Exercise 111.21.2. Let $k$ be your favorite algebraically closed field. Let $n \geq 1$. Let $k[x_1, \ldots , x_ n]$ be the polynomial ring. Set $\mathfrak m = (x_1, \ldots , x_ n)$. Let $k[x_1, \ldots , x_ n] \subset A$ be a finite type extension of domains. Set $d = \dim (A)$.

Show that $d - 1 \geq \dim (A/\mathfrak m A) \geq d - n$ if $A/\mathfrak mA \not= 0$.

Show by example that every value can occur.

Show by example that $\mathop{\mathrm{Spec}}(A/\mathfrak m A)$ can have irreducible components of different dimensions.

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