Artin-Mazur formal group

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.

Let $X$ be a smooth proper $n$ dimensional variety over an algebraically closed field $k$ of positive characteristic $p$. Define the functor $\Phi: Art_k\to Grp$ by $\Phi(S)=\mathrm{ker}(H^n_{et}(X\otimes S, \mathbb{G}_m)\to H^n_{et}(X, \mathbb{G}_m))$. 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** .

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** $\widehat{Br}$.

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

Revised on July 27, 2011 18:05:13
by Toby Bartels
(64.89.62.147)