Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question
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

1 Answer 1

up vote 11 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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.