Let $X$ be an integral normal flat finite type scheme over $\mathbb{Z}$.

Does there exist a proper surjective generically finite morphism of schemes $Y\to X$ with $Y$ an integral regular finite type scheme over $\mathbb{Z}$?

I could not find such a result in the literature.


This is Theorem 8.2 in de Jong's original paper [dJ].

[dJ] de Jong, A. J., Smoothness, semi-stability and alterations. Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996). ZBL0916.14005.

  • $\begingroup$ Thank you. I don't know how I didn't see this (I of course looked at de Jong's original paper, but not carefully enough it seems). $\endgroup$ – Kriss Aug 6 '18 at 18:45

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