Lemma 10.99.3. Suppose that $R \to S$ is a flat and local ring homomorphism of Noetherian local rings. Denote $\mathfrak m$ the maximal ideal of $R$. Suppose $f_1, \ldots , f_ c$ is a sequence of elements of $S$ such that the images $\overline{f}_1, \ldots , \overline{f}_ c$ form a regular sequence in $S/{\mathfrak m}S$. Then $f_1, \ldots , f_ c$ is a regular sequence in $S$ and each of the quotients $S/(f_1, \ldots , f_ i)$ is flat over $R$.

**Proof.**
Induction and Lemma 10.99.2.
$\square$

## 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Your email address will not be published.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.

## Comments (0)