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Let consider $\mathbb{A}^2$ as general affine scheme. Which meaning do have "points" of the shape $(x,y) \in \mathbb{A}^2$? For example the special case $(0,0)$?

That's almost clear that for choosen (therefore fixed) field $k$ we can identify the points $(x,y)$ for $x, y \in k$ with $k$-points $Spec(k) = {*} \to \mathbb{A}^2$. Concretly: $(x,y)\cong (t_1-x, t_2-y)$ is the maximal ideal (thererfore element of Spec) with indeterminats $t_1, t_2$.

But the meaning of this "points" of the affine scheme $\mathbb{A}^2$ without mentioning a fixed field isn't clear to me.

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An affine scheme is always the spectrum of some ring. Usually, $\Bbb A^2$ is shorthand for $\operatorname{Spec} (k[x, y])$ for some field $k$, where the $^2$ in $\Bbb A^2$ comes from the fact that there are two indeterminates.

You can define $\Bbb A^2_R$ for any ring $R$ in a similar manner (being the spectrum of $R[x, y]$). But you are right that without mentioning a ground ring, the notation $\Bbb A^2$ makes little sense. And if $R$ is not a field, then you run into problems defining this point $(0,0)$, as $(x, y)$ wouldn't be a maximal ideal of $R[x, y]$, and therefore geometrically not represent a point the way you're used to over fields (i.e. it's not a closed point).

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  • $\begingroup$ So this only make sense in case of $\mathbb{A}^2 _k = \operatorname{Spec} (k[x, y])$ with field $k$? Here the points $(a,b)$ with $a, b \in k$ are as I understand it correctly identified with maximal ideal $(x-a, y-b)$, aren't they? And the naive identification $k^2 \cong \mathbb{A}^2 _k$ is only true if $k$ algebraic closed, so for arbitrary $k$ the vector space $k^2$ is just "embedded" in $\mathbb{A}^2 _k$? $\endgroup$
    – user267839
    Oct 17, 2017 at 11:08
  • $\begingroup$ @KarlPeter That's right. In fact, at least in the case of $\Bbb A^1$, there is a nice classification of the (topological) map $\Bbb A^1_{\bar k}\to \Bbb A^1_k$, where $\bar k$ is the algebraic closure of $k$: It is the quotient space generated by the Galois group of $\bar k$ over $k$, which by definition of Galois group means that a copy of $k$ sits canonically inside it. For instance, $\Bbb A^1_{\Bbb R}$ has as points the real numbers, and conjugate pairs of complex numbers. $\endgroup$
    – Arthur
    Oct 17, 2017 at 11:22

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