Spectral Sequences

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1 · Prerequisites

Abstract Algebra

Binary Operations, Monoids, Groups and Subgroups

Category Theory

Homological Algebra

Chain Complexes and Homology

Number Theory

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Lemma

Spectral Sequences

3 · Items

Spectral sequence subquotient and local lifting calculus

Increasing and decreasing filtration of an object

Exhaustive separated bounded and finite filtration

Associated graded object of a filtered object

Associated graded quotients are well defined subquotients

Filtered chain complex

Filtered chain map

The associated graded of a filtered complex is a bigraded complex

Induced filtration on homology

The associated graded of filtered homology is a subquotient of chain level data

R cycles and r boundaries of an increasingly filtered complex

R boundaries embed in r cycles

R page of the spectral sequence of a filtered complex

The filtered differential induces d r on the r page

The rth differential has bidegree minus r r minus one

The rth differential squares to zero

The next page is the homology of the current page

E zero is the associated graded complex

E one is homology of the associated graded complex

Limiting cycles boundaries and e infinity

Homological spectral sequence

Cohomological spectral sequence

Morphism of spectral sequences

A filtered chain map induces a morphism of spectral sequences

Spectral sequence stabilization at a bidegree

Collapse at a page

Degree reasons force stabilization in a bounded region

Collapse from one column at page s ≥ 1 or one row at page s ≥ 2

Abutment to a filtered object

Bounded filtered complex spectral sequence abuts to filtered homology

Edge homomorphisms of a first quadrant spectral sequence

Edge homomorphisms are natural

Extension problem of a convergent spectral sequence

Abelian-group model for spectral-sequence computations

Isomorphic associated graded objects need not give isomorphic filtered objects

Collapse does not in general split the abutment

The cohomological filtered complex construction

Shifting a filtered complex reindexes its spectral sequence

E infinity is the abutment object

E r plus one is literally equal to the homology of e r

A spectral sequence collapses when one differential is zero

Collapse solves all extension problems

The filtration on homology is h n of the filtered subcomplex

E zero and e one page conventions are universal across sources