Because of the attack on the computer network of the University of Duisburg-Essen, some content (in particular some image files) cannot be accessed because it is stored on central servers of the university.

# Lecture Course: Adic Spaces (WS 17/18)

## Lecture Course: Adic Spaces (WS 17/18)

The $p$-adic valuation, for a prime number $p$, of the field of rational numbers is an important tool in number theory, and it is therefore an interesting question whether over fields which are complete with respect to a non-archimedean valuation (such as the field $\mathbb Q_p$ of $p$-adic numbers or suitable extension fields of it), there is a good analogue of the theory of complex geometry (which is the study of the geometric structure of zero sets of families of convergent power series in several variables).

It turns out that such a theory of *rigid analytic geometry* can in fact be developed (and has numerous interesting applications to arithmetic). Classical rigid geometry was developed by Tate in the early 1960s. Later, Berkovich gave a variant of the theory. In the early 1990s, Huber developed the theory of adic spaces, so far the most general and powerful version of rigid analytic geometry.

In this course, we will study the basics of Huber’s theory (and the connection with the other variants of rigid analytic geometry mentioned above).

Although their invention dates back quite some time, adic spaces have seen a rise in interest in recent years. One important reason is that they are an important ingredient in Scholze’s theory of perfectoid spaces, a notion which led to (sometimes quite spectacular) progress on a variety of questions in arithmetic geometry and related areas. There will also be a lecture course on perfectoid spaces next term, taught by A. Chatzistamatiou. We expect that attending both of these courses will give a good first impression of these exciting developments.

**Date and time:** Tue, 2pm – 4pm, N-U-4.04. **On Tuesday, Dec. 12, there will be no lecture.**

**Exercise group:** Mon, 2pm-4pm, S-U-3.02.

**Prerequisites:** Basics of algebraic geometry (you should have seen the notions of locally ringed space and of scheme). Being familiar with non-archimedean absolute values would also be helpful.

## Problem sheets

Due date | ||

Sheet 1 | Oct. 20 | |

Sheet 2 | Oct. 27 | |

Sheet 3 (corrected Nov. 1) | Nov. 3 | |

Sheet 4 | Nov. 10 | |

Sheet 5 | Nov. 17 | |

Sheet 6 | Nov. 24 | |

Sheet 7 | Dec. 1 | |

Sheet 8 | Dec. 8 | |

Sheet 9 | Dec. 15 | |

Sheet 10 (corrected Jan. 11) | Dec. 22 | |

Sheet 11 | Jan. 12 | |

Sheet 12 | Jan. 19 |

## References

B. Bhatt, Learning seminar on Adic Spaces, Winter 2017, notes by Matt Stevenson

B. Conrad, Number theory learning seminar 2014-2015 on perfectoid spaces

R. Huber, Continuous valuations, Math. Z. (1993)

R. Huber, A generalization of formal schemes and rigid-analytic varieties, Math. Z. (1994)

R. Huber, Etale cohomology of rigid-analytic varieties and adic spaces, Vieweg 1996.

K. Kedlaya, Course notes (on classical rigid geometry, Berkovich spaces and some other related topics)

F. Martin, Adic spaces

P. Schneider Basic notions of rigid analytic geometry, appeared in: Galois representations in arithmetic algebraic geometry (Durham, 1996), 369-378, London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press 1998

Y. Tian, Introduction to Rigid Geometry

T. Wedhorn, Introduction to Adic Spaces, Available online

J. Weinstein, Arizona Winter School 2017: Adic Spaces