Understanding Tangent Bundles and Tangent Spaces in P-adic Geometry

🔎P-adic geometry introduces differential tools to help us understand geometry using tangent bundles and tangent spaces.

🌍The study of p-adic geometry allows us to explore interesting and exciting phenomena in the world of geometry.

📐We focus on tangent bundles and tangent spaces in a specific example, with potential generalizations.

🧠Hodge-Tate period maps play a significant role in understanding the geometry of p-adic spaces.

⚠️The naive scheme-theoretic approach to differentiation and tangent spaces doesn't work in p-adic geometry.

What is p-adic geometry?

—P-adic geometry studies geometry using p-adic numbers, introducing differential tools to understand geometric phenomena.

What are tangent bundles and tangent spaces?

—Tangent bundles and tangent spaces provide a framework to analyze the behavior and properties of geometric objects at each point.

How do Hodge-Tate period maps relate to p-adic geometry?

—Hodge-Tate period maps encode information about the geometry of p-adic spaces, allowing for deeper understanding and analysis.

Why doesn't the naive scheme-theoretic approach work in p-adic geometry?

—The naive scheme-theoretic approach fails in p-adic geometry due to the lack of continuous derivations on certain perfectoid algebras.

Are there potential generalizations of the discussed concepts?

—Yes, the concepts and techniques explored in this video can be further generalized to study various phenomena in different settings.

00:00Introduction and overview of the video content.

03:40Exploration of tangent bundles and tangent spaces in p-adic geometry.

07:20Discussion of Hodge-Tate period maps and their significance in understanding p-adic spaces.

10:35Analysis of why the naive scheme-theoretic approach fails in p-adic geometry.

13:40Conclusion and potential generalizations of the concepts discussed.