This question is a strengthening of this question (answered negatively), and arose due to David Speyer's answer here.

Geometrically, this asks if every Noetherian connected affine scheme can be realized as a closed subscheme of a Noetherian integral affine scheme.

To briefly justify this question: the linked answer above shows that the approach to the original question is doomed to fail. Moreover, I would argue that this question is inherently more natural, in the following sense: given Noetherian rings $R_1, \ldots R_n$, there is a Noetherian domain $D$ surjecting onto each of $R_1, \ldots, R_n$ iff there exists a Noetherian domain $D'$ surjecting onto $R_1 \times \ldots \times R_n$. Thus the original question is isomorphic to a direct product of copies of this one.

Since I expect answers will reduce to this case anyway, let me make a mild regularity assumption and only consider reduced (i.e. $R_0 + S_1$) rings. Note however that even a slight improvement on this, like normality ($R_1 + S_2$), would render the problem trivial.

  • $\begingroup$ Could you embed the primary components disjointly in a common space, then start gluing that space to itself? $\endgroup$ – Allen Knutson Jul 16 '14 at 16:47
  • $\begingroup$ @AllenKnutson: If I understand correctly, the first step seems to be much of the hurdle $\endgroup$ – zcn Jul 17 '14 at 2:33
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    $\begingroup$ Well, you can define the common space to be the disjoint union, and then all the trouble is in the second step :). I feel like this is at least as hard as constructing the Witt vectors (of course, that construction has been done, so we can use it for our purposes). Suppose that you we given the ring $W(k)/p^n$, where $k$ is an infinite perfect field of char. $p$ and $W(k)$ is the Witt vectors, but you didn't know about $W(k)$. Is there any way to write $W(k)/p^n$ as a quotient of a noetherian domain more easily than inventing $W(k)$? $\endgroup$ – David E Speyer Jul 17 '14 at 16:25
  • $\begingroup$ @DavidSpeyer: Re disjoint union: of course :) As to the ring of Witt vectors, I'm afraid I can't say, though that seems to be going in the direction of a construction. Actually, my current guess is that there is a counterexample $\endgroup$ – zcn Jul 17 '14 at 18:24

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