nLab suspension of a chain complex

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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

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

In a category of chain complexes Ch (𝒜)Ch_\bullet(\mathcal{A}), the suspension object of a chain complex C C_\bullet is the complex

ΣC =C[1] \Sigma C_\bullet = C[1]_\bullet

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

C[1] nC n1 C[1]_n \coloneqq C_{n-1}

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

d n C[1]d n1 C. d^{C[1]}_n \coloneqq - d^C_{n-1} \,.

Generally, for pp \in \mathbb{Z}, C[p]C[p] is the chain complex with

C[p] nC np C[p]_n \coloneqq C_{n-p}
d n C[p](1) pd np C. d^{C[p]}_n \coloneqq (-1)^p d^C_{n-p} \,.

Last revised on October 3, 2019 at 13:44:04. See the history of this page for a list of all contributions to it.