nLab truncation of a chain complex

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Contents

Definition

For C C_\bullet a chain complex, the truncation (τ C) (\tau_{\leq} C)_\bullet at some nn \in \mathbb{N} is the chain complex defined by

(τ nC) i={0 |i>n C n/B n |i=n C n |i<n, (\tau_n C)_i = \left\{ \array{ 0 & | i \gt n \\ C_n/B_n & | i = n \\ C_n & | i \lt n } \right. \,,

where B n=im(d n)B_n = im(d_n).

For connective chain complexes this is the notion of truncated object in an (infinity,1)-category realized in the (infinity,1)-category of chain complexes.

References

Last revised on September 13, 2016 at 12:48:02. See the history of this page for a list of all contributions to it.