The Stacks project

Lemma 21.39.12. Let $\mathcal{C}$ be a category (endowed with chaotic topology). Let $\mathcal{O} \to \mathcal{O}'$ be a map of sheaves of rings on $\mathcal{C}$. Assume

  1. there exists a cosimplicial object $U_\bullet $ in $\mathcal{C}$ as in Lemma 21.39.7, and

  2. $L\pi _!\mathcal{O} \to L\pi _!\mathcal{O}'$ is an isomorphism.

For $K$ in $D(\mathcal{O})$ we have

\[ L\pi _!(K) = L\pi _!(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}') \]

in $D(\textit{Ab})$.

Proof. Note: in this proof $L\pi _!$ denotes the left derived functor of $\pi _!$ on abelian sheaves. Since $L\pi _!$ commutes with colimits, it suffices to prove this for bounded above complexes of $\mathcal{O}$-modules (compare with argument of Derived Categories, Proposition 13.29.2 or just stick to bounded above complexes). Every such complex is quasi-isomorphic to a bounded above complex whose terms are direct sums of $j_{U!}\mathcal{O}_ U$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, see Modules on Sites, Lemma 18.28.8. Thus it suffices to prove the lemma for $j_{U!}\mathcal{O}_ U$. By assumption

\[ S_\bullet = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U) \]

is a simplicial set homotopy equivalent to the constant simplicial set on a singleton. Set $P_ n = \mathcal{O}(U_ n)$ and $P'_ n = \mathcal{O}'(U_ n)$. Observe that the complex associated to the simplicial abelian group

\[ X_\bullet : n \longmapsto \bigoplus \nolimits _{s \in S_ n} P_ n \]

computes $L\pi _!(j_{U!}\mathcal{O}_ U)$ by Lemma 21.39.7. Since $j_{U!}\mathcal{O}_ U$ is a flat $\mathcal{O}$-module we have $j_{U!}\mathcal{O}_ U \otimes ^\mathbf {L}_\mathcal {O} \mathcal{O}' = j_{U!}\mathcal{O}'_ U$ and $L\pi _!$ of this is computed by the complex associated to the simplicial abelian group

\[ X'_\bullet : n \longmapsto \bigoplus \nolimits _{s \in S_ n} P'_ n \]

As the rule which to a simplicial set $T_\bullet $ associated the simplicial abelian group with terms $\bigoplus _{t \in T_ n} P_ n$ is a functor, we see that $X_\bullet \to P_\bullet $ is a homotopy equivalence of simplicial abelian groups. Similarly, the rule which to a simplicial set $T_\bullet $ associates the simplicial abelian group with terms $\bigoplus _{t \in T_ n} P'_ n$ is a functor. Hence $X'_\bullet \to P'_\bullet $ is a homotopy equivalence of simplicial abelian groups. By assumption $P_\bullet \to P'_\bullet $ is a quasi-isomorphism (since $P_\bullet $, resp. $P'_\bullet $ computes $L\pi _!\mathcal{O}$, resp. $L\pi _!\mathcal{O}'$ by Lemma 21.39.7). We conclude that $X_\bullet $ and $X'_\bullet $ are quasi-isomorphic as desired. $\square$


Comments (0)

There are also:

  • 1 comment(s) on Section 21.39: Homology on a category

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 08RX. Beware of the difference between the letter 'O' and the digit '0'.