The Stacks project

Example 10.70.5. Let $R$ be a ring. Let $P = R[t_1, \ldots , t_ n]$ be the polynomial algebra. Let $I = (t_1, \ldots , t_ n) \subset P$. Let $a = t_1$. With notation as in Definition 10.70.1 there is an isomorphism

\[ P[x_2, \ldots , x_ n]/(t_1x_2 - t_2, \ldots , t_1x_ n - t_ n) \longrightarrow \textstyle {P[\frac{I}{a}] = P[\frac{I}{t_1}]} \]

sending $x_ i$ to $t_ i/t_1$. We leave it to the reader to show that this map is well defined. Since $I$ is generated by $t_1, \ldots , t_ n$ we see that our map is surjective. To see that our map is injective, the reader can argue that the source and target of our map are $t_1$-torsion free and that the map is an isomorphism after inverting $t_1$, see Lemma 10.70.2. Alternatively, the reader can use the description of the Rees algebra in Example 10.70.4. We omit the details.

Comments (0)

There are also:

  • 1 comment(s) on Section 10.70: Blow up algebras

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Your email address will not be published.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 0G8R. Beware of the difference between the letter 'O' and the digit '0'.