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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


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