The Stacks project

Lemma 37.59.1. Let $X \to S$ be a finite type morphism of affine schemes. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$. The following are equivalent

  1. for some closed immersion $i : X \to \mathbf{A}^ n_ S$ the object $Ri_*E$ of $D(\mathcal{O}_{\mathbf{A}^ n_ S})$ is $m$-pseudo-coherent, and

  2. for all closed immersions $i : X \to \mathbf{A}^ n_ S$ the object $Ri_*E$ of $D(\mathcal{O}_{\mathbf{A}^ n_ S})$ is $m$-pseudo-coherent.

Proof. Say $S = \mathop{\mathrm{Spec}}(R)$ and $X = \mathop{\mathrm{Spec}}(A)$. Let $i$ correspond to the surjection $\alpha : R[x_1, \ldots , x_ n] \to A$ and let $X \to \mathbf{A}^ m_ S$ correspond to $\beta : R[y_1, \ldots , y_ m] \to A$. Choose $f_ j \in R[x_1, \ldots , x_ n]$ with $\alpha (f_ j) = \beta (y_ j)$ and $g_ i \in R[y_1, \ldots , y_ m]$ with $\beta (g_ i) = \alpha (x_ i)$. Then we get a commutative diagram

\[ \xymatrix{ R[x_1, \ldots , x_ n, y_1, \ldots , y_ m] \ar[d]^{x_ i \mapsto g_ i} \ar[rr]_-{y_ j \mapsto f_ j} & & R[x_1, \ldots , x_ n] \ar[d] \\ R[y_1, \ldots , y_ m] \ar[rr] & & A } \]

corresponding to the commutative diagram of closed immersions

\[ \xymatrix{ \mathbf{A}^{n + m}_ S & \mathbf{A}^ n_ S \ar[l] \\ \mathbf{A}^ m_ S \ar[u] & X \ar[u] \ar[l] } \]

Thus it suffices to show that under a closed immersion

\[ f : \mathbf{A}^ m_ S \to \mathbf{A}^{n + m}_ S \]

an object $E$ of $D(\mathcal{O}_{\mathbf{A}^ m_ S})$ is $m$-pseudo-coherent if and only if $Rf_*E$ is $m$-pseudo-coherent. This follows from Derived Categories of Schemes, Lemma 36.12.5 and the fact that $f_*\mathcal{O}_{\mathbf{A}^ m_ S}$ is a pseudo-coherent $\mathcal{O}_{\mathbf{A}^{n + m}_ S}$-module. The pseudo-coherence of $f_*\mathcal{O}_{\mathbf{A}^ m_ S}$ is straightforward to prove directly, but it also follows from Derived Categories of Schemes, Lemma 36.10.2 and More on Algebra, Lemma 15.81.3. $\square$


Comments (0)


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Your email address will not be published.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 09VC. Beware of the difference between the letter 'O' and the digit '0'.