Theorem 77.13.6 (0DIQ): Derived Grothendieck Existence Theorem

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Part 4: Algebraic Spaces
Chapter 77: Flatness on Algebraic Spaces
Section 77.13: Grothendieck's Existence Theorem, bis
Theorem 77.13.6: Derived Grothendieck Existence Theorem (cite)

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Theorem 77.13.6 (Derived Grothendieck Existence Theorem). In Situation 77.13.1 there exists a pseudo-coherent $K$ in $D(\mathcal{O}_ X)$ such that $K_ n = K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_{X_ n}$ for all $n$ compatibly with the maps $\varphi _ n$ .

Proof. Apply Lemmas 77.13.2, 77.13.3, 77.13.4 to get a pseudo-coherent object $K$ of $D(\mathcal{O}_ X)$ . Choosing affine $U$ in Lemma 77.13.5 it follows immediately that $K$ restricts to $K_ n$ over $X_ n$ . $\square$

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