The Stacks project

Lemma 48.15.6. Let $Y$ be a quasi-compact and quasi-separated scheme. Let $i : X \to Y$ be a Koszul-regular closed immersion. Let $a$ be the right adjoint of $Ri_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ of Lemma 48.3.1. Then there is an isomorphism

\[ \wedge ^ r\mathcal{N}[-r] \longrightarrow a(\mathcal{O}_ Y) \]

where $\mathcal{N} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{C}_{X/Y}, \mathcal{O}_ X)$ is the normal sheaf of $i$ (Morphisms, Section 29.31) and $r$ is its rank viewed as a locally constant function on $X$.

Proof. Recall, from Lemmas 48.9.7 and 48.9.3, that $a(\mathcal{O}_ Y)$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ whose pushforward to $Y$ is $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(i_*\mathcal{O}_ X, \mathcal{O}_ Y)$. Thus the result follows from Lemma 48.15.5. $\square$

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