Timeline for Relative identity component for group algebraic spaces

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Dec 10, 2014 at 17:16	comment	added	Question Mark		Thanks to both of you! @user74230: in your first parenthetical, you need to also assume that $X$ is connected. On a different note, I wonder if these type of results are purely topological? Is it possible that a similar result could be proved in the generality of locally spectral spaces? After all, both flatness and finite presentation assumptions have a strong topological aspect to them (generizing + Chevalley's theorem); perhaps the assumption of reduced geometric fibers is not so topological though...
Dec 10, 2014 at 17:00	vote	accept	Question Mark
Dec 10, 2014 at 12:09	comment	added	user74230		As noted there, the EGA proof applies almost verbatim for stacks, up to IV$_3$ Lemma 15.5.6 (any $X$ flat and lfp over a dvr $R$ with reduced connected special fiber also has connected generic fiber), whose proof in EGA is scheme-specific (set of generizations of a point is a local Spec). Romagny's Lemma B.2 gives a simpler proof of this ("reduced fiber trick") that works beyond schemes (omitting the easy reduction to the quasi-compact case for a statement about idempotents, q-c needed to pass localization through H$^0$); the same trick is in an earlier paper of Kisin, and perhaps elsewhere.
Dec 10, 2014 at 8:19	history	answered	Laurent Moret-Bailly	CC BY-SA 3.0