2
$\begingroup$

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

$\endgroup$
  • 2
    $\begingroup$ Well, $\mathbb{A}^1 \cup_{\mathbb{A}^1-0} \mathbb{A}^1$ is smooth and non-separated. $\endgroup$ – J.C. Ottem May 1 '11 at 9:15
16
$\begingroup$

- 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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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