nLab
Artin-Mazur formal group

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 $\Phi (S)=\ker (H_{\mathrm{et}}^n(X\otimes S,{𝔾}_m)\to H_{\mathrm{et}}^n(X,{𝔾}_m))$. 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 $\widehat{\mathrm{Pic}}$.

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

Related concepts

• formal group

• height of a variety

References

• Michael Artin, Barry Mazur, Formal Groups Arising from Algebraic Varieties, numdam, MR56:15663