8.3 Spectral sequences of a double complex

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 double complex. This approach often leads to surprising connections and simplifies seemingly difficult homological problems.

Double Complex and Filtrations

Definition and Structure of Double Complexes

• Double complex consists of a collection of abelian groups or modules $C_{p,q}$ arranged in a grid indexed by integers $p$ and $q$
• Equipped with horizontal differentials $d_h: C_{p,q} \to C_{p+1,q}$ and vertical differentials $d_v: C_{p,q} \to C_{p,q+1}$
• Satisfies the conditions $d_h \circ d_h = 0$, $d_v \circ d_v = 0$, and $d_h \circ d_v + d_v \circ d_h = 0$ (anticommutativity)
• Can be visualized as a commutative diagram with rows and columns connected by differentials

Total Complex and Its Properties

• Total complex $\text{Tot}(C)$ of a double complex $C$ is a single complex constructed by "flattening" the double complex
• Defined as $\text{Tot}(C)_n = \bigoplus_{p+q=n} C_{p,q}$ with differential $d = d_h + d_v$
• Differential $d$ satisfies $d \circ d = 0$ 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 $F_p C = \bigoplus_{i \geq p} C_{i,*}$
  • Obtained by considering the rows of the double complex from the $p$-th row onwards
  • Satisfies $F_p C \subseteq F_{p-1} C$ for all $p$
• Column filtration on a double complex $C$ is a sequence of subcomplexes $F_q C = \bigoplus_{j \geq q} C_{*,j}$
  • Obtained by considering the columns of the double complex from the $q$-th column onwards
  • Satisfies $F_q C \subseteq F_{q-1} C$ 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

• First quadrant spectral sequence is a spectral sequence $\{E_r^{p,q}, d_r\}$ with $E_r^{p,q} = 0$ for $p < 0$ or $q < 0$
• Arises naturally from a double complex $C$ with $C_{p,q} = 0$ for $p < 0$ or $q < 0$ (first quadrant double complex)
• Pages of the spectral sequence are computed iteratively using the differentials $d_r: E_r^{p,q} \to E_r^{p+r,q-r+1}$
• Each page $E_r$ is the homology of the previous page $E_{r-1}$ with respect to the differential $d_{r-1}$

Convergence of Double Complex Spectral Sequence

• Spectral sequence of a first quadrant double complex $C$ converges to the homology of the total complex $\text{Tot}(C)$
• Convergence means that there exists an $r_0$ such that for all $r \geq r_0$, the pages $E_r$ stabilize (i.e., $E_r \cong E_{r+1} \cong \cdots$)
• The stable page $E_\infty$ is isomorphic to the associated graded module of the homology of $\text{Tot}(C)$ 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 $d_r = 0$ and $d_i = 0$ for all $i > r$
• Collapsing implies that $E_r \cong E_{r+1} \cong \cdots \cong E_\infty$
• 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., $C_{p,q} = 0$ for $p > p_0$ or $q > q_0$)