The Stacks project

Lemma 57.10.2. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{D}$ be a triangulated category. Let $F, F' : D^ b(\mathcal{A}) \longrightarrow \mathcal{D}$ be exact functors of triangulated categories. Assume

  1. the functors $F \circ i$ and $F' \circ i$ are isomorphic where $i : \mathcal{A} \to D^ b(\mathcal{A})$ is the inclusion functor, and

  2. for all $X, Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ we have $\mathop{\mathrm{Ext}}\nolimits ^ q_\mathcal {D}(F(X), F(Y)) = 0$ for $q < 0$ (for example if $F$ is fully faithful).

Then $F$ and $F'$ are siblings.

Proof. Let $K \in D^ b(\mathcal{A})$. We will show $F(K)$ is isomorphic to $F'(K)$. We can represent $K$ by a bounded complex $A^\bullet $ of objects of $\mathcal{A}$. After replacing $K$ by a translation we may assume $A^ i = 0$ for $i > 0$. Choose $n \geq 0$ such that $A^{-i} = 0$ for $i > n$. The objects

\[ M_ i = (A^{-i} \to \ldots \to A^0)[-i],\quad i = 0, \ldots , n \]

form a Postnikov system in $D^ b(\mathcal{A})$ for the complex $A^\bullet = A^{-n} \to \ldots \to A^0$ in $D^ b(\mathcal{A})$. See Derived Categories, Example 13.41.2. Since both $F$ and $F'$ are exact functors of triangulated categories both

\[ F(M_ i) \quad \text{and}\quad F'(M_ i) \]

form a Postnikov system in $\mathcal{D}$ for the complex

\[ F(A^{-n}) \to \ldots \to F(A^0) = F'(A^{-n}) \to \ldots \to F'(A^0) \]

Since all negative $\mathop{\mathrm{Ext}}\nolimits $s between these objects vanish by assumption we conclude by uniqueness of Postnikov systems (Derived Categories, Lemma 13.41.6) that $F(K) = F(M_ n[n]) \cong F'(M_ n[n]) = F'(K)$. $\square$

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Your email address will not be published.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 0FZU. Beware of the difference between the letter 'O' and the digit '0'.