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Let us fix a field $k$ (algebraically closed for convenience).

Taking $\mathrm{Spec}$ is an equivalence of categories

$$ \text{$k$-algebras} \overset{\simeq}{\longrightarrow} \text{affine $k$-schemes} $$ whose quasi-inverse is taking global sections. At the same time, we also have an equivalence given by taking $\mathrm{SpecMax}$

$$ \text{affine $k$-algebras} \overset{\simeq}{\longrightarrow} \text{affine algebraic $k$-varieties} $$ with the same quasi-inverse.

Question: I cannot wrap my head around with the discrepancy of taking $\mathrm{Spec}$ in the first functor and $\mathrm{SpecMax}$ in the other. Why taking $\mathrm{SpecMax}$ for affine schemes is needed? How does this equivalence relate with the restriction of $\mathrm{Spec}$ to affine schemes?

I know this has to be related with the sentence "for algebraic schemes it is convenient to ignore non-closed points" that I have read several times but I cannot see why this is "convenient" at all.

Minkowski
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  • What is $\operatorname{SpecMax}$? The set of maximal ideals? That would not be a scheme. – red_trumpet Mar 14 '25 at 10:38
  • @red_trumpet I meant "variety", I just changed it. I suppose that I am also confused because I have two definitions of "affine algebraic variety" in mind that I maybe I don't know to reconcile: one is a ringed space $(X, \mathcal{O}_X)$ where X is an algebraic set, and the other is an affine scheme $(Y,\mathcal{O}_Y)$ where $\mathcal{O}_Y(Y)$ is an affine algebra (finitely generated and reduced). – Minkowski Mar 14 '25 at 10:46
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    Can I ask: how you are defining "affine variety"? For one thing, different authors have different conventions about whether varieties are reduced, irreducible, etc. For another, I feel that an affine variety sometimes has a different meaning in the scheme-theoretic context compared to the classical context. – Joe Mar 14 '25 at 10:49
  • @Minkowski, are you following any specific book or note? If so, please mention it – Learner Mar 14 '25 at 11:00
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    You are right, there are two definitions of "affine variety", one related to the maximal spectrum, the other as a scheme. They are not the same, but their categories are equivalent. This is explained in Hartshorne's Algebraic Geometry, Proposition 2.6, Proposition 4.10 and the following definition. – red_trumpet Mar 14 '25 at 11:00
  • This thread may be interesting: https://math.stackexchange.com/questions/3906932/grothendieck-point-of-view-of-algebraic-geometry/3997811#3997811 – hm2020 Mar 14 '25 at 11:07
  • @Minkowski - If your ring $A$ is a finitely generated commutative $k$-algebra where $k$ is a Dedekind domain, there is a locally ringed space ${X^m, \mathcal{O}{X^m}}$ where $X^m$ is the set of maximal ideals in $A$ and where $\Gamma(X^m, \mathcal{O}{X^m})=A$. Hence you recover $A$ from this locally ringed space. This locally ringed space has stalks that may be non-reduced rings. – hm2020 Mar 14 '25 at 11:10
  • @red_trumpet so if I understand correctly, the composition of the functor SpecMax that I described before with the equivalence you mentioned is precisely the restriction of the first functor Spec to affine $k$-algebras, right? – Minkowski Mar 14 '25 at 12:23
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    @Learner I am ready the excellent books by J.S. Milne "Algebraic Geometry" https://www.jmilne.org/math/CourseNotes/AG500.pdf and "Basic theory of affine group schemes" https://www.jmilne.org/math/CourseNotes/AGS.pdf – Minkowski Mar 14 '25 at 12:26
  • @Joe please see my reply to red_trumpet – Minkowski Mar 14 '25 at 12:27

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