Seminar on Arithmetic Geometry

In this lecture series we aim to introduce the audience to the theory of kimberlites and its applications. The theory of kimberlites attempts to single out within Scholze’s category of v-sheaves those objects that “behave” as formal schemes. In this way, kimberlites provide well-behaved candidates for integral models of locally spatial diamonds. In the first two talks we discuss the foundations of the theory, and in the second two talks we discuss its applications to study moduli spaces of B_dR-lattices and moduli spaces of p-adic shtukas.

Talk 1: The purpose of this talk is to discuss the point-set topology subtleties that arise from working with non-analytic adic spaces within Scholze’s framework of v-sheaves. We recall miscellaneous aspects of the theory of adic spaces, perfectoid spaces, diamonds and v-sheaves. We introduce the small diamond and big diamond functors. We define the “diamond” (or olivine) spectrum of a Huber pair and describe it. We explain explicitly the example of the olivine spectrum of valuation rings. Finally, we use this topological considerations to “explain” the reversal of closure relations betwenn |Bun_G| and |Isoc_G|.

Ian Glesaon (Berkeley, USA)

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