nLab
splicing of short exact sequences

Contents

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

Definition

Given a sequence of short exact sequences of the form

0X ni nY np nX n+10 0 \to X_n \overset{i_n}{\longrightarrow} Y_n \overset{p_n}{\longrightarrow} X_{n+1} \to 0

for nn \in \mathbb{Z}, then their splicing is the horizontal composite sequence

Y n1 Y n Y n+1 p n1 i n p n i n+1 X n X n+1 \array{ \cdots && \overset{}{\longrightarrow} && Y_{n-1} && \longrightarrow && Y_n && \longrightarrow && Y_{n+1} && \longrightarrow && \cdots \\ && && & {}_{\mathllap{p_{n-1}}}\searrow && \nearrow_{\mathrlap{ i_n } } && {}_{\mathllap{p_n}}\searrow && \nearrow_{\mathrlap{ i_{n+1} }} \\ && && && X_n && && X_{n+1} }

which is, evidently, a long exact sequence.

More generally, there is splicing of interlocking systems of long exact sequences. See at Exact couple of a tower of fibrations.

Last revised on December 1, 2020 at 14:59:09. See the history of this page for a list of all contributions to it.