Dimension of the maximal spectrum

The closed sets of $\mathrm{Spec}A$ are the sets $V(I)$ of primes containing an ideal $I$. An ascending chain of ideals stabilizes because $A$ is noetherian, so descending chains of closed sets stabilize. Every subspace of a noetherian topological space is noetherian: the closures in the ambient space of a descending chain of subspace-closed sets stabilize, and intersection with the subspace recovers each set.

The ambient closures of a strict chain of irreducible closed subsets of $\mathrm{\Omega }$ form a strict chain of irreducible closed subsets of $\mathrm{Spec}A$: closure preserves irreducibility, and intersection with $\mathrm{\Omega }$ recovers each member. Each irreducible closed subset of $\mathrm{Spec}A$ is $V(\mathfrak{p})$ for a prime $\mathfrak{p}$: if $V(I)$ is irreducible, its radical is prime, since $ab$ in the radical implies $V(I)\subseteq V(a)\cup V(b)$. Inclusion of such closed sets reverses inclusion of primes. Hence every chain has length at most $d$.