Krull dimension of $\mathbb{Z}[x_1,\dots,x_n] • The object…

Krull dimension of $\mathbb{Z}[x_1,\dots,x_n]

• The objective is to demonstrate that the Krull dimension of the polynomial ring $\mathbb{Z}[x_1, \dots, x_n]$ is $n + 1$.

• A known theorem states that for a Noetherian ring $A$, the dimension of the polynomial ring $A[x_1, \dots, x_n]$ can be calculated using the formula: $\dim(A[x_1, \dots, x_n]) = n + \dim(A)$.

• The proof utilizes Hilbert polynomials and the fact that if $F$ is a field, then the Krull dimension of $F[x_1, \dots, x_n]$ is $n$.

• A maximal ideal $\mathfrak{m}$ in $\mathbb{Z}[x_1, \dots, x_n]$ is considered, along with a prime number $p$ that is contained in $\mathfrak{m}$, which can be identified by examining $\mathfrak{m} \cap \mathbb{Z}$.

• A short exact sequence is constructed:
$$0 \longrightarrow \mathbb{Z}[x_1, \dots, x_n] \overset{p \cdot}{\longrightarrow} \mathbb{Z}[x_1, \dots, x_n] \longrightarrow \mathbb{F}_p[x_1, \dots, x_n] \longrightarrow 0,$$
where the map $p \cdot$ represents multiplication by the prime $p$.

• The sequence is localized at the maximal ideal $\mathfrak{m}$, leading to:
$$0 \longrightarrow \mathbb{Z}[x_1, \dots, x_n]_{\mathfrak{m}} \overset{p \cdot}{\longrightarrow} \mathbb{Z}[x_1, \dots, x_n]_{\mathfrak{m}} \longrightarrow \mathbb{F}_p[x_1, \dots, x_n]_{\mathfrak{m}} \longrightarrow 0.$$

• Hilbert polynomials are employed to derive the equation:
$$P(\mathbb{Z}[x_1, \dots, x_n]_{\mathfrak{m}}, t) = P(\mathbb{Z}[x_1, \dots, x_n]_{\mathfrak{m}}, t) \cdot t + P(\mathbb{F}_p[x_1, \dots, x_n]_{\mathfrak{m}}, t).$$

• The term $P(\mathbb{Z}[x_1, \dots, x_n]_{\mathfrak{m}}, t) \cdot t$ on the right side indicates that the dimension of the polynomial ring increases by one for each additional variable, represented by the factor $t$.

• The validity of the proof relies on comprehending the localization process and the significance of Hilbert polynomials in connecting the dimensions of the involved rings.

• The final conclusion is that the Krull dimension of $\mathbb{Z}[x_1, \dots, x_n]$ is confirmed to be $n + 1$.