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Venkataramana
Venkataramana
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If $k$ has characteristic zero, then $H^1(k,H)$ is finite (Borel-Serre: https://mathscinet.ams.org/mathscinet-getitem?mr=181643) and the map $G(k)/H(k) \rightarrow (G/H)(k)$ has open image (can also be proved by the implicit function theorem). Moreover, a Baire category argument shownshows that this map is a homeomorphism onto the image. In positive characteristic, there is an old paper by Bernstein Zelevinsky (https://mathscinet.ams.org/mathscinet-getitem?mr=425030; see the appendix) which says (I think) that the map is a homeo onto the image, and that the image is closed.

If $k$ has characteristic zero, then $H^1(k,H)$ is finite (Borel-Serre: https://mathscinet.ams.org/mathscinet-getitem?mr=181643) and the map $G(k)/H(k) \rightarrow (G/H)(k)$ has open image (can also be proved by the implicit function theorem). Moreover, a Baire category argument shown that this map is a homeomorphism onto the image. In positive characteristic, there is an old paper by Bernstein Zelevinsky (https://mathscinet.ams.org/mathscinet-getitem?mr=425030; see the appendix) which says (I think) that the map is a homeo onto the image, and that the image is closed.

If $k$ has characteristic zero, then $H^1(k,H)$ is finite (Borel-Serre: https://mathscinet.ams.org/mathscinet-getitem?mr=181643) and the map $G(k)/H(k) \rightarrow (G/H)(k)$ has open image (can also be proved by the implicit function theorem). Moreover, a Baire category argument shows that this map is a homeomorphism onto the image. In positive characteristic, there is an old paper by Bernstein Zelevinsky (https://mathscinet.ams.org/mathscinet-getitem?mr=425030; see the appendix) which says (I think) that the map is a homeo onto the image, and that the image is closed.

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Venkataramana
Venkataramana
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If $k$ has characteristic zero, then $H^1(k,H)$ is finite and(Borel-Serre: https://mathscinet.ams.org/mathscinet-getitem?mr=181643) and the map $G(k)/H(k) \rightarrow (G/H)(k)$ has open image (can also be proved by the implicit function theorem). Moreover, a Baire category argument shown that this map is a homeomorphism onto the image. In positive characteristic, there is an old paper by Bernstein Zelevinsky (https://mathscinet.ams.org/mathscinet-getitem?mr=425030; see the appendix) which says (I think) that the map is a homeo onto the image, and that the image is closed.

If $k$ has characteristic zero, then $H^1(k,H)$ is finite and the map $G(k)/H(k) \rightarrow (G/H)(k)$ has open image (can also be proved by the implicit function theorem). Moreover, a Baire category argument shown that this map is a homeomorphism onto the image. In positive characteristic, there is an old paper by Bernstein Zelevinsky which says (I think) that the map is a homeo onto the image, and that the image is closed.

If $k$ has characteristic zero, then $H^1(k,H)$ is finite (Borel-Serre: https://mathscinet.ams.org/mathscinet-getitem?mr=181643) and the map $G(k)/H(k) \rightarrow (G/H)(k)$ has open image (can also be proved by the implicit function theorem). Moreover, a Baire category argument shown that this map is a homeomorphism onto the image. In positive characteristic, there is an old paper by Bernstein Zelevinsky (https://mathscinet.ams.org/mathscinet-getitem?mr=425030; see the appendix) which says (I think) that the map is a homeo onto the image, and that the image is closed.

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Venkataramana
Venkataramana
  • 11.2k
  • 1
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  • 67

If $k$ has characteristic zero, then $H^1(k,H)$ is finite and the map $G(k)/H(k) \rightarrow (G/H)(k)$ has open image (can also be proved by the implicit function theorem). Moreover, a Baire category argument shown that this map is a homeomorphism onto the image. In positive characteristic, there is an old paper by Bernstein Zelevinsky which says (I think) that the map is a homeo onto the image, and that the image is closed.