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I apologize for the vagueness of my question.
In the case of morphisms between schemes (that locally look like an $A$-algebra maybe), the textbook (Vakil's FOAG, July 31, 2023 version) often uses the "coordinate" descriptions.
For example, on page 300, the Segre embedding is described by $$\mathbb{P}_A^m\times_A\mathbb{P}_A^n\to \mathbb{P}_A^{mn+m+m}$$ sending $$([x_0:\cdots:x_m],[y_0:\cdots:y_n])\to [z_{00}:\cdots:z_{mn}]= [x_0y_0:\cdots:x_my_n]$$ I think this is because we are allowed to do so in the affine case (as the projective morphism here comes from glueing).
Usually, such coordinates are identified with its corresponding maximal ideals. However, a scheme usually has more points than those closed ones. And the set of closed points is not usually dense $(\operatorname{Spec}k[x]_{(x)}$, even though it is not locally of finite type).
Even if it is a dense subset, with Zariski topology, I am not sure if one can recover a scheme morphism by its value on a denset subset. So why is this description enough?
Any help is appreciated! Thank you very much.

Mizutsuki
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  • You're not the first person to wonder about this concept - this early MSE question explains it somewhat. Please let me know if that's enough to resolve your question, or if there's further clarification you're after. – KReiser Nov 17 '23 at 16:19
  • @KReiser Thank you! This is my interpretation: A morphism between affine schemes $\operatorname{Spec}A[x_1,\cdots,x_n]\to \operatorname{Spec}A[y_1,\cdots,y_m]$ corresponds to a ring map $\varphi:A[y_1,\cdots,y_m]\to A[x_1,\cdots,x_n]$, which is basically determined by the image of $y_i$'s, i.e. it is enough to specify $\varphi(y_i)=f_i\in A[x_1,\cdots,x_n]$. Then this map will map the "points" $(a_1,\cdots,a_n)$ to $(f_1(a_1,\cdots,a_n),\cdots,f_m(a_1,\cdots,a_n)$. Hence by describing the morphism in coordinate language, we specified the ring map by deciding where $y_i$ goes. Is it true? – Mizutsuki Nov 18 '23 at 09:24
  • Yep, that's correct. – KReiser Nov 18 '23 at 15:23
  • @KReiser Thank you! But for a general scheme morphism, it is not always enough to specify what it does on closed points, right? I cannot come up with an example... – Mizutsuki Nov 19 '23 at 04:02
  • Again, this has been addressed before on MSE. I really do suggest looking around - there are many nice questions and answers here. – KReiser Nov 19 '23 at 05:30

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