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Sorry for one more stupid AG question. I need schemes that are regular, excellent and separated. Are these three conditions independent?

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Well, $\mathbb{A}^1 \cup_{\mathbb{A}^1-0} \mathbb{A}^1$ is smooth and non-separated. –  J.C. Ottem May 1 '11 at 9:15
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1 Answer

up vote 10 down vote accepted

- Separated, excellent, regular: Spec$(k)$.

- Separated, excellent, not regular: Spec$(k[\epsilon]/\epsilon^2)$.

- Separated, not excellent, regular: See http://en.wikipedia.org/wiki/Excellent_ring

- Separated, not excellent, not regular: Spec$(k[\epsilon_1,\epsilon_2,\ldots]/\langle\epsilon_1^2,\epsilon_2^2,\ldots\rangle$.

- Not separated, excellent, regular: Glue Spec$(\mathbb{Z})$ to itself along the complement of a closed point.

To get the other three, take the disjoint union of the fifth example with any of the second, third, or fourth examples.

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