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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