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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asked May 1, 2011 at 8:58
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2$\begingroup$ Well, $\mathbb{A}^1 \cup_{\mathbb{A}^1-0} \mathbb{A}^1$ is smooth and non-separated. $\endgroup$– J.C. OttemCommented May 1, 2011 at 9:15
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1 Answer
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- 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.
answered May 1, 2011 at 9:36