I am trying the show the following:

(10) Show that polynomials $F\in k[x_1,\dotsc,x_n]$ are continuous maps $f\colon\mathbb{A}^n\to\mathbb{A}^1$ (for the Zariski topology on $\mathbb{A}^n$ and on $\mathbb{A}^1$).

$\textbf{My Attempt:}$

In this case, to show that this map is continuous we need to show that the preimage of a closed set is closed. We know $f$ is a function that maps the zeroes of a certain affine set to the zeroes of another affine set. I'm not sure how to proceed further. Would there be a simple topological argument hidden in here?

  • $\begingroup$ What are the closed subsets of $A^1$? What are their preimages under $f$? $\endgroup$ – Mariano Suárez-Álvarez Mar 21 '18 at 0:14
  • $\begingroup$ @MarianoSuárez-Álvarez the closed subsets of $A^1$ are a finite number of points, and their pre-images are the surfaces corresponding to the zero sets of $A^n$? $\endgroup$ – user119264 Mar 21 '18 at 0:17
  • $\begingroup$ Is the preimage of point of A1 closed in An? $\endgroup$ – Mariano Suárez-Álvarez Mar 21 '18 at 0:40

let $Z$ be a closed set. Then $f(v) \in Z$ if and only if $f(z)$ belongs to the zero locus of some collection of polynomials $h_1,\dots h_n$. But then $0=h_1\circ f=\dots =h_n\circ f$. But these are all polynomials, so $f^{-1}(Z)$ must be closed.

I've written it this way, since it generalizes, but really the point is that we can argue for a single point, since if the preimage $f^{-1}(a)$ is closed, then the preimage of a finite set of points is a finite union of closed sets, which is closed.

Now note that

$f(v)=a$ if and only if $f(v)-a=0$, which is a specialization of $h_1 \circ f$ when $h_1(x)=x-a$.

  • $\begingroup$ can you explain why you used the composition of $h_n$ with $f$? $\endgroup$ – user119264 Mar 21 '18 at 0:47
  • $\begingroup$ hmmm... well the way I see it, you belong to a closed set $Z$ if and only if you are in the zero locus of some polynomials. These are $h_i$. $\endgroup$ – Andres Mejia Mar 21 '18 at 0:49
  • $\begingroup$ I see. This argument is clear $\endgroup$ – user119264 Mar 21 '18 at 0:54

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