# Embedding of a scheme into a regular scheme

Hello,

Is it true that if I have a scheme $X$ which is, say, Noetherian, of finite Krull dimension, and semi-separated (intersection of two open affines is again open affine), then I can find a locally closed embedding of it into a scheme of the same type, which is, in addition, regular?

I ask just of curiosity, or minor desire not to say "quasi-projective" when I want to have enough locally free objects.

Thank you, Sasha

• This isn't even true for specs of local rings if I recall correctly. I'll try to find an example in a little bit. – Karl Schwede Oct 17 '12 at 12:42
• Such schemes are divisorial (ie they have an ample family of line bundles) if they are quasi-separated and quasi-compact. I trust there must be many non-divisorial schemes, although I don't have an example at hand. – Damian Rössler Oct 17 '12 at 21:20
• @Damian, this is interesting. As there are proper varieties with trivial Picard group, there are counterexamples with proper varieties. How to prove that a closed scheme in a regular scheme is divisorial ? – Qing Liu Oct 18 '12 at 7:22
• @Qing Liu. A regular is divisorial (see SGA6, II, Cor. 2.2.7.1) and subscheme of a divisorial scheme is also divisorial (this follows from the definition). – Damian Rössler Oct 18 '12 at 8:19
• Hartshorne, Exercise III.5.9 gives an example as Roessler suggests. – Jason Starr Oct 18 '12 at 16:23

But there are affine noetherian schemes (integral of dimension $2$) which are not universally catenary. See an example of Nagata in Matsumura, 14.E, or a slightly simpler one in EGA IV.5.6.11 (it consists in identifying two points of respective codimension 1 and 2 in an affine regular scheme of dimension 2).