Lemma 15.76.3 (0A1V)—The Stacks project

Lemma 15.76.3. Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime ideal. Let $K^\bullet $ be a pseudo-coherent complex of $R$-modules. Assume that for some $i \in \mathbf{Z}$ the maps

\[ H^ i(K^\bullet ) \otimes _ R \kappa (\mathfrak p) \longrightarrow H^ i(K^\bullet \otimes _ R^{\mathbf{L}} \kappa (\mathfrak p)) \quad \text{and}\quad H^{i - 1}(K^\bullet ) \otimes _ R \kappa (\mathfrak p) \longrightarrow H^{i - 1}(K^\bullet \otimes _ R^{\mathbf{L}} \kappa (\mathfrak p)) \]

are surjective. Then there exists an $f \in R$, $f \not\in \mathfrak p$ such that

$\tau _{\geq i + 1}(K^\bullet \otimes _ R R_ f)$ is a perfect object of $D(R_ f)$ with tor amplitude in $[i + 1, \infty ]$,
$H^ i(K^\bullet )_ f$ is a finite free $R_ f$-module, and
there is a canonical direct sum decomposition

\[ K^\bullet \otimes _ R R_ f \cong \tau _{\leq i - 1}(K^\bullet \otimes _ R R_ f) \oplus H^ i(K^\bullet )_ f[-i] \oplus \tau _{\geq i + 1}(K^\bullet \otimes _ R R_ f) \]

in $D(R_ f)$.

Proof. We get (1) from Lemma 15.76.2 as well as a splitting $K^\bullet \otimes _ R R_ f = \tau _{\leq i}K^\bullet \otimes _ R R_ f \oplus \tau _{\geq i + 1}K^\bullet \otimes _ R R_ f$ in $D(R_ f)$. Applying Lemma 15.76.2 once more to $\tau _{\leq i}K^\bullet \otimes _ R R_ f$ we obtain (after suitably choosing $f$) a splitting $\tau _{\leq i}K^\bullet \otimes _ R R_ f = \tau _{\leq i - 1}K^\bullet \otimes _ R R_ f \oplus H^ i(K^\bullet )_ f$ in $D(R_ f)$ as well as the conclusion that $H^ i(K)_ f$ is a flat perfect module, i.e., finite projective. $\square$

The Stacks project

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