statement
Writing E2 implies H proves convergence
Sources
- Weibel, Homological Algebra, Section 5.7
- Sharifi, Homological Algebra, Section 4.3
- Weibel, 5.7.2 (finite-diagonal Cartan-Eilenberg totalization)
- Stacks Project, Tag 015G (Cartan-Eilenberg resolutions)
- Weibel, 5.7.2
- Sharifi, Section 4.3
- Weibel, Definition 5.7.1 and Lemma 5.7.2, printed pp.145–146; cohomology variant 5.7.9
- Sharifi, Theorem 4.3.2
- Weibel, Definition 5.7.4 and cohomology variant 5.7.9, printed pp.147 and 149–150
- Weibel, 5.7.3
- Weibel, Exercises 5.7.2–3 and cohomology variant 5.7.9; completed comparison argument
- Stacks Project, Lemmas 12.25.1 and 12.25.3, with explicit dual filtration conversion
- Weibel, 5.7.9
- Sharifi, Theorem 4.3.7
- Stacks Project, Tags 015M-015N
- Weibel, Section 5.7
- Stacks Project, Tag 015H
- Stacks Project, Tags 015H and 015M
- Weibel, 5.8.3
- Stacks Project, Tag 015N
- Sharifi, Theorem 4.3.8
- Weibel, Theorem 5.8.3
- Weibel, Sections 5.2 and 5.8
- Weibel, Theorem 5.8.3, low-degree sequence, printed p.151
- Stacks Project, Tags 015J-015N
- Weibel, Theorem 5.8.3, dual form
- Weibel, Sections 5.6-5.7
- Weibel, 5.6.4
- Weibel, Sections 5.2 and 5.6
- Weibel, 5.7.8
- Stacks Project, Tag 07AA (bounded-above first-variable Ext spectral sequence)
- Stacks Project, Tag 0AVG (bounded-below second-variable Ext spectral sequence)
- Weibel, 6.8.2
- Weibel, Sections 6.3 and 6.8; finite-support choice-free induction argument supplied here
- Weibel, Theorem 6.8.2
- Sharifi, Hochschild-Serre spectral sequence
- Weibel, 6.8.3
- Weibel, Sections 5.2-5.5
- Weibel, Section 5.2
- Sharifi, Definition 4.3.1
- Weibel, Sections 6.1 and 6.8