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.



Do arithmetic schemes have non-singular alterations?
This is Theorem 8.2 in de Jong's original paper [dJ].


[dJ] de Jong, A. J., [*Smoothness, semi-stability and alterations*](http://dx.doi.org/10.1007/BF02698644). Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996). [ZBL0916.14005](https://zbmath.org/?q=an:0916.14005).


