Let $S$ be a locally noetherian scheme and let $G$ be a separated and smooth $S$-group algebraic space of finite presentation. Does there exist an open sub-(group algebraic space) $G^0 \subset G$ characterized by the requirement that $G\_s^0$ be the identity component of $G\_s$ for every $s \in S$? (Note that the $S$-fibers of $G$ are automatically schemes.)


In the scheme case, the positive answer (with less restrictive assumptions) follows from the results of EGA IV, section 15.6 (or of <http://stacks.math.columbia.edu/tag/055K>). It seems that the positive answer in the algebraic space case is used implicitly in the proof of Thm. 1 of section 6.6 of "Neron models" of Bosch, Lutkebohmert, and Raynaud (seemingly it is used to justify the applicability of Thm. 2 in the first paragraph of the said proof on p. 163). Thus I wonder if a positive answer to my question already appears somewhere in the literature?



Relative identity component for group algebraic spaces
There is even a reference for algebraic stacks: see M. Romagny, manuscripta math. 136, 1–32 (2011), especially section 2.2.


