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Artin-Mazur formal group

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 XX be a smooth proper nn dimensional variety over an algebraically closed field kk of positive characteristic pp. Define the functor Φ:Art kGrp\Phi: Art_k\to Grp by Φ(S)=ker(H et n(XS,𝔾 m)H et n(X,𝔾 m))\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 .

Examples

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

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

References

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