# Shifting the nonzero homology

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.

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