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



Is every regular (excellent) scheme separated?
**-** 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.


