Runge immersion

Although I learned mainly about introductory stuffs of class field theory, I will write about my interest, rigid geometry. I have tried to prove one of the most fundamental, but important theorem of Gerritzen-Grauert.

Affinoid subdominans, which I won’t explain in detail because many special concepts in rigid geometry is required to do so, play a central role in rigid geometry. They are, roughly speaking, open basis. To get to know the structure of rigid varieties, we have to know about that of affinoid sub domains.

Gerritzen-Grauert theorem tells us the rough structure of them. Actually the most typical example of an affinoid sub domain is called a “rational domain,” which I won’t explain the definition either but is defined so explicitly. this theorem claims all affinoid subdominans can be written as finite union of rational domains.

To prove this main theorem, several novel concepts are required. I started reading the other book than one I did in a seminar, called “BGR”, 3 days ago. It is a little hard to read somehow, so I’ve just read 4 pages, with 10 pages of my notebook. I’m learning about the certain immersion of affinoid varieties, called a Runge immersion. this will play an important role in the proof according to the textbook. I will write here about how they are made use of in the proof in the next article possibly (perhaps I may do different things in the next…)

Anyway, I think the main purpose, to explain GG theorem briefly, has just completed here. Since I don’t expect anyone sees this blog, I will write like no one except for those who already knew them would understand.