The Stacks project

Lemma 13.16.3. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined.

  1. We have $R^ iF = 0$ for $i < 0$,

  2. $R^0F$ is left exact,

  3. the map $F \to R^0F$ is an isomorphism if and only if $F$ is left exact.

Proof. Let $A$ be an object of $\mathcal{A}$. Let $A[0] \to K^\bullet $ be any quasi-isomorphism. Then it is also true that $A[0] \to \tau _{\geq 0}K^\bullet $ is a quasi-isomorphism. Hence in the category $A[0]/\text{Qis}^{+}(\mathcal{A})$ the quasi-isomorphisms $s : A[0] \to K^\bullet $ with $K^ n = 0$ for $n < 0$ are cofinal. Thus it is clear that $H^ i(RF(A[0])) = 0$ for $i < 0$. Moreover, for such an $s$ the sequence

\[ 0 \to A \to K^0 \to K^1 \]

is exact. Hence if $F$ is left exact, then $0 \to F(A) \to F(K^0) \to F(K^1)$ is exact as well, and we see that $F(A) \to H^0(F(K^\bullet ))$ is an isomorphism for every $s : A[0] \to K^\bullet $ as above which implies that $H^0(RF(A[0])) = F(A)$.

Let $0 \to A \to B \to C \to 0$ be a short exact sequence of $\mathcal{A}$. By Lemma 13.12.1 we obtain a distinguished triangle $(A[0], B[0], C[0], a, b, c)$ in $K^{+}(\mathcal{A})$. From the long exact cohomology sequence (and the vanishing for $i < 0$ proved above) we deduce that $0 \to R^0F(A) \to R^0F(B) \to R^0F(C)$ is exact. Hence $R^0F$ is left exact. Of course this also proves that if $F \to R^0F$ is an isomorphism, then $F$ is left exact. $\square$

Comments (1)

Comment #7814 by Anonymous on

In the second sentence of the second paragraph of the proof, I think it should read "... we obtain a distinguished triangle ... in " instead of "... we obtain a distinguished triangle ... in ".

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Your email address will not be published.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 05TD. Beware of the difference between the letter 'O' and the digit '0'.