The Stacks project

Lemma 52.17.2. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. If the inverse system $H^0(U, \mathcal{F}_ n)$ has Mittag-Leffler, then the canonical maps

\[ \widetilde{M/I^ nM}|_ U \to \mathcal{F}_ n \]

are surjective for all $n$ where $M$ is as in (

Proof. Surjectivity may be checked on the stalk at some point $y \in Y \setminus Z$. If $y$ corresponds to the prime $\mathfrak q \subset A$, then we can choose $f \in \mathfrak a$, $f \not\in \mathfrak q$. Then it suffices to show

\[ M_ f \longrightarrow H^0(U, \mathcal{F}_ n)_ f = H^0(D(f), \mathcal{F}_ n) \]

is surjective as $D(f)$ is affine (equality holds by Properties, Lemma 28.17.1). Since we have the Mittag-Leffler property, we find that

\[ \mathop{\mathrm{Im}}(M \to H^0(U, \mathcal{F}_ n)) = \mathop{\mathrm{Im}}(H^0(U, \mathcal{F}_ m) \to H^0(U, \mathcal{F}_ n)) \]

for some $m \geq n$. Using the long exact sequence of cohomology we see that

\[ \mathop{\mathrm{Coker}}(H^0(U, \mathcal{F}_ m) \to H^0(U, \mathcal{F}_ n)) \subset H^1(U, \mathop{\mathrm{Ker}}(\mathcal{F}_ m \to \mathcal{F}_ n)) \]

Since $U = X \setminus V(\mathfrak a)$ this $H^1$ is $\mathfrak a$-power torsion. Hence after inverting $f$ the cokernel becomes zero. $\square$

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Your email address will not be published.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