The Stacks project

Lemma 83.13.5. In Situation 83.3.3. If $K, K' \in D(\mathcal{C}_{total})$. Assume

  1. $K$ is cartesian,

  2. $\mathop{\mathrm{Hom}}\nolimits (K_ i[i - 1], K'_ i) = 0$ for $i > 1$.

Then any map $\{ K_ n \to K'_ n\} $ between the associated simplicial systems of $K$ and $K'$ comes from a map $K \to K'$ in $D(\mathcal{C}_{total})$.

Proof. Let $\{ K_ n \to K'_ n\} _{n \geq 0}$ be a morphism of simplicial systems of the derived category. Consider the objects $X_ n$ and Postnikov system $Y_ n$ associated to $K$ of Lemma 83.13.3. By (1) and Lemma 83.12.9 we have $X_ n = g_{n!}K_ n$. In particular, the map $K_0 \to K'_0$ induces a morphism $X_0 \to K'$. Since $\{ K_ n \to K'_ n\} $ is a morphism of systems, a computation (omitted) shows that the composition

\[ X_1 \to X_0 \to K' \]

is zero. As $Y_0 = X_0$ and as $Y_1$ fits into a distinguished triangle

\[ Y_1 \to X_1 \to Y_0 \to Y_1[1] \]

we conclude that there exists a morphism $Y_1[1] \to K'$ whose composition with $X_0 = Y_0 \to Y_1[1]$ is the morphism $X_0 \to K'$ given above. Suppose given a map $Y_ n[n] \to K'$ for $n \geq 1$. From the distinguished triangle

\[ X_{n + 1}[n] \to Y_ n[n] \to Y_{n + 1}[n + 1] \to X_{n + 1}[n + 1] \]

we get an exact sequence

\[ \mathop{\mathrm{Hom}}\nolimits (Y_{n + 1}[n + 1], K') \to \mathop{\mathrm{Hom}}\nolimits (Y_ n[n], K') \to \mathop{\mathrm{Hom}}\nolimits (X_{n + 1}[n], K') \]

As $X_{n + 1}[n] = g_{n + 1!}K_{n + 1}[n]$ the last group is equal to

\[ \mathop{\mathrm{Hom}}\nolimits (K_{n + 1}[n], K'_{n + 1}) \]

which is zero by assumption (2). By induction we get a system of maps $Y_ n[n] \to K'$ compatible with transition maps and reducing to the given map on $Y_0$. This produces a map

\[ \gamma : K = \text{hocolim} Y_ n[n] \longrightarrow K' \]

This map in any case has the property that the diagram

\[ \xymatrix{ X_0 \ar[rd] \ar[r] & K \ar[d]^\gamma \\ & K' } \]

is commutative. Restricting to $\mathcal{C}_0$ we deduce that the map $\gamma _0 : K_0 \to K'_0$ is the same as the first map $K_0 \to K'_0$ of the morphism of simplicial systems. Since $K$ is cartesian, this easily gives that $\{ \gamma _ n\} $ is the map of simplicial systems we started out with. $\square$


Comments (0)


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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0D9K. Beware of the difference between the letter 'O' and the digit '0'.