Lemma 59.76.4. Let $X$ be a quasi-compact and quasi-separated scheme. Let $\Lambda $ be a Noetherian ring. Let $K \in D(X_{\acute{e}tale}, \Lambda )$ and $b \in \mathbf{Z}$ such that $H^ b(K)$ is constructible. Then there exist a sheaf $\mathcal{F}$ which is a finite direct sum of $j_{U!}\underline{\Lambda }$ with $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ affine and a map $\mathcal{F}[-b] \to K$ in $D(X_{\acute{e}tale}, \Lambda )$ inducing a surjection $\mathcal{F} \to H^ b(K)$.

**Proof.**
Represent $K$ by a complex $\mathcal{K}^\bullet $ of sheaves of $\Lambda $-modules. Consider the surjection

By Modules on Sites, Lemma 18.30.6 we may choose a surjection $\bigoplus _{i \in I} j_{U_ i!} \underline{\Lambda } \to \mathop{\mathrm{Ker}}(\mathcal{K}^ b \to \mathcal{K}^{b + 1})$ with $U_ i$ affine. For $I' \subset I$ finite, denote $\mathcal{H}_{I'} \subset H^ b(K)$ the image of $\bigoplus _{i \in I'} j_{U_ i!} \underline{\Lambda }$. By Lemma 59.71.8 we see that $\mathcal{H}_{I'} = H^ b(K)$ for some $I' \subset I$ finite. The lemma follows taking $\mathcal{F} = \bigoplus _{i \in I'} j_{U_ i!} \underline{\Lambda }$. $\square$

## 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Your email address will not be published.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).

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.

## Comments (0)

There are also: