The Stacks project

Definition 13.9.9. Let $\mathcal{A}$ be an additive category. A termwise split exact sequence of complexes of $\mathcal{A}$ is a complex of complexes

\[ 0 \to A^\bullet \xrightarrow {\alpha } B^\bullet \xrightarrow {\beta } C^\bullet \to 0 \]

together with given direct sum decompositions $B^ n = A^ n \oplus C^ n$ compatible with $\alpha ^ n$ and $\beta ^ n$. We often write $s^ n : C^ n \to B^ n$ and $\pi ^ n : B^ n \to A^ n$ for the maps induced by the direct sum decompositions. According to Homology, Lemma 12.14.10 we get an associated morphism of complexes

\[ \delta : C^\bullet \longrightarrow A^\bullet [1] \]

which in degree $n$ is the map $\pi ^{n + 1} \circ d_ B^ n \circ s^ n$. In other words $(A^\bullet , B^\bullet , C^\bullet , \alpha , \beta , \delta )$ forms a triangle

\[ A^\bullet \to B^\bullet \to C^\bullet \to A^\bullet [1] \]

This will be the triangle associated to the termwise split sequence of complexes.

Comments (3)

Comment #294 by arp on

Typo: In the definition of , I think it should say in degree it is given by the map (i.e. replace with ).

Comment #3250 by William Chen on

To make the terminology consistent with the next proposition (Tag 05SS), maybe this should be called a "termwise split exact sequence of complexes" ?

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