The Stacks project

Lemma 37.56.1. 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. $\square$

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