The Stacks project

Proposition 52.28.7. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module and let $s \in \Gamma (X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$ and denote $\mathcal{I} \subset \mathcal{O}_ X$ the corresponding sheaf of ideals. Let $\mathcal{V}$ be the set of open subschemes of $X$ containing $Y$ ordered by reverse inclusion. Assume that for all $x \in X \setminus Y$ we have

\[ \text{depth}(\mathcal{O}_{X, x}) + \dim (\overline{\{ x\} }) > 2 \]

Then the completion functor

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{Coh}(\mathcal{O}_ V) \longrightarrow \textit{Coh}(X, \mathcal{I}) \]

is an equivalence on the full subcategories of finite locally free objects.

Proof. To prove fully faithfulness it suffices to prove that

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \Gamma (V, \mathcal{L}^{\otimes m}) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{L}^{\otimes m}|_{Y_ n}) \]

is an isomorphism for all $m$, see Lemma 52.15.2. This follows from Lemma 52.28.2.

Essential surjectivity. Let $(\mathcal{F}_ n)$ be a finite locally free object of $\textit{Coh}(X, \mathcal{I})$. Then for $y \in Y$ we have $\mathcal{F}_ y^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_{n, y}$ is is a finite free $\mathcal{O}_{X, y}^\wedge $-module. Let $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ be a prime with $\mathfrak p \not\in V(\mathcal{I}_ y^\wedge )$. Then $\mathfrak p$ lies over a prime $\mathfrak p_0 \subset \mathcal{O}_{X, y}$ which corresponds to a specialization $x \leadsto y$ with $x \not\in Y$. By Local Cohomology, Lemma 51.11.3 and some dimension theory (see Varieties, Section 33.20) we have

\[ \text{depth}((\mathcal{O}_{X, y}^\wedge )_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) = \text{depth}(\mathcal{O}_{X, x}) + \dim (\overline{\{ x\} }) - \dim (\overline{\{ y\} }) \]

Thus our assumptions imply the assumptions of Proposition 52.28.5 are satisfied and we find that $(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ X$-module $\mathcal{F}$. It then follows that $\mathcal{F}_ y$ is finite free for all $y \in Y$ and hence $\mathcal{F}$ is finite locally free in an open neighbourhood $V$ of $Y$. This finishes the proof. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 52.28: Application to Lefschetz theorems

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Your email address will not be published.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 0EL7. Beware of the difference between the letter 'O' and the digit '0'.