The Stacks project

Lemma 15.26.3. Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \geq 0$ and $I = \text{Fit}_ k(M)$. For every $a \in I$ with $R' = R[\frac{I}{a}]$ the strict transform

\[ M' = (M \otimes _ R R')/a\text{-power torsion} \]

has $\text{Fit}_ k(M') = R'$.

Proof. First observe that $\text{Fit}_ k(M \otimes _ R R') = IR' = aR'$. The first equality by Lemma 15.8.4 part (3) and the second equality by Algebra, Lemma 10.70.2. From Lemma 15.8.8 and exactness of localization we see that $M'_{\mathfrak p'}$ can be generated by $\leq k$ elements for every prime $\mathfrak p'$ of $R'$. Then $\text{Fit}_ k(M') = R'$ for example by Lemma 15.8.6. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 15.26: Blowing up and flatness

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Your email address will not be published.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