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?
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