Spectral sequence

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Spectral sequences were invented by Jean Leray as an approach to computing sheaf cohomology.

Historical development[edit]

Definition[edit]

A (cohomology) spectral sequence (starting at ${\displaystyle E_a}$) in an abelian category ${\displaystyle A}$ consists of the following data:

1. A family ${\displaystyle \{E_r^{pq}\}}$ of objects of ${\displaystyle A}$ defined for all integers ${\displaystyle p,q}$ and ${\displaystyle r\ge a}$
2. Morphisms ${\displaystyle d_r^{pq}:E_r^{pq}\to E_r^{p+r,q-r+1}}$ that are differentials in the sense that ${\displaystyle d_r\circ d_r=0}$, so that the lines of "slope" ${\displaystyle -r/(r+1)}$ in the lattice ${\displaystyle E_r^{**}}$ form chain complexes (we say the differentials "go to the right")
3. Isomorphisms between ${\displaystyle E_{r+1}^{pq}}$ and the homology of ${\displaystyle E_r^{**}}$ at the spot ${\displaystyle E_r^{pq}}$:

${\displaystyle E_{r+1}^{pq}\simeq \ker (d_r^{pq})/\text{image}(d_r^{p-r,q+r+1})}$

Convergence[edit]

Examples[edit]

1. The Leray spectral sequence
2. The Grothendieck spectral sequence