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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