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truncation of a chain complex

Definition

For $C_\bullet$ a chain complex, the truncation $(\tau_{\leq} C)_\bullet$ at some $n \in \mathbb{N}$ is the chain complex defined by

$$(\tau_n C)_i = \left\{ \array{ 0 & | i \gt n \\ C_n/B_n & | i = n \\ C_n & | i \lt n } \right. \,,$$

where $B_n = im(d_n)$.

For connective chain complexes this is the notion of truncated object in an (infinity,1)-category realized in the (infinity,1)-category of chain complexes.

References

• Charles Weibel, An Introduction to Homological Algebra