Local structure of algebraic monoids

TL;DR

This paper investigates the local structure of irreducible algebraic monoids at idempotents, revealing their fiber bundle structures and providing criteria for properties like normality and smoothness.

Contribution

It characterizes the local structure of algebraic monoids at idempotents and establishes their description as induced varieties over kernels and abelian varieties.

Findings

Irreducibility of stabilizers and centralizers in normal monoids

Criteria for normality and smoothness of algebraic monoids

Description of monoids as induced varieties over kernels and abelian varieties

Abstract

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.