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I'm a student in Algebraic Geometry. I've read chapter 2 and 3 of Hartshorne. I want to study the theory of Condensed Mathematics and Analytic Geometry by Scholze and Clausen.

What are the basic prerequisites for understanding the theory?

How much of Topos Theory is needed? Just the basic definitions? Can you give me some references?

Is the theory of $\infty$-Categories and/or Derived Algebraic Geometry needed?

Many Thanks

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    $\begingroup$ Although you might need some topos theory and $\infty$-categories (no derived geometry is necessary), they are not that essential and the prerequisites are covered in the Masterclass: math.ku.dk/english/calendar/events/condensed-mathematics $\endgroup$
    – Yai0Phah
    Commented Nov 25, 2021 at 18:43
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    $\begingroup$ Perhaps you'll want to first take a look at some more classical approaches to analytic geometry (rigid geometry, formal schemes), the applications that they have, and the categorical difficulties that arise when studying them. $\endgroup$ Commented Nov 28, 2021 at 19:17
  • $\begingroup$ @JeroenvanderMeer Can you give me some reference in formal schemes besides Hartshorne? Thank you $\endgroup$ Commented Nov 28, 2021 at 21:04

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I am not an expert, but I think it's actually relatively elementary compared to other things in the field. To get started reading condensed.pdf (and watching parts of the masterclass) you'll want at least

  • a good foundation in Hartshorne 2 and 3
  • a basic understanding of locally compact abelian groups (including at least the statement of Pontryagin duality)
  • a solid understanding of the derived category
  • some basic definitions and generalities related to sites
  • perhaps a few basic definitions of adic spaces
  • a willingness to black-box $\infty$-category stuff and pretend like it's the normal derived category.

In particular, if I understand correctly, condensed.pdf doesn't make use of any $\infty$-category stuff that couldn't already be phrased in terms of the triangulated version of the derived category.

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