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



Embedding of a scheme into a regular scheme
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). 


