The Stacks project

Lemma 86.14.1. Let $A \to B$ be a morphism in $\textit{WAdm}^{Noeth}$ (Formal Spaces, Section 85.17). The following are equivalent:

  1. $A \to B$ satisfies the equivalent conditions of Lemma 86.11.1 and there exists an ideal of definition $I \subset B$ such that $B$ is rig-smooth over $(A, I)$, and

  2. $A \to B$ satisfies the equivalent conditions of Lemma 86.11.1 and for all ideals of definition $I \subset A$ the algebra $B$ is rig-smooth over $(A, I)$.

Proof. Let $I$ and $I'$ be ideals of definitions of $A$. Then there exists an integer $c \geq 0$ such that $I^ c \subset I'$ and $(I')^ c \subset I$. Hence $B$ is rig-smooth over $(A, I)$ if and only if $B$ is rig-smooth over $(A, I')$. This follows from Definition 86.4.1, the inclusions $I^ c \subset I'$ and $(I')^ c \subset I$, and the fact that the naive cotangent complex $\mathop{N\! L}\nolimits _{B/A}^\wedge $ is independent of the choice of ideal of definition of $A$ by Remark 86.11.2. $\square$


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Your email address will not be published.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