David Rydh and I have been dscussing residual gerbes and stratifcations by gerbes. We have now shown that if X is an algebraic stack whose inertia I_X is quasi-compact over X, then X has a canonical stratification by locally closed algebraic substacks which are gerbes… but this stratification is indexed by a possibly infinite well ordered set. This is the stratification of type (a) of Lemma Tag 06RF.
As the example in this post shows we cannot always expect to find a finite stratification. To me an intriguing question is what possible order types one can obtain from the canonical stratification of these algebraic stacks. My first guess is that the index should in any case always be countable (but I do not even have a heuristic argument for this).
The result above relies on a very general “generic flatness” result which also allows one to prove the existence of residual gerbes at any point of an algebraic stack whose inertia is quasi-compact.
My next goal is to revise the chapter on formal deformation theory.