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suspension of a chain complex

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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} \,.

Revised on November 23, 2013 21:09:41 by Tim Porter (2.26.12.120)