The Analytic Topology Suffices for the B_dR^+-Grassmannian

For a reductive group G, its BdR+-affine Grassmannian is defined as the étale (equivalently, v-) sheafification of the presheaf quotient LG/L+G of the BdR-loop group LG by the BdR+ -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 BdR+-affine Grassmannian agrees with the analytic sheafification of the presheaf quotient LG/L+G. The talk is based on joint work with Alex Youcis.