Lemma 14.23.3. Let \mathcal{A} be an abelian category. Let A be an object of \mathcal{A} and let k be an integer. We have H_ i(s(K(A, k))) = A if i = k and 0 else.
Proof. First, let us prove this if k = 0. In this case we have K(A, 0)_ n = A for all n. Furthermore, all the maps in this simplicial abelian group are \text{id}_ A, in other words K(A, 0) is the constant simplicial object with value A. The boundary maps d_ n = \sum _{i = 0}^ n (-1)^ i \text{id}_ A = 0 if n odd and = \text{id}_ A if n is even. Thus s(K(A, 0)) looks like this
\ldots \to A \xrightarrow {0} A \xrightarrow {1} A \xrightarrow {0} A \to 0
and the result is clear.
Next, we prove the result for all k by induction. Given the result for k consider the short exact sequence
0 \to K(A, k) \to E \to K(A, k + 1) \to 0
from Lemma 14.22.4. By Lemma 14.22.1 the associated sequence of chain complexes is exact. By Lemma 14.23.2 we see that s(E) is acyclic. Hence the result for k + 1 follows from the long exact sequence of homology, see Homology, Lemma 12.13.6. \square
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