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This quote is from Matsumura's Commutative Ring Theory, page 100: "The principal ideal theorem corresponds to the familiar and obvious-looking proposition of geometrical and physical intuition (which is strictly speaking not always true) that 'adding one equation can decrease the dimension of the space of solutions by at most one'."

One consequence of the principal ideal theorem is that if $P$ is a prime ideal of height $r$ in a Noetherian ring $A$, then $P$ is the minimal prime divisor of some ideal $I=(a_1,\cdots,a_r)$ and that the height of $P/(a_1,\cdots,a_i)$ is precisely equal to $r-i$.

Question 1: How can we make a connection between the quote from Matsumura and the above consequence of the principal ideal theorem? Which is the solution space?

Question 2: Are there any examples where adding one equation drops the dimension by more than 1?

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  1. The solution space associated with a function $f$ is, of course, the zero-set $\{ x : f (x) = 0 \}$, or more formally, $\{ \mathfrak{p} \in \operatorname{Spec} A : f \in \mathfrak{p} \}$.
  2. Consider $A = \mathbb{C} [x,y,z] / (x z, y z, z^2 - z)$. This is a 2-dimensional noetherian ring: its spectrum is the disjoint union of the affine plane and a point. Of course, if we look at the equation $z - 1 = 0$, we end up with a 0-dimensional ring (namely $\mathbb{C}$).
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  • $\begingroup$ +1 Dear Zhen Lin, can you please explicitly state what is the "affine plane" and the "point" that you mention? $\endgroup$ – Manos Jun 9 '13 at 20:25
  • $\begingroup$ The plane $z = 0$ and the point $(x, y, z) = (0, 0, 1)$. $\endgroup$ – Zhen Lin Jun 9 '13 at 20:40

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