nLab
edge morphism

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

Given a spectral sequence {E r s,t,d r}\{E_r^{s,t},d_r\} with a “vanishing edge” in the sense that all its terms vanish for ss or tt 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.

Specificaly, Given a first-quadrant (cohomological) spectral sequence (E r p,q)(E_r^{p,q}) there are natural morphisms

E 2 n,0E n E_2^{n,0} \longrightarrow E^n

and

E nE 2 0,n. 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

0E 2 1,0E 1E 2 0,1d 2E 2 2,0E 2 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,(0qn)=0 E_r^{p,(0 \leq q \leq n)} = 0

then E 2 p,0E pE^{p,0}_2 \simeq E^p for p<np \lt n and also

0E 2 1,0E 1E 2 0,1d 2E 2 2,0E 2 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

Last revised on July 8, 2017 at 16:06:14. See the history of this page for a list of all contributions to it.