The Stacks project

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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