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 $D(R_ f)$.
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