nLab edge morphism

Idea

Given a spectral sequence $\{E_r^{s,t},d_r\}$ with a “vanishing edge” in the sense that all its terms vanish for $s$ or $t$ smaller or larger some fixed value, then the fact that all differentials starting or ending on that edge necessarily vanish implies that all terms on the edge project onto or inject into the corresponding terms on the infinity-page, respectively. These are called the edge homomorphisms.

Specifically, given a first-quadrant (cohomological) spectral sequence $(E_r^{p,q})$ there are natural morphisms

$$E_2^{n,0} \longrightarrow E^n$$

and

$$E^n \longrightarrow E_2^{0,n} \,.$$

These are called the edge morphisms or edge maps of the spectral sequence.

Properties

The edge morphisms sit in an exact sequence of the form

$$0 \to E_2^{1,0} \to E^1 \to E_2^{0,1} \stackrel{d_2}{\to} E_2^{2,0} \to E^2$$

This is often called the exact sequence of terms of low degree or five term exact sequence.

e.g. (Cartan_Eilenberg XV, 5, Tamme, 0 2.3.2)

If

$$E_r^{p,(0 \leq q \leq n)} = 0$$

then $E^{p,0}_2 \simeq E^p$ for $p \lt n$ and also

$$0 \to E_2^{1,0} \to E^1 \to E_2^{0,1} \stackrel{d_2}{\to} E_2^{2,0} \to E^2$$

is exact.

e.g. (Cartan_Eilenberg XV, 5, Tamme, 0 2.3.3)

References

• Henri Cartan, Samuel Eilenberg, Homological algebra, 1956

• Günter Tamme, section 0 2.3 of Introduction to Étale Cohomology