D = C[i]
g = f[i]
The shifted complex $D$ is defined by $D_j = C_{i+j}$ for all $j$ and the sign of the differential is changed if $i$ is odd.
The shifted complex map $g$ is defined by $g_j = f_{i+j}$ for all $j$.
The shift defines a natural automorphism on the category of complexes. Topologists often call the shifted complex $C[1]$ the suspension of $C$.
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In order to shift the complex one step, and not change the differential, one can do the following.
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The shift operator is functorial, as illustrated below.
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The source of this document is in Complexes/ChainComplexDoc.m2:912:0.