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The $B_{\mathrm{dR}}^+$-affine Grassmannian was introduced by Scholze in thecontext of the geometric local Langlands program in mixed characteristic and isthe Fargues-Fontaine curve analogue of the equal characteristicBeilinson-Drinfeld affine Grassmannian. For a reductive group $G$, it isdefined as the \'{e}tale (equivalently, $v$-) sheafification of the presheafquotient $LG/L^+G$ of the $B_{\mathrm{dR}}$-loop group $LG$ by the$B_{\mathrm{dR}}^+$-loop subgroup $L^+G$. We combine algebraization andapproximation techniques with the known cases of the Grothendieck-Serreconjecture to show that the analytic topology suffices for this sheafification,more precisely, that the $B_{\mathrm{dR}}^+$-affine Grassmannian agrees withthe analytic sheafification of the aforementioned presheaf quotient $LG/L^+G$.

The analytic topology suffices for the $B_{\mathrm{dR}}^+$-Grassmannian