The Stacks project

Lemma 21.19.8. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \subset \textit{Ab}(\mathcal{C})$ denote the Serre subcategory consisting of torsion abelian sheaves. Then the functor $D(\mathcal{A}) \to D_\mathcal {A}(\mathcal{C})$ is an equivalence.

Proof. A key observation is that an injective abelian sheaf $\mathcal{I}$ is divisible. Namely, if $s \in \mathcal{I}(U)$ is a local section, then we interpret $s$ as a map $s : j_{U!}\mathbf{Z} \to \mathcal{I}$ and we apply the defining property of an injective object to the injective map of sheaves $n : j_{U!}\mathbf{Z} \to j_{U!}\mathbf{Z}$ to see that there exists an $s' \in \mathcal{I}(U)$ with $ns' = s$.

For a sheaf $\mathcal{F}$ denote $\mathcal{F}_{tor}$ its torsion subsheaf. We claim that if $\mathcal{I}^\bullet $ is a complex of injective abelian sheaves whose cohomology sheaves are torsion, then

\[ \mathcal{I}^\bullet _{tor} \to \mathcal{I}^\bullet \]

is a quasi-isomorphism. Namely, by flatness of $\mathbf{Q}$ over $\mathbf{Z}$ we have

\[ H^ p(\mathcal{I}^\bullet ) \otimes _\mathbf {Z} \mathbf{Q} = H^ p(\mathcal{I}^\bullet \otimes _\mathbf {Z} \mathbf{Q}) \]

which is zero because the cohomology sheaves are torsion. By divisibility (shown above) we see that $\mathcal{I}^\bullet \to \mathcal{I}^\bullet \otimes _\mathbf {Z} \mathbf{Q}$ is surjective with kernel $\mathcal{I}^\bullet _{tor}$. The claim follows from the long exact sequence of cohomology sheaves associated to the short exact sequence you get.

To prove the lemma we will construct right adjoint $T : D(\mathcal{C}) \to D(\mathcal{A})$. Namely, given $K$ in $D(\mathcal{C})$ we can represent $K$ by a K-injective complex $\mathcal{I}^\bullet $ whose cohomology sheaves are injective, see Injectives, Theorem 19.12.6. Then we set $T(K) = \mathcal{I}^\bullet _{tor}$, in other words, $T$ is the right derived functor of taking torsion. The functor $T$ is a right adjoint to $i : D(\mathcal{A}) \to D_\mathcal {A}(\mathcal{C})$. This readily follows from the observation that if $\mathcal{F}^\bullet $ is a complex of torsion sheaves, then

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(\mathcal{F}^\bullet , I^\bullet _{tor}) = \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Ab}(\mathcal{C}))}(\mathcal{F}^\bullet , I^\bullet ) \]

in particular $\mathcal{I}^\bullet _{tor}$ is a K-injective complex of $\mathcal{A}$. Some details omitted; in case of doubt, it also follows from the more general Derived Categories, Lemma 13.30.3. Our claim above gives that $L = T(i(L))$ for $L$ in $D(\mathcal{A})$ and $i(T(K)) = K$ if $K$ is in $D_\mathcal {A}(\mathcal{C})$. Using Categories, Lemma 4.24.4 the result follows. $\square$

Comments (4)

Comment #4538 by Herman Rohrbach on

Typo: in the definition of the right adjoint , the source of the functor is given as , which I think should be .

Comment #4539 by Herman Rohrbach on

Nevermind, I think I see the point of the way it is phrased.

Comment #5042 by Taro konno on

Sorry , What is the definition of torsion sheaf of Stacks Project ? (I'm sorry for my stupid question.)

Comment #5044 by on

An abelian sheaf on a topological space is torsion if all its stalks are torsion. An abelian sheaf on a site is torsion if any section is locally torsion, i.e., if there is a covering of such that is torsion for all .

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 0DD7. Beware of the difference between the letter 'O' and the digit '0'.