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

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    $\begingroup$ 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. $\endgroup$ – Karl Schwede Oct 17 '12 at 12:42
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    $\begingroup$ 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. $\endgroup$ – Damian Rössler Oct 17 '12 at 21:20
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    $\begingroup$ @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 ? $\endgroup$ – Qing Liu Oct 18 '12 at 7:22
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    $\begingroup$ @Qing Liu. A regular is divisorial (see SGA6, II, Cor. and subscheme of a divisorial scheme is also divisorial (this follows from the definition). $\endgroup$ – Damian Rössler Oct 18 '12 at 8:19
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    $\begingroup$ Hartshorne, Exercise III.5.9 gives an example as Roessler suggests. $\endgroup$ – Jason Starr Oct 18 '12 at 16:23

A noetherian regular scheme is universally catenary (Matsumura, 14.B, 16.D), so any subscheme of a regular scheme is universally catenary.

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).

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    $\begingroup$ As Emma says somewhere in Austen's novel, silly things do cease to be silly if they are done by sensible people in an impudent way. Identifying points of different codimension was surely not what she had in mind, but the observation surely applies! $\endgroup$ – Mariano Suárez-Álvarez Oct 18 '12 at 7:51

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