Timeline for A basic question on a base change of a homogeneous space of a linear algebraic group

5 events

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Feb 10, 2019 at 2:25	history	edited	YCor		edited tags
Jun 24, 2017 at 16:05	comment	added	nfdc23		Tangent space arguments don't work when $H$ is not smooth. For a robust theory of smooth affine groups in characteristic $p>0$ we need to confront non-separable surjective homomorphisms and geometrically transitive actions whose scheme-theoretic stabilizer at any point is not smooth (so the orbit map is not a submersion). Hence, one needs a positive answer (including a good existence result for $G/H$!) even when $H$ isn't smooth. Chevalley's "stable line" trick to construct $G/H$ works for non-smooth $H$ by considering scheme-theoretic normalizers, and flatness + descent theory do the rest.
Jun 24, 2017 at 15:29	comment	added	Mikhail Borovoi		@Venkataramana: Exactly! The differential of $\bar\lambda$ is bijective at $x_0$ and by homogeneity at any point, and the result follows.
Jun 24, 2017 at 15:04	comment	added	Venkataramana		Is not the map $\overline{\lambda}$ a dominant map? Thus the image contains an open subset of the variety $(G/H)_K$ and by homogeneity is all of the variety (for the same reason, it is an open map).
Jun 24, 2017 at 14:18	history	asked	Mikhail Borovoi	CC BY-SA 3.0