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In Vakil's FOAG, p218, he states Chevalley's theorem as follows

7.4.2 Chevally's Theroem---If $\pi:X \to Y$ is a finite type morphism of Noetherian schemes, the image of any constructible set is constructible. In particular, the image of $\pi$ is constructible.

And the following is 7.4.P.Exercise at page 221, in which Chevally's theorem is used. enter image description here

In my understanding, the scheme $X$ is a open subscheme $D(g_1,...,g_q)$ of the closed subscheme $\mathrm{Spec}(k[W_1,...,W_m,...X_1,..,X_n]/(f_1,...,f_p))$ of $\mathbb{A}^{n+m}$. Is that right? But then I can't figure out why $\pi$ in the commutative diagram is a morphism of finite type, which is necessary to use Chevalley's theorem. Could you help me? Thanks in advance.

Mugenen
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1 Answers1

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The inclusion $X \hookrightarrow \mathbb{A}^{m+n}$ is of finite type. Observe that we may factor this inclusion by $D(g_1,\ldots,g_q) \hookrightarrow V(f_1,\ldots,f_p) \hookrightarrow \operatorname{Spec}k[W_1,\ldots,W_m,X_1,\ldots,X_n]$. First, the inclusion $V(f_1,\ldots,f_p) \hookrightarrow \operatorname{Spec}k[W_1,\ldots,W_m,X_1,\ldots,X_n]$ is of finite-type as $A :=k[W_1,\ldots,W_m,X_1,\ldots,X_n]/(f_1,\ldots,f_p)$ is a finitely generated $k[W_1,\ldots,W_m,X_1,\ldots,X_n]$-algebra. Second, each $D(g_i)$ is affine and its coordinate ring is $A_{g_i}$ which is of finite-type over $A$. This yields that the inclusion $D(g_1,\ldots,g_q) \hookrightarrow V(f_1,\ldots,f_p)$ is locally of finite-type. The fact that there are finite $g_i$'s yields finite-type. Composition of finite-type morphisms is finite-type. Then $X \hookrightarrow \mathbb{A}^{m+n}$ is of finite-type. Finally the projection $\mathbb{A}^{m+n} \to \mathbb{A}^n$ is trivially of finite-type.

The argument actually concludes that every locally closed immersion $Z \hookrightarrow Y$ is locally of finite-type.

juan diego rojas
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