The Stacks project

Lemma 15.63.2. Let $R$ be a ring. Let $K^\bullet $ be a bounded above complex of flat $R$-modules with tor-amplitude in $[a, b]$. Then $\mathop{\mathrm{Coker}}(d_ K^{a - 1})$ is a flat $R$-module.

Proof. As $K^\bullet $ is a bounded above complex of flat modules we see that $K^\bullet \otimes _ R M = K^\bullet \otimes _ R^{\mathbf{L}} M$. Hence for every $R$-module $M$ the sequence

\[ K^{a - 2} \otimes _ R M \to K^{a - 1} \otimes _ R M \to K^ a \otimes _ R M \]

is exact in the middle. Since $K^{a - 2} \to K^{a - 1} \to K^ a \to \mathop{\mathrm{Coker}}(d_ K^{a - 1}) \to 0$ is a flat resolution this implies that $\text{Tor}_1^ R(\mathop{\mathrm{Coker}}(d_ K^{a - 1}), M) = 0$ for all $R$-modules $M$. This means that $\mathop{\mathrm{Coker}}(d_ K^{a - 1})$ is flat, see Algebra, Lemma 10.74.8. $\square$


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Your email address will not be published.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 0653. Beware of the difference between the letter 'O' and the digit '0'.