Corollary

Let $S$ be a multiplicative subset of $A$, $A' = S^{-1}A$ and $h$ the canonical homomorphism $i_A^S$; then $^a h$ is a homeomorphism of $X' = \mathrm{Spec}(A')$ onto the subspace of $X = \mathrm{Spec}(A)$ consisting of the prime ideals of $A$ which do not meet $S$.

Let $f' = f/s$, where $f \in A, s \in S$; then $X'_{f'} = X'_{f/1}$ since $s/1$ is invertible in $A'$. We know already that $^a h$ is injective and that, for all $p' \in X'$, the relations $f/1 \in p'$ and $f \in h^{-1}(p') = ^a h(p')$ are equivalent and hence the conditions $p' \in X'_{f/1}$ and $^a h(p') \in X_f$ are equivalent; this shows that $^a h(X'_{f'})$ is equal to $X_f \cap ^a h(X')$, whence the first assertion, since the $X, (resp.\ X'_{f'})$ form a base of the topology of $X$ (resp. $X$). The second assertion follows from , no. 5, Proposition 11 (ii)