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

  1. $\tau _{\geq i + 1}(K^\bullet \otimes _ R R_ f)$ is a perfect object of $D(R_ f)$ with tor amplitude in $[i + 1, \infty ]$,

  2. $H^ i(K^\bullet )_ f$ is a finite free $R_ f$-module, and

  3. 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A1V. Beware of the difference between the letter 'O' and the digit '0'.