Definition 12.5.7. Let $\mathcal{A}$ be an additive category. Consider a sequence of morphisms

in $\mathcal{A}$. We say such a sequence is a *complex* if the composition of any two consecutive (drawn) arrows is zero. If $\mathcal{A}$ is abelian then we say a complex of the first type above is *exact at $y$* if $\mathop{\mathrm{Im}}(x \to y) = \mathop{\mathrm{Ker}}(y \to z)$ and we say a complex of the second kind is *exact at $x_ i$* where $1 < i < n$ if $\mathop{\mathrm{Im}}(x_{i - 1} \to x_ i) = \mathop{\mathrm{Ker}}(x_ i \to x_{i + 1})$. We a sequence as above is *exact* or is an *exact sequence* or is an *exact complex* if it is a complex and exact at every object (in the first case) or exact at $x_ i$ for all $1 < i < n$ (in the second case). There are variants of these notions for sequences of the form

A *short exact sequence* is an exact complex of the form

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