The Stacks project

Lemma 15.72.4. Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime ideal. Let $i \in \mathbf{Z}$. Let $K^\bullet $ be a pseudo-coherent complex of $R$-modules such that $H^ i(K^\bullet \otimes _ R^{\mathbf{L}} \kappa (\mathfrak p)) = 0$. Then there exists an $f \in R$, $f \not\in \mathfrak p$ and a canonical direct sum decomposition

\[ K^\bullet \otimes _ R R_ f = \tau _{\geq i + 1}(K^\bullet \otimes _ R R_ f) \oplus \tau _{\leq i - 1}(K^\bullet \otimes _ R R_ f) \]

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

Proof. This is an often used special case of Lemma 15.72.2. A direct proof is as follows. We may assume that $K^\bullet $ is a bounded above complex of finite free $R$-modules. Let us inspect what is happening in degree $i$:

\[ \ldots \to K^{i - 2} \to R^{\oplus l} \to R^{\oplus m} \to R^{\oplus n} \to K^{i + 2} \to \ldots \]

Let $A$ be the $m \times l$ matrix corresponding to $K^{i - 1} \to K^ i$ and let $B$ be the $n \times m$ matrix corresponding to $K^ i \to K^{i + 1}$. The assumption is that $A \bmod \mathfrak p$ has rank $r$ and that $B \bmod \mathfrak p$ has rank $m - r$. In other words, there is some $r \times r$ minor $a$ of $A$ which is not in $\mathfrak p$ and there is some $(m - r) \times (m - r)$-minor $b$ of $B$ which is not in $\mathfrak p$. Set $f = ab$. Then after inverting $f$ we can find direct sum decompositions $K^{i - 1} = R^{\oplus l - r} \oplus R^{\oplus r}$, $K^ i = R^{\oplus r} \oplus R^{\oplus m - r}$, $K^{i + 1} = R^{\oplus m - r} \oplus R^{\oplus n - m + r}$ such that the module map $K^{i - 1} \to K^ i$ kills of $R^{\oplus l - r}$ and induces an isomorphism of $R^{\oplus r}$ onto the corresponding summand of $K^ i$ and such that the module map $K^ i \to K^{i + 1}$ kills of $R^{\oplus r}$ and induces an isomorphism of $R^{\oplus m - r}$ onto the corresponding summand of $K^{i + 1}$. Thus $K^\bullet $ becomes quasi-isomorphic to

\[ \ldots \to K^{i - 2} \to R^{\oplus l - r} \to 0 \to R^{\oplus n - m + r} \to K^{i + 2} \to \ldots \]

and everything is clear. $\square$


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Your email address will not be published.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 068U. Beware of the difference between the letter 'O' and the digit '0'.