My question is about the passage (11.1) in the book of Laumon and Moret-Bailly on algebraic stacks. There we have a scheme $S$, an algebraic stack $\mathscr{X}$ over $S$, and a point $\xi$ of $\mathscr{X}$. The residual gerbe $\mathscr{G}\_\xi$ at $\xi$ is defined as follows: choose an $S$-field $K$ and an $S$-morphism $x\colon \mathrm{Spec}(K) \rightarrow \mathscr{X}$ representing $\xi$, and let $\mathscr{G}\_\xi \subset \mathscr{X}$ be the smallest substack (for the fppf topology) through which $x$ factors. The claim is that this $\mathscr{G}\_\xi$ does not depend on the choices of $K$ and $x$. How does one verify this? 


I am especially having trouble in the case when $K'$ is a big overfield of $K$, when I cannot figure out how to make contact with the fppf topology that is used in defining $\mathscr{G}\_\xi$. It seems to me that $\mathscr{G}\_\xi$ could shrink after replacing $K$ by this $K'$ in the defining procedure.



Is the residual gerbe really independent of the choice of a representative?
IIRC a reference for a discussion is David Rydh's paper \'Etale d\'evissage, descent and pushouts of stacks, Appendix B. (Of course grghxy also already answered your question too, my apologies to grghxy.)


You can look at [Tag 06ML](http://stacks.math.columbia.edu/tag/06ML) for an answer to your question if you already know the thing as defined in Laumon and Moret-Bailly is algebraic. This is not hard.


To see that it is algebraic (following Rydh and in the quasi-separated case, i.e., in the case of algebraic stacks as discussed in Laumon and Moret-Bailly), you can look [Section Tag 06UH](http://stacks.math.columbia.edu/tag/06UH). Essentially the idea is to reduce to the case where $x$ is the generic point and $\mathcal{X}$ is an integral algebraic stack, then reduce to the case where $\mathcal{X}$ is a gerbe (flattening stratification for inertia), then pull back from the algebraic space that the algebraic stack is a gerbe over. Cheers!


