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Suppose $(\text{Spec}(R), \tilde{R})$ is a scheme locally of finite type over $\mathbb{C}$. We want to show that $R$ is a finitely generated $\mathbb{C}$-algebra.

Since $(\text{Spec}(R), \tilde{R})$ is a scheme locally of finite type over $\mathbb{C}$, we have a ring homomorphism $$\rho_{\text{Spec}(R)} : \mathbb{C} \longrightarrow \Gamma(\text{Spec}(R), \tilde{R})$$ and the existence of an open cover $\text{Spec}(R) = \bigcup_{i \in I} U_i$, such that for each $U_i$ in the open cover, $(U_i, \tilde{R} \vert_{U_i})$ is isomorphic as a ringed space to $(\text{Spec}(R_i), \tilde{R_i})$, with each $R_i$ a finitely generated $\mathbb{C}$ algebra. Suppose that $R$ is not a finitely generated $\mathbb{C}$-algebra. Then it is not possible to express every element as a polynomial of some finite set $\{r_1, r_2, ..., r_n \} \subset R$. We note that for each $R_i$, there exists a finite set $\{r_{i_1}, r_{i_2}, ..., r_{i_j} \} \subset R_i$ such that every element in $R_i$ can be expressed as a polynomial in these elements. Since there is an isomorphism or ringed spaces between the $U_i$ in the cover of $\text{Spec}(R)$ and the $R_i$, it follows that we can express every element in the $U_i$ as a polynomail in some finite set $\{ \varphi(r_{i_1}), \varphi(r_{i_2}), ..., \varphi(r_{i_j}) \}$. Thus, $\text{Spec}(R)$ is covered by an open cover of finitely generated $\mathbb{C}$-algebras.

Can we therefore conclude that $\text{Spec}(R)$ is a finitely generated $\mathbb{C}$-algebra? And does this yield a sufficient platform to prove that $R$ is indeed a finitely generated $\mathbb{C}$-algebra?

user26857
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    I do not understand the difference between the two questions, but I guess you want to prove that if Spec($B$) $\to$ Spec($A$) is (locally) of finite type then $A \to B$ is of finite type. This is proved in many introductory books to the theory of schemes, for instance in EGA I.6.2.5 (Springer edition). – A.G Jul 21 '16 at 14:19
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    Note that $\mathrm{Spec}(R)$ is quasi-compact (see http://math.stackexchange.com/questions/1872170/proof-that-an-affine-scheme-is-quasi-compact ) so you can suppose that your open covering $(U_i)$ is finite. – paf Jul 31 '16 at 11:45
  • @paf Is a space a finitely generated $\mathbb{C}$-algebra if it is finitely covered by finitely generated $\mathbb{C}$-algebras? –  Aug 03 '16 at 07:20
  • The 'trick' is to refine the open cover to one of finitely many distinguished opens. Algebraically, this reduces to the case where you get finitely generated $\mathbb{C}$-algebras $R_{f_i}$ where $f_1,\ldots,f_s\in R$ generate the unit ideal. Are you now able to prove that $R$ is finitely generated? – Ben Aug 04 '16 at 12:47

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