Spectral sequences of double complexes are powerful tools for computing homology. They allow us to break down complex calculations into manageable steps, using filtrations to analyze the structure piece by piece.
By studying the pages of a spectral sequence, we can uncover hidden relationships between different parts of a . This approach often leads to surprising connections and simplifies seemingly difficult homological problems.
Double Complex and Filtrations
Definition and Structure of Double Complexes
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Double complex consists of a collection of abelian groups or modules Cp,q arranged in a grid indexed by integers p and q
Equipped with horizontal differentials dh:Cp,q→Cp+1,q and vertical differentials dv:Cp,q→Cp,q+1
Satisfies the conditions dh∘dh=0, dv∘dv=0, and dh∘dv+dv∘dh=0 (anticommutativity)
Can be visualized as a commutative diagram with rows and columns connected by differentials
Total Complex and Its Properties
Tot(C) of a double complex C is a single complex constructed by "flattening" the double complex
Defined as Tot(C)n=⨁p+q=nCp,q with d=dh+dv
Differential d satisfies d∘d=0 due to the anticommutativity condition in the double complex
Allows studying the homology of the double complex by considering the homology of the total complex
Filtrations on Double Complexes
Row filtration on a double complex C is a sequence of subcomplexes FpC=⨁i≥pCi,∗
Obtained by considering the rows of the double complex from the p-th row onwards
Satisfies FpC⊆Fp−1C for all p
Column filtration on a double complex C is a sequence of subcomplexes FqC=⨁j≥qC∗,j
Obtained by considering the columns of the double complex from the q-th column onwards
Satisfies FqC⊆Fq−1C for all q
Filtrations provide a way to study the double complex by considering its "slices" along rows or columns
Spectral Sequences and Convergence
First Quadrant Spectral Sequence
is a spectral sequence {Erp,q,dr} with Erp,q=0 for p<0 or q<0
Arises naturally from a double complex C with Cp,q=0 for p<0 or q<0 (first quadrant double complex)
Pages of the spectral sequence are computed iteratively using the differentials dr:Erp,q→Erp+r,q−r+1
Each page Er is the homology of the previous page Er−1 with respect to the differential dr−1
Convergence of Double Complex Spectral Sequence
Spectral sequence of a first quadrant double complex C converges to the homology of the total complex Tot(C)
means that there exists an r0 such that for all r≥r0, the pages Er stabilize (i.e., Er≅Er+1≅⋯)
The stable page E∞ is isomorphic to the associated graded module of the homology of Tot(C) with respect to a certain filtration
Convergence provides a way to compute the homology of the total complex by studying the spectral sequence pages
Collapse of Spectral Sequence
Spectral sequence of a double complex is said to collapse at the r-th page if dr=0 and di=0 for all i>r
Collapsing implies that Er≅Er+1≅⋯≅E∞
If the spectral sequence collapses at the r-th page, the homology of the total complex can be read off directly from the r-th page
Collapsing at an early page simplifies the computation of the homology of the total complex
Occurs in various situations, such as when the double complex has only a few non-zero terms or when certain vanishing conditions are satisfied (e.g., Cp,q=0 for p>p0 or q>q0)