The Stacks project

Lemma 37.56.4. Let $S$ be an affine scheme. Let $V \subset S$ be a standard open. Let $X \to V$ be a finite type morphism of affine schemes. Let $U \subset X$ be an affine open. Let $E$ be an object of $D(\mathcal{O}_ X)$. If the equivalent conditions of Lemma 37.56.1 are satisfied for the pair $(X \to V, E)$, then the equivalent conditions of Lemma 37.56.1 are satisfied for the pair $(U \to S, E|_ U)$.

Proof. Write $S = \mathop{\mathrm{Spec}}(R)$, $V = D(f)$, $X = \mathop{\mathrm{Spec}}(A)$, and $U = D(g)$. Assume the equivalent conditions of Lemma 37.56.1 are satisfied for the pair $(X \to V, E)$.

Choose $R_ f[x_1, \ldots , x_ n] \to A$ surjective. Write $R_ f = R[x_0]/(fx_0 - 1)$. Then $R[x_0, x_1, \ldots , x_ n] \to A$ is surjective, and $R_ f[x_1, \ldots , x_ n]$ is pseudo-coherent as an $R[x_0, \ldots , x_ n]$-module. Thus we have

\[ X \to \mathbf{A}^ n_ V \to \mathbf{A}^{n + 1}_ S \]

and we can apply Derived Categories of Schemes, Lemma 36.12.5 to conclude that the pushforward $E'$ of $E$ to $\mathbf{A}^{n + 1}_ S$ is $m$-pseudo-coherent.

Choose an element $g' \in R[x_0, x_1, \ldots , x_ n]$ which maps to $g \in A$. Consider the surjection $R[x_0, \ldots , x_{n + 1}] \to R[x_0, \ldots , x_ n, 1/g']$. We obtain

\[ \xymatrix{ X \ar[d] & U \ar[d] \ar[l] \ar[dr] \\ \mathbf{A}^{n + 1}_ S & D(g')\ar[l] \ar[r] & \mathbf{A}^{n + 2}_ S } \]

where the lower left arrow is an open immersion and the lower right arrow is a closed immersion. We conclude as before that the pushforward of $E'|_{D(g')}$ to $\mathbf{A}^{n + 2}_ S$ is $m$-pseudo-coherent. Since this is also the pushforward of $E|_ U$ to $\mathbf{A}^{n + 2}_ S$ we conclude the lemma is true. $\square$


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Your email address will not be published.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