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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)