in a lot of algebraic geometry books I've heard that working over $\mathrm{Spec}(A)$ is better than working over $\mathrm{Spmax}(A)$ in the case where you consider a variety over a non-agebraically closed field $k$. Of course, the prime spectrum suits better because of the functoriality, but in a lot of books dealing with arithmetic of curves (in affine charts, without sheaves) there are discussions and proofs of facts dealing with non-algebraically closed fields (even zeta functions) using only maximal ideals of reduced $k$-algebras (zero sets of polynomials).

Therefore I'm searching for a simple example where prime ideals are better than the maximal ideals. Furthermore I would like to know how should I think about non-maximal prime ideals (points that are not geometric). I think that it would be useful, too, if someone could explains what's the difference between $\mathrm{Spec}(k)$ and $\mathrm{Spec}(k')$.

Thanks in advance.

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  • $\begingroup$ What is $k'$? Is it the algebraic closure of $k$? $\endgroup$ – Manos Jan 23 '14 at 21:54
  • $\begingroup$ @Manos any extension $\endgroup$ – user40276 Jan 23 '14 at 22:00
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    $\begingroup$ The difference between $Spec(k)$ and $Spec(k')$ is exactly the same as that between $k$ and $k'$ ! But this is a tautology and proves nothing for or against the use of nonmaximal prime ideals... $\endgroup$ – Georges Elencwajg Jan 23 '14 at 22:34
  • $\begingroup$ @GeorgesElencwajg but this is just a point $\endgroup$ – user40276 Jan 23 '14 at 22:36
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    $\begingroup$ Dear user: $spec(k)$ is not just a point but a point plus a sheaf of local rings on that point. In this trivial situation this sheaf is just the datum of the field $k$, but the beauty of scheme theory is that the scheme $Spec(k)$ remembers the field $k$ in its structure. $\endgroup$ – Georges Elencwajg Jan 23 '14 at 23:09

If $k$ is an algebraically closed field and $A=k[T_1,...,T_n]/\mathfrak p\,$ (with $\mathfrak p$ a prime ideal) is a finitely generated domain over $k$, you should think of the affine scheme $\mathrm{Spec}(A)$ as the variety $V=V(\mathfrak p)\subset \mathbb A^n(k) $ to which you adjoin formally a point $\eta_W $ for every irreducible subvariety $W\subset V$.

If $k$ is not algebraically closed the advantage of schemes over varieties is more convincing: the schemes $\mathrm{Spec}(\mathbb R[X,Y]/(X,Y))$ and $\mathrm{Spec}(\mathbb R[X,Y]/(X^2+Y^2))$ are completely different (they don't even have the same dimension) whereas the varieties $V(X,Y)\subset \mathbb R^2$ and $V(X^2+Y^2)\subset \mathbb R^2$ are both equal to a single point.

But the really convincing superiority of schemes over varieties is attested by the amazing results obtained through the use of scheme theory in the last half century by geometers and arithmeticians like Grothendieck, Hironaka, Deligne, Faltings, Wiles and a good proportion of Fields medalists and Abel prize recipients in that period.

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  • $\begingroup$ Thanks for the answer, but if a point is not generic, I cannot see the utility of adding a point that is not geometric (for instance some closed point not geometric). Anyway, why wouldn't use a stratification instead of adding such new points? $\endgroup$ – user40276 Jan 24 '14 at 12:17
  • $\begingroup$ I know this is an old question, but I'm not sure the middle paragraph answers the question here, since for nonalgebraically closed fields the variety as a subset of affine space is completely different than the max spec. Max spec of $\Bbb{R}[X,Y]/(X^2+Y^2)$ has a pile of complex points that distinguish it from the single point of the max spec of $\Bbb{R}\cong \Bbb{R}[X,Y]/(X,Y)$. $\endgroup$ – jgon Jan 8 '19 at 0:27

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