The Stacks project

Lemma 16.2.5 (Elkik). Let $R \to A$ be a ring map of finite presentation. The singular ideal $H_{A/R}$ is the radical of the ideal generated by strictly standard elements in $A$ over $R$ and also the radical of the ideal generated by elementary standard elements in $A$ over $R$.

Proof. Assume $a$ is strictly standard in $A$ over $R$. We claim that $A_ a$ is smooth over $R$, which proves that $a \in H_{A/R}$. Namely, let $A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$, $c$, and $a' \in A$ be as in Definition 16.2.3. Write $I = (f_1, \ldots , f_ m)$ so that the naive cotangent complex of $A$ over $R$ is given by $I/I^2 \to \bigoplus A\text{d}x_ i$. Assumption (16.2.3.4) implies that $(I/I^2)_ a$ is generated by the classes of $f_1, \ldots , f_ c$. Assumption (16.2.3.3) implies that the differential $(I/I^2)_ a \to \bigoplus A_ a\text{d}x_ i$ has a left inverse, see Lemma 16.2.4. Hence $R \to A_ a$ is smooth by definition and Algebra, Lemma 10.134.13.

Let $H_ e, H_ s \subset A$ be the radical of the ideal generated by elementary, resp. strictly standard elements of $A$ over $R$. By definition and what we just proved we have $H_ e \subset H_ s \subset H_{A/R}$. The inclusion $H_{A/R} \subset H_ e$ follows from Lemma 16.2.2. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 16.2: Singular ideals

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Your email address will not be published.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 07CA. Beware of the difference between the letter 'O' and the digit '0'.