The Stacks project

Remark 15.40.7. The assertion of Lemma 15.40.6 is quite strong. Namely, suppose that we have a diagram

\[ \xymatrix{ & B \\ A \ar[r] & A' \ar[u] } \]

of local homomorphisms of Noetherian complete local rings where $A \to A'$ induces an isomorphism of residue fields $k = A/\mathfrak m_ A = A'/\mathfrak m_{A'}$ and with $B \otimes _{A'} k$ formally smooth over $k$. Then we can extend this to a commutative diagram

\[ \xymatrix{ C \ar[r] & B \\ A \ar[r] \ar[u] & A' \ar[u] } \]

of local homomorphisms of Noetherian complete local rings where $A \to C$ is formally smooth in the $\mathfrak m_ C$-adic topology and where $C \otimes _ A k \cong B \otimes _{A'} k$. Namely, pick $A \to C$ as in Lemma 15.40.6 lifting $B \otimes _{A'} k$ over $k$. By formal smoothness we can find the arrow $C \to B$, see Lemma 15.37.5. Denote $C \otimes _ A^\wedge A'$ the completion of $C \otimes _ A A'$ with respect to the ideal $C \otimes _ A \mathfrak m_{A'}$. Note that $C \otimes _ A^\wedge A'$ is a Noetherian complete local ring (see Algebra, Lemma 10.97.5) which is flat over $A'$ (see Algebra, Lemma 10.99.11). We have moreover

  1. $C \otimes _ A^\wedge A' \to B$ is surjective,

  2. if $A \to A'$ is surjective, then $C \to B$ is surjective,

  3. if $A \to A'$ is finite, then $C \to B$ is finite, and

  4. if $A' \to B$ is flat, then $C \otimes _ A^\wedge A' \cong B$.

Namely, by Nakayama's lemma for nilpotent ideals (see Algebra, Lemma 10.20.1) we see that $C \otimes _ A k \cong B \otimes _{A'} k$ implies that $C \otimes _ A A'/\mathfrak m_{A'}^ n \to B/\mathfrak m_{A'}^ nB$ is surjective for all $n$. This proves (1). Parts (2) and (3) follow from part (1). Part (4) follows from Algebra, Lemma 10.99.1.


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Your email address will not be published.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