# nLab splicing of short exact sequences

Contents

### Context

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

## Definition

Given a sequence of short exact sequences of the form

$0 \to X_n \overset{i_n}{\longrightarrow} Y_n \overset{p_n}{\longrightarrow} X_{n+1} \to 0$

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

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

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.