The Stacks project

Lemma 48.11.4. Let $f : Y \to X$ be a finite pseudo-coherent morphism of schemes (a finite morphism of Noetherian schemes is pseudo-coherent). The functor $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ Y, -)$ maps $D_\mathit{QCoh}^+(\mathcal{O}_ X)$ into $D_\mathit{QCoh}^+(f_*\mathcal{O}_ Y)$. If $X$ is quasi-compact and quasi-separated, then the diagram

\[ \xymatrix{ D_\mathit{QCoh}^+(\mathcal{O}_ X) \ar[rr]_ a \ar[rd]_{R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ Y, -)} & & D_\mathit{QCoh}^+(\mathcal{O}_ Y) \ar[ld]^\Phi \\ & D_\mathit{QCoh}^+(f_*\mathcal{O}_ Y) } \]

is commutative, where $a$ is the right adjoint of Lemma 48.3.1 for $f$ and $\Phi $ is the equivalence of Derived Categories of Schemes, Lemma 36.5.4.

Proof. (The parenthetical remark follows from More on Morphisms, Lemma 37.60.9.) Since $f$ is pseudo-coherent, the $\mathcal{O}_ X$-module $f_*\mathcal{O}_ Y$ is pseudo-coherent, see More on Morphisms, Lemma 37.60.8. Thus $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ Y, -)$ maps $D_\mathit{QCoh}^+(\mathcal{O}_ X)$ into $D_\mathit{QCoh}^+(f_*\mathcal{O}_ Y)$, see Derived Categories of Schemes, Lemma 36.10.8. Then $\Phi \circ a$ and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ Y, -)$ agree on $D_\mathit{QCoh}^+(\mathcal{O}_ X)$ because these functors are both right adjoint to the restriction functor $D_\mathit{QCoh}^+(f_*\mathcal{O}_ Y) \to D_\mathit{QCoh}^+(\mathcal{O}_ X)$. To see this use Lemmas 48.3.5 and 48.11.2. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 48.11: Right adjoint of pushforward for finite morphisms

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Your email address will not be published.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 0AX2. Beware of the difference between the letter 'O' and the digit '0'.