0
$\begingroup$

I want to prove the following property of polynomial map:

If $\varphi:V\to W$ is a polynomial map of affine varieties $V\subset \mathbb{A}^n,W\subset \mathbb{A}^m$ and $X\subset W$ is an algebraic subset of $W$, suppose $X\subset \varphi(V)$, then we have $\varphi^{-1}(X)$ is also an algebraic subset of $V$.

This is a well-known property, and you can also regard it as the exercise 2.7 from Fulton's Curve Book. I've search it on Google, every lecture notes prove it as a simple consequence of properties in Zariski topology. But up to now Fulton not yet introduce the concept of Zariski topology. So I'm asking that is there a proof of this simple property without using Zariski topology?

$\endgroup$
1
$\begingroup$

Suppose $X$ is the vanishing locus of the polynomials $f_1, \dots, f_n$ on $W$. Then $\varphi^{-1}(X)$ is the vanishing locus of the polynomials $f_1 \circ \varphi, \dots , f_n \circ \varphi$ on $V$, and hence is algebraic too.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.