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
\[ \mathop{\mathrm{Ker}}(\mathcal{K}^ b \to \mathcal{K}^{b + 1}) \longrightarrow H^ b(K) \]
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 $\mathcal{F} = \bigoplus _{i \in I'} j_{U_ i!} \underline{\Lambda }$. $\square$
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