UW Student Algebraic Geometry Seminar: Alex Scheffelin

Formal schemes were introduced by Grothendieck to geometrize the idea of taking the completion of a ring which corresponds to taking “formal” analytic / power series functions around a closed subset. They find a lot of use in deformation theory in particular, being related to Artin algebrization and Grothendieck’s existence theorem. Unfortunately, the theory of formal schemes is often understudied. While the basics of the theory exist in EGA, and to an even lesser extent in Hartshrone’s book, a lot of the theory of formal schemes exists in scattered papers throughout the literature. Furthermore, as they are often used to solve particular problems, the definitions can vary very slightly, making them a topic unfamiliar for many young algebraic geometers. In this talk we will define formal schemes, and take the time to highlight some of their properties unintuitive aspects of their theory, but also highlight how this can be very useful as well. I will provide complete proofs of various results which I do not know are compiled in a single source to show how one can work with them.