The analytic topology suffices for the $B_{\mathrm{dR}}^+$-Grassmannian
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| Format: | Preprint |
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2023
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| _version_ | 1866913574609223680 |
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| author | Cesnavicius, Kestutis Youcis, Alex |
| author_facet | Cesnavicius, Kestutis Youcis, Alex |
| contents | The $B_{\mathrm{dR}}^+$-affine Grassmannian was introduced by Scholze in the context of the geometric local Langlands program in mixed characteristic and is the Fargues-Fontaine curve analogue of the equal characteristic Beilinson-Drinfeld affine Grassmannian. For a reductive group $G$, it is defined as the étale (equivalently, $v$-) sheafification of the presheaf quotient $LG/L^+G$ of the $B_{\mathrm{dR}}$-loop group $LG$ by the $B_{\mathrm{dR}}^+$-loop subgroup $L^+G$. We combine algebraization and approximation techniques with known cases of the Grothendieck-Serre conjecture to show that the analytic topology suffices for this sheafification, more precisely, that the $B_{\mathrm{dR}}^+$-affine Grassmannian agrees with the analytic sheafification of the aforementioned presheaf quotient $LG/L^+G$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_11710 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The analytic topology suffices for the $B_{\mathrm{dR}}^+$-Grassmannian Cesnavicius, Kestutis Youcis, Alex Algebraic Geometry Number Theory Primary 14G45, Secondary 14L15, 14M15 The $B_{\mathrm{dR}}^+$-affine Grassmannian was introduced by Scholze in the context of the geometric local Langlands program in mixed characteristic and is the Fargues-Fontaine curve analogue of the equal characteristic Beilinson-Drinfeld affine Grassmannian. For a reductive group $G$, it is defined as the étale (equivalently, $v$-) sheafification of the presheaf quotient $LG/L^+G$ of the $B_{\mathrm{dR}}$-loop group $LG$ by the $B_{\mathrm{dR}}^+$-loop subgroup $L^+G$. We combine algebraization and approximation techniques with known cases of the Grothendieck-Serre conjecture to show that the analytic topology suffices for this sheafification, more precisely, that the $B_{\mathrm{dR}}^+$-affine Grassmannian agrees with the analytic sheafification of the aforementioned presheaf quotient $LG/L^+G$. |
| title | The analytic topology suffices for the $B_{\mathrm{dR}}^+$-Grassmannian |
| topic | Algebraic Geometry Number Theory Primary 14G45, Secondary 14L15, 14M15 |
| url | https://arxiv.org/abs/2303.11710 |