The Stacks project

Lemma 15.63.3. Let $R$ be a ring. Let $K^\bullet $ be an object of $D(R)$. Let $a, b \in \mathbf{Z}$. The following are equivalent

  1. $K^\bullet $ has tor-amplitude in $[a, b]$.

  2. $K^\bullet $ is quasi-isomorphic to a complex $E^\bullet $ of flat $R$-modules with $E^ i = 0$ for $i \not\in [a, b]$.

Proof. If (2) holds, then we may compute $K^\bullet \otimes _ R^\mathbf {L} M = E^\bullet \otimes _ R M$ and it is clear that (1) holds. Assume that (1) holds. We may replace $K^\bullet $ by a projective resolution. Let $n$ be the largest integer such that $K^ n \not= 0$. If $n > b$, then $K^{n - 1} \to K^ n$ is surjective as $H^ n(K^\bullet ) = 0$. As $K^ n$ is projective we see that $K^{n - 1} = K' \oplus K^ n$. Hence it suffices to prove the result for the complex $(K')^\bullet $ which is the same as $K^\bullet $ except has $K'$ in degree $n - 1$ and $0$ in degree $n$. Thus, by induction on $n$, we reduce to the case that $K^\bullet $ is a complex of projective $R$-modules with $K^ i = 0$ for $i > b$.

Set $E^\bullet = \tau _{\geq a}K^\bullet $. Everything is clear except that $E^ a$ is flat which follows immediately from Lemma 15.63.2 and the definitions. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 15.63: Tor dimension

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Your email address will not be published.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 0654. Beware of the difference between the letter 'O' and the digit '0'.