The Stacks project

Lemma 50.17.4. Let $i : Z \to X$ be a closed immersion of schemes which is regular of codimension $c$. Then $\mathop{\mathrm{Ext}}\nolimits ^ q_{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{E}) = 0$ for $q < c$ for $\mathcal{E}$ locally free on $X$ and $\mathcal{F}$ any $\mathcal{O}_ Z$-module.

Proof. By the local to global spectral sequence of $\mathop{\mathrm{Ext}}\nolimits $ it suffices to prove this affine locally on $X$. See Cohomology, Section 20.43. Thus we may assume $X = \mathop{\mathrm{Spec}}(A)$ and there exists a regular sequence $f_1, \ldots , f_ c$ in $A$ such that $Z = \mathop{\mathrm{Spec}}(A/(f_1, \ldots , f_ c))$. We may assume $c \geq 1$. Then we see that $f_1 : \mathcal{E} \to \mathcal{E}$ is injective. Since $i_*\mathcal{F}$ is annihilated by $f_1$ this shows that the lemma holds for $i = 0$ and that we have a surjection

\[ \mathop{\mathrm{Ext}}\nolimits ^{q - 1}_{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{E}/f_1\mathcal{E}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ q_{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{E}) \]

Thus it suffices to show that the source of this arrow is zero. Next we repeat this argument: if $c \geq 2$ the map $f_2 : \mathcal{E}/f_1\mathcal{E} \to \mathcal{E}/f_1\mathcal{E}$ is injective. Since $i_*\mathcal{F}$ is annihilated by $f_2$ this shows that the lemma holds for $q = 1$ and that we have a surjection

\[ \mathop{\mathrm{Ext}}\nolimits ^{q - 2}_{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{E}/f_1\mathcal{E} + f_2\mathcal{E}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^{q - 1}_{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{E}/f_1\mathcal{E}) \]

Continuing in this fashion the lemma is proved. $\square$

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Your email address will not be published.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 0G5I. Beware of the difference between the letter 'O' and the digit '0'.