Suppose there is a finite chain complex $$C_{n} \rightarrow \cdots \rightarrow C_{d}\rightarrow \cdots \rightarrow C_{0}\,,$$ such that $H_{i}(C_{\bullet})$ is vanished except for $i=d$.

Are there any general processes of modifying the complex (still index from $n$ to $0$) so that the homology is unvanished only in $d-1$ or $d+1$-th place and the nonzero homology is same as $H_{d}(C_{\bullet})$? The ultimate goal is shifting the nonzero homology to the head or tail.

Many thanks.

  • 3
    $\begingroup$ I mean you could reindex. What exactly do you mean by modifying the complex? $\endgroup$ – jgon Jun 15 at 21:16

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