# Reference for normalization of an algebraic stack?

Is there a standard reference on stacks which discusses (relative) normalization?

This older question seemed to link to someone's notes, but the link is now broken. In any case, it would be nice to have a reference which is a book or a paper.

I've tried the stacks project, Martin Olsson's book on stacks, and Laumon/Moret-Bailly's "Champs Algebriques", and none of them seem to have anything on normalization of stacks (at least not in the index).

• I guess this doesn't classify as a "standard reference", but there is a recent paper of Kenneth Ascher and Dori Bejleri arxiv.org/pdf/1702.06107.pdf that has an appendix about normalizations of algebraic stacks. Jun 9, 2017 at 2:27
• Maybe Section 2.3 in arxiv.org/pdf/1703.00488.pdf could be of some help to you. Jun 12, 2017 at 12:20
• Section 5.1 of arxiv.org/pdf/1812.04425.pdf has a short discussion of relative normalization of Artin stacks Feb 17 at 10:33

Quoting

Vistoli, Angelo

Intersection theory on algebraic stacks and on their moduli spaces.

Invent. Math. 97 (1989), no. 3, 613–670.

to be found here http://dx.doi.org/10.1007/BF01388892 :

Definition 1.18 p.623

`Definition. The normalization $F$ of a reduced stack $F$ with a presentation $R \rightrightarrows U$ is the stack associated with the groupoid $\overline{R} \rightrightarrows \overline{U}$. The normalization of a general stack is the normalization of its associated reduced stack.

If $F$ is a stack, the canonical morphism $\overline{F} \to F$ is representable. If $F$ is of finite type over a universally japanese ring (see EGA IV, 23.1.1) then $\overline{F} \to F$ is also finite.'

Here $\overline{R}$ and $\overline{U}$ denote the normalizations of $R$ and $U$, respectively.

• From what I can see this doesn't seem to discuss relative normalization? Jun 9, 2017 at 19:17
• No, it doesn't, but your question wasn't limited to the relative case, was it ? Jun 21, 2017 at 8:24