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

$\ldots \to x \to y \to z \to \ldots \quad \text{or}\quad x_1 \to x_2 \to \ldots \to x_ n$

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

$\ldots \to x_{-3} \to x_{-2} \to x_{-1} \quad \text{and}\quad x_1 \to x_2 \to x_3 \to \ldots$

A short exact sequence is an exact complex of the form

$0 \to A \to B \to C \to 0.$

There are also:

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