Spectral sequence ( $(E_{r}^{p, q}, d_{r})$ )

and the differential between them is an isomorphism, both disappear on the next page. In the homological convention, the differential instead has bidegree

(- r, r - 1)

and total degree minus one. Specifying the convention prevents reversing the arrows when reading a spectral-sequence diagram.

Künneth spectral sequence ( $E_{2}^{p, q} = Tor_{- p}^{R} (H^{q} P, M) \Rightarrow H^{p + q} (P \otimes_{R} M)$ )

 0  0

For a bounded complex

P

of projective right modules and a left module

M

of finite projective dimension, a finite projective resolution of

M

yields a bounded double cochain complex. Its two spectral sequences give the displayed abutment. Over a noncommutative ring, the handedness of the two tensor factors is essential.

Two-column degeneration of a Künneth spectral sequence ( $E_{2} = E_{\infty}$ )

 0  0

If only the columns

p = - 1, 0

are nonzero, a cohomological differential of bidegree

(r, 1 - r)

has no possible nonzero source-target pair for

r \geq 2

. Projective boundaries in a complex of projectives give this condition because every cohomology module has projective dimension at most one.

Convergence of a spectral sequence ( $E_{\infty}^{p, q} ≅ F^{p} H^{p + q} / F^{p + 1} H^{p + q}$ )

 0  0

The stable page identifies the associated graded module of a filtration of the abutment. For a finite filtration this is an exhaustive, separated finite extension description. It does not itself determine a canonical splitting. Unbounded filtrations require further convergence hypotheses.

Bounded filtered-complex convergence theorem ( $E_{1}^{p, q} = H^{p + q} (F^{p} C / F^{p + 1} C) \Rightarrow H^{p + q} (C)$ )

 0  0

A finite filtration of a cochain complex produces a convergent spectral sequence. The induced abutment filtration is the image of

H^{n} (F^{p} C) \to H^{n} (C)

. Finiteness in each degree suffices; a bounded range of cochain degrees is not required.

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A spectral sequence is a mathematical tool used in algebraic topology, homological algebra, and related fields to compute homology or cohomology groups that may be difficult to compute directly. It provides a method to systematically approximate these groups through a sequence of pages (typically indexed by integers) and associated differentials.

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Spectral sequence ( $(E_{r}^{p, q}, d_{r})$ )

Spectral sequence differential ( $d_{r} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1}$ )

Künneth spectral sequence ( $E_{2}^{p, q} = Tor_{- p}^{R} (H^{q} P, M) \Rightarrow H^{p + q} (P \otimes_{R} M)$ )

Two-column degeneration of a Künneth spectral sequence ( $E_{2} = E_{\infty}$ )

Convergence of a spectral sequence ( $E_{\infty}^{p, q} ≅ F^{p} H^{p + q} / F^{p + 1} H^{p + q}$ )

Bounded filtered-complex convergence theorem ( $E_{1}^{p, q} = H^{p + q} (F^{p} C / F^{p + 1} C) \Rightarrow H^{p + q} (C)$ )

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