## Student Arithmetic Geometry Seminar

•  Nick Rekuski, MSU
•  Perfectoid Fields and Tilting
•  09/19/2019
•  10:00 AM - 11:30 AM
•  C329 Wells Hall

In this talk we will introduce perfectoid fields and tilting. Perfectoid fields provide the the correct base scheme for perfectoid spaces. Tilting is a fundamental tool that will let us lift characteristic $0$ results to characteristic $p$ results. For example, if $K$ is a characteristic $0$ perfectoid field and $K^{\flat}$ is a tilt of $K$ then $K^{\flat}$ is a characteristic $p$ field; $K^{\circ}/K^{\circ\circ}\cong K^{\flat \circ}/K^{\flat\circ\circ}$; if $[L:K]$ is finite then $[L^{\flat}:K^{\flat}]=[L:K]$ (in particular, $L$ is perfectoid); and there is an equivalence of categories between finite étale covers of $K$ and finite étale covers of $K^{\flat}$ via $L\mapsto L^{\flat}$. This talk will not require any material beyond first-year graduate algebra. However, the sophistication required may be higher. To make this talk as accessible as possible, we will include numerous examples.

## Contact

Department of Mathematics
Michigan State University
619 Red Cedar Road
C212 Wells Hall
East Lansing, MI 48824

Phone: (517) 353-0844
Fax: (517) 432-1562

College of Natural Science