Lemma 50.23.3. Let $X \to S$ be a morphism of schemes. Let $Z \to X$ be a closed immersion of finite presentation whose conormal sheaf $\mathcal{C}_{Z/X}$ is locally free of rank $c$. Then there is a canonical map

$\gamma ^ p : \Omega ^ p_{Z/S} \to \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S})$

which is locally given by the maps $\gamma ^ p_{f_1, \ldots , f_ c}$ of Remark 50.23.1.

Proof. The assumptions imply that given $x \in Z \subset X$ there exists an open neighbourhood $U$ of $x$ such that $Z$ is cut out by $c$ elements $f_1, \ldots , f_ c \in \mathcal{O}_ X(U)$. Thus it suffices to show that given $f_1, \ldots , f_ c$ and $g_1, \ldots , g_ c$ in $\mathcal{O}_ X(U)$ cutting out $Z \cap U$, the maps $\gamma ^ p_{f_1, \ldots , f_ c}$ and $\gamma ^ p_{g_1, \ldots , g_ c}$ are the same. To do this, after shrinking $U$ we may assume $g_ j = \sum a_{ji} f_ i$ for some $a_{ji} \in \mathcal{O}_ X(U)$. Then we have $c_{f_1, \ldots , f_ c} = \det (a_{ji}) c_{g_1, \ldots , g_ c}$ by Derived Categories of Schemes, Lemma 36.6.12. On the other hand we have

$\text{d}(g_1) \wedge \ldots \wedge \text{d}(g_ c) \equiv \det (a_{ji}) \text{d}(f_1) \wedge \ldots \wedge \text{d}(f_ c) \bmod (f_1, \ldots , f_ c)\Omega ^ c_{X/S}$

Combining these relations, a straightforward calculation gives the desired equality. $\square$

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