The Stacks project

Lemma 21.46.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. Let $K$ be an object of $D(\mathcal{O})$.

  1. If $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}$ is bounded above, then $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$ is uniformly bounded above for all $n$.

  2. If $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}$ as an object of $D(\mathcal{O}/\mathcal{I})$ has tor amplitude in $[a, b]$, then $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$ as an object of $D(\mathcal{O}/\mathcal{I}^ n)$ has tor amplitude in $[a, b]$ for all $n$.

Proof. Proof of (1). Assume that $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}$ is bounded above, say $H^ i(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}) = 0$ for $i > b$. Note that we have distinguished triangles

\[ K \otimes _\mathcal {O}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1} \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^{n + 1} \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1}[1] \]

and that

\[ K \otimes _\mathcal {O}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1} = \left( K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}\right) \otimes _{\mathcal{O}/\mathcal{I}}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1} \]

By induction we conclude that $H^ i(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n) = 0$ for $i > b$ for all $n$.

Proof of (2). Assume $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}$ as an object of $D(\mathcal{O}/\mathcal{I})$ has tor amplitude in $[a, b]$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}/\mathcal{I}^ n$-modules. Then we have a finite filtration

\[ 0 \subset \mathcal{I}^{n - 1}\mathcal{F} \subset \ldots \subset \mathcal{I}\mathcal{F} \subset \mathcal{F} \]

whose successive quotients are sheaves of $\mathcal{O}/\mathcal{I}$-modules. Thus to prove that $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$ has tor amplitude in $[a, b]$ it suffices to show $H^ i(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n \otimes _{\mathcal{O}/\mathcal{I}^ n}^\mathbf {L} \mathcal{G})$ is zero for $i \not\in [a, b]$ for all $\mathcal{O}/\mathcal{I}$-modules $\mathcal{G}$. Since

\[ \left(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n\right) \otimes _{\mathcal{O}/\mathcal{I}^ n}^\mathbf {L} \mathcal{G} = \left(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}\right) \otimes _{\mathcal{O}/\mathcal{I}}^\mathbf {L} \mathcal{G} \]

for every sheaf of $\mathcal{O}/\mathcal{I}$-modules $\mathcal{G}$ the result follows. $\square$

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 0942. Beware of the difference between the letter 'O' and the digit '0'.