The Stacks project

Lemma 74.42.10. Let $S$ be a scheme. Let $f : X \to Y$ be a representable proper morphism of Noetherian algebraic spaces over $S$. Let $\mathcal{J}, \mathcal{K} \subset \mathcal{O}_ Y$ be quasi-coherent sheaves of ideals. Assume $f$ is an isomorphism over $V = Y \setminus V(\mathcal{K})$. Set $\mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_ X$. Let $(\mathcal{G}_ n)$ be an object of $\textit{Coh}(Y, \mathcal{J})$, let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module, and let $\beta : (f^*\mathcal{G}_ n) \to \mathcal{F}^\wedge $ be an isomorphism in $\textit{Coh}(X, \mathcal{I})$. Then there exists a map

\[ \alpha : (\mathcal{G}_ n) \longrightarrow (f_*\mathcal{F})^\wedge \]

in $\textit{Coh}(Y, \mathcal{J})$ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$.

Proof. Since $f$ is a proper morphism we see that $f_*\mathcal{F}$ is a coherent $\mathcal{O}_ Y$-module (Cohomology of Spaces, Lemma 67.20.2). Thus the statement of the lemma makes sense. Consider the compositions

\[ \gamma _ n : \mathcal{G}_ n \to f_*f^*\mathcal{G}_ n \to f_*(\mathcal{F}/\mathcal{I}^ n\mathcal{F}). \]

Here the first map is the adjunction map and the second is $f_*\beta _ n$. We claim that there exists a unique $\alpha $ as in the lemma such that the compositions

\[ \mathcal{G}_ n \xrightarrow {\alpha _ n} f_*\mathcal{F}/\mathcal{J}^ nf_*\mathcal{F} \to f_*(\mathcal{F}/\mathcal{I}^ n\mathcal{F}) \]

equal $\gamma _ n$ for all $n$. Because of the uniqueness and ├ętale descent for $\textit{Coh}(Y, \mathcal{J})$ it suffices to prove this ├ętale locally on $Y$. Thus we may assume $Y$ is the spectrum of a Noetherian ring. As $f$ is representable we see that $X$ is a scheme as well. Thus we reduce to the case of schemes, see proof of Cohomology of Schemes, Lemma 30.25.3. $\square$


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Your email address will not be published.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 08BD. Beware of the difference between the letter 'O' and the digit '0'.