# Contents

## Idea

Every variety in positive characteristic has a formal group attached to it. This group is often related to arithmetic properties of the variety such as being ordinary or supersingular.

## Definition

Let $X$ be a smooth proper $n$ dimensional variety over an algebraically closed field $k$ of positive characteristic $p$. Define the functor $\Phi :{\mathrm{Art}}_{k}\to \mathrm{Grp}$ by $\Phi \left(S\right)=\mathrm{ker}\left({H}_{\mathrm{et}}^{n}\left(X\otimes S,{𝔾}_{m}\right)\to {H}_{\mathrm{et}}^{n}\left(X,{𝔾}_{m}\right)\right)$. It is a fundamental result of the paper of Artin and Mazur that under these hypotheses the functor is prorepresentable by a one-dimensional formal group. This is known as the Artin-Mazur formal group .

## Examples

For a curve, this group is often called the formal Picard group $\stackrel{^}{\mathrm{Pic}}$.

For a surface, this group is called the formal Brauer group $\stackrel{^}{\mathrm{Br}}$.