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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- 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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