Lemma 57.10.4. Let $\mathcal{A}$ be an abelian category with enough negative objects. Let $X \in D^ b(\mathcal{A})$. Let $b \in \mathbf{Z}$ with $H^ i(X) = 0$ for $i > b$. Then there exists a map $N[-b] \to X$ such that the induced map $N \to H^ b(X)$ is surjective and $\mathop{\mathrm{Hom}}\nolimits (H^ b(X), N) = 0$.
Proof. Using the truncation functors we can represent $X$ by a complex $A^ a \to A^{a + 1} \to \ldots \to A^ b$ of objects of $\mathcal{A}$. Choose $N$ in $\mathcal{A}$ such that there exists a surjection $t : N \to A^ b$ and such that $\mathop{\mathrm{Hom}}\nolimits (A^ b, N) = 0$. Then the surjection $t$ defines a map $N[-b] \to X$ as desired. $\square$
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Your email address will not be published.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)