Lemma 15.75.7 (0BCD)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.75: Lifting complexes
Lemma 15.75.7 (cite)

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Lemma 15.75.7. Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime. Let $K \in D(R)$ be perfect. Set $d_ i = \dim _{\kappa (\mathfrak p)} H^ i(K \otimes _ R^\mathbf {L} \kappa (\mathfrak p))$ . Then $d_ i < \infty$ and only a finite number are nonzero. Then there exists an $f \in R$ , $f \not\in \mathfrak p$ and a complex

$\ldots \to 0 \to R_ f^{\oplus d_ a} \to R_ f^{\oplus d_{a + 1}} \to \ldots \to R_ f^{\oplus d_{b - 1}} \to R_ f^{\oplus d_ b} \to 0 \to \ldots$

representing $K \otimes _ R^\mathbf {L} R_ f$ in $D(R_ f)$ .

Proof. Observe that $K \otimes _ R^\mathbf {L} \kappa (\mathfrak p)$ is perfect as an object of $D(\kappa (\mathfrak p))$ , see Lemma 15.74.9. Hence only a finite number of $d_ i$ are nonzero and they are all finite. Applying Lemma 15.75.6 we get a complex representing $K$ having the desired shape over the local ring $R_\mathfrak p$ . We have $R_\mathfrak p = \mathop{\mathrm{colim}}\nolimits R_ f$ for $f \in R$ , $f \not\in \mathfrak p$ (Algebra, Lemma 10.9.9). We conclude by Lemma 15.74.17. Some details omitted. $\square$

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