STAGE Seminar

Alex Smith

Reconstructing a formal group from its Cartier module

Abstract: Given a commutative ring K, there is a natural non-commutative ring structure on

1 + t x K[[x]][t] whose addition is given by multiplication and whose multiplication is difficult to describe explicitly. This ring E is called the Cartier ring, and it can be given as the natural endomorphism ring of a certain functor.

Every formal group over K corresponds in a natural way to a certain E module obeying a short list of conditions. We will prove that this construction gives a functor that embeds the category of formal groups over K as a full subcategory of the set of E modules. In particular, we will give a method for reconstruction a formal group from its Cartier module.

Finally, we will give generators and most of the relations for the Cartier ring.

The talk will be accompanied by breakfast.