Lemma 48.15.6 (0BR0)—The Stacks project

Processing math: 100%

Table of contents
Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.15: Right adjoint of pushforward in examples
Lemma 48.15.6 (cite)

numberstags

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. $\square$

Comments (0)

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).

Comments (0)

Post a comment