Exercise 111.50.5. Let $k$ be a field, let $f_1, \ldots , f_ c \in k[x_1, \ldots , x_ n]$, and let $A = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$. Assume that $f_ j(0, \ldots , 0) = 0$. This means that $\mathfrak m = (x_1, \ldots , x_ n)A$ is a maximal ideal. Prove that the local ring $A_\mathfrak m$ is regular if the rank of the matrix

$(\partial f_ j/ \partial x_ i)|_{(x_1, \ldots , x_ n) = (0, \ldots , 0)}$

is $c$. What is the dimension of $A_\mathfrak m$ in this case? Show that the converse is false by giving an example where $A_\mathfrak m$ is regular but the rank is less than $c$; what is the dimension of $A_\mathfrak m$ in your example?

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.

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 0D1T. Beware of the difference between the letter 'O' and the digit '0'.