# nLab suspension of a chain complex

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Idea

In a category of chain complexes ${\mathrm{Ch}}_{•}\left(𝒜\right)$, the suspension object of a chain complex ${C}_{•}$ is the complex

$\Sigma {C}_{•}=C\left[1{\right]}_{•}$\Sigma C_\bullet = C[1]_\bullet

(or sometimes denoted $C\left[-1{\right]}_{•}$, depending on an unessential choice of sign convention) obtained by shifting the degrees up by one:

$C\left[1{\right]}_{n}≔{C}_{n-1}$C[1]_n \coloneqq C_{n-1}

with the differential the original one but equipped with a sign:

${d}_{n}^{X\left[1\right]}≔-{d}_{n-1}^{X}\phantom{\rule{thinmathspace}{0ex}}.$d^{X[1]}_n \coloneqq - d^X_{n-1} \,.

Generally for $p\in ℤ$ $C\left[p\right]$ is the chain complex with

$C\left[p{\right]}_{n}≔{C}_{n-p}$C[p]_n \coloneqq C_{n-p}
${d}_{n}^{X\left[p\right]}≔\left(-1{\right)}^{p}{d}_{n-p}^{X}\phantom{\rule{thinmathspace}{0ex}}.$d^{X[p]}_n \coloneqq (-1)^p d^X_{n-p} \,.

Revised on September 24, 2012 12:13:59 by Urs Schreiber (89.204.139.197)