It's maybe silly to ask "Why are there three types of non-Archimedean geometry?", that would be like asking why there are three (and even more) different Weil cohomologies. So I have to clarify my question.

Let $X$ be a scheme on a $p$-adic ring (i.e. an extension of $\mathbb{Z}_{p}$).

- What are the relationships between the rigid, berkovich, and adic analytification of $X$?

- For each of them, are the GAGA statements exactly the same as in the complex case or are there (subtle) changes?

- What are the cohomologies on these analytification? Are they Weil cohomologies? Are there comparison isomorphisms?

- Finally, a question that may be too general, how do you know in the context if you have to work with one analytification more than another?

Feel free to answer even if you don't have the answer to all my questions :)

It may seem silly to ask "Why are there three types of non-Archimedean geometry?", that would be like asking why there are three (and even more) different Weil cohomologies. So I have to clarify my question.

Let $X$ be a scheme on a $p$-adic ring (i.e. an extension of $\mathbb{Z}_{p}$).

- What are the relationships between the rigid, berkovich, and adic analytification of $X$?

- For each of them, are the GAGA statements exactly the same as in the complex case or are there (subtle) changes?

- What are the cohomologies on these analytification? Are they Weil cohomologies? Are there comparison isomorphisms?

- Finally, a question that may be too general, how do you know in the context if you have to work with one analytification more than another?

Feel free to answer even if you don't have the answer to all my questions :)

It's maybe silly to ask "Why are there three types of non-Archimedean geometry?", that would be like asking why there are three (and even more) different Weil cohomologies. So I have to clarify my question.

Let $X$ be a scheme on a $p$-adic ring (i.e. an extension of $\mathbb{Z}_{p}$).

- What are the relationships between the rigid, berkovich, and perfectoid analytification of $X$?

- For each of them, are the GAGA statements exactly the same as in the complex case or are there (subtle) changes?

- What are the cohomologies on these analytification? Are they Weil cohomologies? Are there comparison isomorphisms?

- Finally, a question that may be too general, how do you know in the context if you have to work with one analytification more than another?

Feel free to answer even if you don't have the answer to all my questions :)

It's maybe silly to ask "Why are there three types of non-Archimedean geometry?", that would be like asking why there are three (and even more) different Weil cohomologies. So I have to clarify my question.

Let $X$ be a scheme on a $p$-adic ring (i.e. an extension of $\mathbb{Z}_{p}$).

- What are the relationships between the rigid, berkovich, and adic analytification of $X$?

- For each of them, are the GAGA statements exactly the same as in the complex case or are there (subtle) changes?

- What are the cohomologies on these analytification? Are they Weil cohomologies? Are there comparison isomorphisms?

- Finally, a question that may be too general, how do you know in the context if you have to work with one analytification more than another?

Feel free to answer even if you don't have the answer to all my questions :)