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. $\square$

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