In 1995, de Jong proved the existence of regular alterations in arbitrary characteristic. I would like to have a little survey of important applications of this theorem, e.g. things you could do if you had resolution of singularities, but for which alterations are sufficient.

I am aware of the 1996 Bourbaki article by Berthelot, however I guess that many new developments have taken place since then.

  • 1
    $\begingroup$ On the old version of MathOverflow, I think we used to have a rule of thumb that "MathOverflow is not for requests for people to write enyclopaedic entries". Could you perhaps narrow down what kinds of application you are looking for? $\endgroup$
    – Yemon Choi
    Commented May 6, 2014 at 14:06
  • $\begingroup$ It's meant to convince a broader audience (not only algebraic geometers) why they should care about alterations ... $\endgroup$
    – pgraf
    Commented May 6, 2014 at 15:16
  • $\begingroup$ Gabber's generalization of de Jong's theorem is now being used a lot; see e.g. arxiv.org/abs/1112.5206 $\endgroup$ Commented May 6, 2014 at 17:12

2 Answers 2


There is a book, Resolution of Singularities — A research textbook in tribute to Oscar Zariski, edited by Hauser, Lipman, Oort and Quiros (Progress in Mathematics, Birkhäuser, 2000). There, you'll get four papers related to alterations : one by Abramovich and Oort that explains De Jong's theorem, one by Geisser that gives alterations, and another by De Jong himself about Dieudonné modules, and finally one by Pop on birational anabelian geometry.


It is (in the form of Gabber's generalisation) applied in Luc Illusie, Yves Laszlo, Fabrice Orgogozo, Travaux de Gabber sur l'uniformisation locale et la cohomologie etale des schemas quasi-excellents https://arxiv.org/abs/1207.3648 to finiteness statements and purity in étale cohomology.


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