Lemma 14.23.5. Let $\mathcal{A}$ be an abelian category. Let $A$ be an object of $\mathcal{A}$ and let $k$ be an integer. We have $N(K(A, k))_ i = A$ if $i = k$ and $0$ else.

Proof. It is clear that $N(K(A, k))_ i = 0$ when $i < k$ because $K(A, k)_ i = 0$ in that case. It is clear that $N(K(A, k))_ k = A$ since $K(A, k)_{k - 1} = 0$ and $K(A, k)_ k = A$. For $i > k$ we have $N(K(A, k))_ i = 0$ by Lemma 14.21.9 and the definition of $K(A, k)$, see Definition 14.22.3. $\square$

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