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- W3172049139 abstract "We prove that the continuous cohomology groups of an $ l $-group $ G $ with coefficients in a smooth $ k $-representation $ pi $ of $ G $ are isomorphic to the $ mathrm{Ext} $-groups $ mathrm{Ext}^i_G(mathbb{1},pi) $ computed in the category of smooth $ k $-representations of $ G $. We apply this to show that if $ pi $ is a supersingular $ overline{mathbb{F}}_p $-representation of $ mathrm{GL}_2(mathbb{Q}_p) $, then the continuous cohomology of $ mathrm{SL}_2(mathbb{Q}_p)$ with values in $ pi $ vanishes. Furthermore, we prove that the continuous cohomology groups of a $ p $-adic reductive group $ G $, with coefficients in an admissible unitary $ mathbb{Q}_p $-Banach space representation $ Pi $, are finite dimensional. We show that the continuous cohomology of $ mathrm{SL}_2(mathbb{Q}_p) $ with values in non-ordinary irreducible $ mathbb{Q}_p $-Banach space representations of $ mathrm{GL}_2(mathbb{Q}_p) $ vanishes." @default.
- W3172049139 created "2021-06-22" @default.
- W3172049139 creator A5060181921 @default.
- W3172049139 date "2021-06-08" @default.
- W3172049139 modified "2023-09-27" @default.
- W3172049139 title "Continuous Cohomology and Ext-Groups" @default.
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