# Geometry of the B_dR affine Grassmannian

As many readers of this blog already know, one key result in modern p-adic geometry is Scholze’s theorem that the $B_{\mathrm{dR}}$-affine Grassmannian is an ind-spatial diamond. The proof of this given in the Berkeley notes is a bit tricky and technical: it uses covering by infinite-dimensional objects in a crucial way, as well as an abstract Artin-type representability criterion.  So I’m very pleased to report that Bence Hevesi has given a beautiful new proof of this theorem in his Bonn master’s thesis. Bence’s proof avoids representability criteria or coverings by huge objects. Instead, his idea is to reduce to $\mathrm{GL}_n$ and then construct explicit charts for closed Schubert cells, using moduli of local shtukas at infinite level. You can read Bence’s outstanding thesis here.

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