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Berkeley Lectures on p-adic Geometry(AMS-207)$
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Peter Scholze and Jared Weinstein

Print publication date: 2020

Print ISBN-13: 9780691202082

Published to Princeton Scholarship Online: January 2021

DOI: 10.23943/princeton/9780691202082.001.0001

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Drinfeld’s lemma for diamonds

Drinfeld’s lemma for diamonds

(p.140) Lecture 16 Drinfeld’s lemma for diamonds
Berkeley Lectures on p-adic Geometry

Peter Scholze

Jared Weinstein

Princeton University Press

This chapter addresses Drinfeld's lemma for diamonds. It proves a local analogue of Drinfeld's lemma, thereby giving a first nontrivial argument involving diamonds. This lecture is entirely about fundamental groups. A diamond is defined to be connected if it is not the disjoint union of two open subsheaves. For a connected diamond, finite étale covers form a Galois category. As such, for a geometric point, one can define a profinite group, such that finite sets are equivalent to finite étale covers. In this proof, the chapter uses the formalism of diamonds rather heavily to transport finite étale maps between different presentations of a diamond as the diamond of an analytic adic space.

Keywords:   Drinfeld's lemma, diamonds, fundamental groups, connected diamond, finite étale covers, Galois category, formalism, finite étale maps, analytic adic space

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