Timeline for 2-fold linear cover of reductive group of type A

7 events

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Mar 20, 2021 at 20:13	comment	added	Mikhail Borovoi		Thus we come to ${\rm Ext}({\rm Res \,GL}_1,\mu_2)$ and ${\rm Ext}({\rm Res\,U}_1,\mu_2)$. I think that Res does not change Ext, and so we come to ${\rm Ext}({\rm GL}_1,\mu_2)$ and ${\rm Ext}({\rm U}_1,\mu_2)$. Each of them has cardinality 2, the nontrivial extensions being the standard double covers $$\rm GL_1\overset 2 \longrightarrow GL_1\quad\text{and}\quad \rm U_1\overset 2 \longrightarrow U_1\,.$$ Among those $H^1(F,\rm U_1)\ne 1$.
Mar 20, 2021 at 19:44	comment	added	Mikhail Borovoi		Let us call ${\rm Ext}(G,\mu_2)$ the set of isomorphism classes of these extensions. I think that the homomorphism $G\to G^{\rm tor}$ induces an isomorphism $${\rm Ext}(G^{\rm tor},\mu_2)\overset{\sim}{\longrightarrow} {\rm Ext}(G,\mu_2),$$ where $G^{\rm tor}=G/(G,G)$ (the quotient by the commutator subgroup), because $(G,G)$ is simply connected.
Mar 20, 2021 at 19:33	history	edited	Mikhail Borovoi		edited tags
Mar 15, 2021 at 14:59	history	edited	Mikhail Borovoi		edited tags
Mar 15, 2021 at 14:49	history	edited	Mikhail Borovoi		edited tags
Mar 15, 2021 at 7:09	comment	added	მამუკა ჯიბლაძე		What kind of extension is $E$?
Mar 15, 2021 at 2:43	history	asked	Windi	CC BY-SA 4.0