Let $A$ be a commutative (unital) ring, and $A[x_1,\ldots,x_n]$ a polynomial ring over it in some finite number of variables. The inclusion $i\colon A \hookrightarrow A[x_1,\ldots,x_n]$ induces (by contraction) a continuous surjection $\mathrm{Spec}(i)\colon \mathrm{Spec}(A[x_1,\ldots,x_n]) \twoheadrightarrow \mathrm{Spec}(A)$ on the prime spectra. Is $\mathrm{Spec}(i)$ a closed map of topological spaces? Does this become the case if $A$ is assumed to be Noetherian and/or an integral domain or a field?
If it's not closed, (under whatever assumptions on $A$), could someone provide a simple counterexample?
I realize this is probably a very stupid question. It seems like the map should be obviously be closed or obviously not be, but I've vacillated as to which. I seem finally to have devised a proof it is closed, but I am suspicious of this quasi-proof, because it seems to make an exercise I've been working on easier than the hint provided would indicate, and also fails to use some of the hypotheses granted for the exercise. Also, if it were true, I would expect to have seen some mention of it on the Internet or in some text, and so far I haven't.