The Stacks project

Lemma 90.3.5. Let $A \to B$ be a ring map in $\mathcal{C}_\Lambda $. The following are equivalent

  1. $f$ is surjective,

  2. $\mathfrak m_ A/\mathfrak m_ A^2 \to \mathfrak m_ B/\mathfrak m_ B^2$ is surjective, and

  3. $\mathfrak m_ A/(\mathfrak m_\Lambda A + \mathfrak m_ A^2) \to \mathfrak m_ B/(\mathfrak m_\Lambda B + \mathfrak m_ B^2)$ is surjective.

Proof. For any ring map $f : A \to B$ in $\mathcal{C}_\Lambda $ we have $f(\mathfrak m_ A) \subset \mathfrak m_ B$ for example because $\mathfrak m_ A$, $\mathfrak m_ B$ is the set of nilpotent elements of $A$, $B$. Suppose $f$ is surjective. Let $y \in \mathfrak m_ B$. Choose $x \in A$ with $f(x) = y$. Since $f$ induces an isomorphism $A/\mathfrak m_ A \to B/\mathfrak m_ B$ we see that $x \in \mathfrak m_ A$. Hence the induced map $\mathfrak m_ A/\mathfrak m_ A^2 \to \mathfrak m_ B/\mathfrak m_ B^2$ is surjective. In this way we see that (1) implies (2).

It is clear that (2) implies (3). The map $A \to B$ gives rise to a canonical commutative diagram

\[ \xymatrix{ \mathfrak m_\Lambda /\mathfrak m_\Lambda ^2 \otimes _{k'} k \ar[r] \ar[d] & \mathfrak m_ A/\mathfrak m_ A^2 \ar[r] \ar[d] & \mathfrak m_ A/(\mathfrak m_\Lambda A + \mathfrak m_ A^2) \ar[r] \ar[d] & 0 \\ \mathfrak m_\Lambda /\mathfrak m_\Lambda ^2 \otimes _{k'} k \ar[r] & \mathfrak m_ B/\mathfrak m_ B^2 \ar[r] & \mathfrak m_ B/(\mathfrak m_\Lambda B + \mathfrak m_ B^2) \ar[r] & 0 } \]

with exact rows. Hence if (3) holds, then so does (2).

Assume (2). To show that $A \to B$ is surjective it suffices by Nakayama's lemma (Algebra, Lemma 10.20.1) to show that $A/\mathfrak m_ A \to B/\mathfrak m_ AB$ is surjective. (Note that $\mathfrak m_ A$ is a nilpotent ideal.) As $k = A/\mathfrak m_ A = B/\mathfrak m_ B$ it suffices to show that $\mathfrak m_ AB \to \mathfrak m_ B$ is surjective. Applying Nakayama's lemma once more we see that it suffices to see that $\mathfrak m_ AB/\mathfrak m_ A\mathfrak m_ B \to \mathfrak m_ B/\mathfrak m_ B^2$ is surjective which is what we assumed. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 90.3: The base category

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Your email address will not be published.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 06S3. Beware of the difference between the letter 'O' and the digit '0'.