Notes on Solid Geometry

TL;DR

This paper provides expanded notes on the theory of solid geometry and analytic stacks, focusing on solid mathematics, inspired by Clausen and Scholze's work, with original discussion on morphisms of solid Huber rings.

Contribution

It introduces original insights into morphisms of solid Huber rings within the framework of solid mathematics and analytic stacks.

Findings

Discussion of smooth, étale, and finite presentation morphisms of solid Huber rings

Clarification of the theory of solid mathematics and analytic stacks

Extension of Clausen and Scholze's work with new technical details

Abstract

These are expanded notes of a seminar held in Columbia university during the Spring and Fall of 2024 about the theory of analytic stacks of Clausen and Scholze, with a focus in the theory of solid mathematics. The seminar is inspired from the Lecture Series of Analytic Stacks of Clausen and Scholze during the winter semester of 2023. All the theory of light condensed mathematics, analytic stacks and the proof of Serre duality must be attributed to Clausen and Scholze, any mistake or misconception is totally due to the author. The only original work in these notes is the discussion of smooth, \'etale and finite presentation morphisms of solid Huber rings in Section 7.