The Stacks project

Lemma 96.19.5. Let

\[ \xymatrix{ \mathcal{V} \ar[d]_ g \ar[r]_ h & \mathcal{U} \ar[d]^ f \\ \mathcal{Y} \ar[r]^ e & \mathcal{X} } \]

be a $2$-fibre product of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ and let $\mathcal{F}$ be an abelian presheaf on $\mathcal{X}$. Then the map $e^{-1}\mathcal{K}^\bullet (f, \mathcal{F}) \to \mathcal{K}^\bullet (g, e^{-1}\mathcal{F})$ of Lemma 96.19.2 is an isomorphism of complexes of abelian presheaves.

Proof. Let $y$ be an object of $\mathcal{Y}$ lying over the scheme $T$. Set $x = e(y)$. We are going to show that the map induces an isomorphism on sections over $y$. Note that

\[ \Gamma (y, e^{-1}\mathcal{K}^\bullet (f, \mathcal{F})) = \Gamma (x, \mathcal{K}^\bullet (f, \mathcal{F})) = \check{\mathcal{C}}^\bullet ( (\mathit{Sch}/T)_{fppf} \times _{x, \mathcal{X}} \mathcal{U} \to (\mathit{Sch}/T)_{fppf}, x^{-1}\mathcal{F}) \]

by Remark 96.19.4. On the other hand,

\[ \Gamma (y, \mathcal{K}^\bullet (g, e^{-1}\mathcal{F})) = \check{\mathcal{C}}^\bullet ( (\mathit{Sch}/T)_{fppf} \times _{y, \mathcal{Y}} \mathcal{V} \to (\mathit{Sch}/T)_{fppf}, y^{-1}e^{-1}\mathcal{F}) \]

also by Remark 96.19.4. Note that $y^{-1}e^{-1}\mathcal{F} = x^{-1}\mathcal{F}$ and since the diagram is $2$-cartesian the $1$-morphism

\[ (\mathit{Sch}/T)_{fppf} \times _{y, \mathcal{Y}} \mathcal{V} \to (\mathit{Sch}/T)_{fppf} \times _{x, \mathcal{X}} \mathcal{U} \]

is an equivalence. Hence the map on sections over $y$ is an isomorphism by Lemma 96.18.1. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 96.19: The relative Čech complex

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Your email address will not be published.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 06XC. Beware of the difference between the letter 'O' and the digit '0'.