Are global complete intersections locally complete intersections?

Question

In https://stacks.math.columbia.edu/tag/00S8 a global complete intersection is defined to be a ring of the form $R=k[x_1,\ldots,x_n]/(f_1,\ldots,f_c)$ of Krull dimension $n-c$. Is such a ring locally a complete intersection? I.e. if $\mathfrak{m}$ is a maximal ideal, is then $R_\mathfrak{m}$ a local complete intersection?

Answer 1

The dimension of a ring is the supremum of the local dimensions. Hence, $\dim R_\mathfrak{m} \leq n-c$ for all maximal ideals $\mathfrak{m}$. On the other hand, $$R_{\mathfrak{m}} = k[x_1,\ldots,x_m]_{\mathfrak{m}}/(f_1,\ldots, f_c),$$ and $\dim k[x_1,\ldots, x_m]_{\mathfrak{m}} = n$ again, so again $R_{\mathfrak{m}}$ is a complete intersection.