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MSR206 Utility Program V 1.76 By ThejermQ:

Zariski topology induced by affine topology

Let $K$ be a field, let $L$ be a finite field extension of $K$ and let $A$ be a finite $\mathbb{Z}$-algebra.
Then the Zariski topology on $\mathrm{Spec}(A)$ is (trivially) induced by the affine topology, right? That is, $\overline{V}_{\mathrm{aff}}=\overline{V}_{\mathrm{zar}}\cap \overline{\mathrm{Aff}}$, for $V$ open in $\mathrm{Spec}(A)$? I expect the equality to be true in general, but I can’t find a proof.
Theorem 4.4 in Milne’s notes says that the Zariski topology is induced by the usual topology if $A$ is a finitely generated $\mathbb{Z}$-algebra, which isn’t the case here. However, I don’t think we need finiteness in the theorem 4.4.

A:

All you really need is the hypothesis that $A$ is finitely generated over $\mathbf{Z}$: this guarantees that the affine topology on $\mathrm{Spec}(A)$ is the Zariski topology. If $A$ is not finitely generated over $\mathbf{Z}$, there are many affine open subsets $U_i$ such that $\bigcap U_i = \emptyset$. The Zariski topology on $\mathrm{Spec}(A)$ is the coarsest topology that covers the affine open subsets of $\mathrm{Spec}(A)$. The subspace topology on $\mathrm{Spec}(A)\backslash \bigcup U_i$ is finer than the Zariski topology, and so the Zariski topology is not induced by the affine topology.
As David Roberts points out, this is a very clear and illuminating explanation for how the Zariski topology is induced.

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