Local structure of algebraic monoids

Details Local structure of algebraic monoids Local structure of algebraic monoids

2007-09-09

arxiv.org/abs/0709.1255v2

Mathematics

Algebraic Geometry

Final version, minor changes, to appear in Moscow Mathematical Journal

Scientific paper

We describe the local structure of an irreducible algebraic monoid $M$ at an idempotent element $e$. When $e$ is minimal, we show that $M$ is an induced variety over the kernel $MeM$ (a homogeneous space) with fibre the two-sided stabilizer $M_e$ (a connected affine monoid having a zero element and a dense unit group). This yields the irreducibility of stabilizers and centralizers of idempotents when $M$ is normal, and criteria for normality and smoothness of an arbitrary $M$. Also, we show that $M$ is an induced variety over an abelian variety, with fiber a connected affine monoid having a dense unit group.

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