The Stacks project

Lemma 14.23.9. Let $\mathcal{A}$ be an abelian category. Let $V$ be a simplicial object of $\mathcal{A}$. The canonical morphism of chain complexes $N(V) \to s(V)$ is a quasi-isomorphism. In other words, the complex $D(V)$ of Lemma 14.23.6 is acyclic.

Proof. Note that the result holds for $K(A, k)$ for any object $A$ and any $k \geq 0$, by Lemmas 14.23.3 and 14.23.5. Consider the hypothesis $IH_{n, m}$: for all $V$ such that $V_ j = 0$ for $j \leq m$ and all $i \leq n$ the map $N(V) \to s(V)$ induces an isomorphism $H_ i(N(V)) \to H_ i(s(V))$.

To start of the induction, note that $IH_{n, n}$ is trivially true, because in that case $N(V)_ n = 0$ and $s(V)_ n = 0$.

Assume $IH_{n, m}$, with $m \leq n$. Pick a simplicial object $V$ such that $V_ j = 0$ for $j < m$. By Lemma 14.22.2 and Definition 14.22.3 we have $K(V_ m, m) = i_{m!} \text{sk}_ mV$. By Lemma 14.21.10 the natural morphism

\[ K(V_ m, m) = i_{m!} \text{sk}_ mV \to V \]

is injective. Thus we get a short exact sequence

\[ 0 \to K(V_ m, m) \to V \to W \to 0 \]

for some $W$ with $W_ i = 0$ for $i = 0, \ldots , m$. This short exact sequence induces a morphism of short exact sequence of associated complexes

\[ \xymatrix{ 0 \ar[r] & N(K(V_ m, m)) \ar[r] \ar[d] & N(V) \ar[r] \ar[d] & N(W) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & s(K(V_ m, m)) \ar[r] & s(V) \ar[r] & s(W) \ar[r] & 0 } \]

see Lemmas 14.23.1 and 14.23.8. Hence we deduce the result for $V$ from the result on the ends. $\square$

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