The Stacks project

Lemma 50.17.5. 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 20.41. 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. $\square$


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