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An affineness criterion for algebraic groups and applications

C. Sancho de Salas, F. Sancho de Salas, J. B. Sancho de Salas

Abstract

We prove that a smooth and connected algebraic group $G$ is affine if and only if any invertible sheaf on any normal $G$-variety is $G$-invariant. For the proof, a key ingredient is the following result: if $G$ is a connected and smooth algebraic group and $\mathcal L$ is a $G$-invariant invertible sheaf on a $G$-variety $X$, then the action of $G$ on $X$ extends to a projective action on the complete linear ${\mathbb P}(H^0(X,{\mathcal L})$. As an application of the affineness criterion, we give a new and simple proof of Chevalley-Barsotti Theorem on the structure of algebraic groups.

An affineness criterion for algebraic groups and applications

Abstract

We prove that a smooth and connected algebraic group is affine if and only if any invertible sheaf on any normal -variety is -invariant. For the proof, a key ingredient is the following result: if is a connected and smooth algebraic group and is a -invariant invertible sheaf on a -variety , then the action of on extends to a projective action on the complete linear . As an application of the affineness criterion, we give a new and simple proof of Chevalley-Barsotti Theorem on the structure of algebraic groups.

Paper Structure

This paper contains 10 sections, 28 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Rigidity of the group of units
  4. Invariance of invertible sheaves
  5. Fixing isomorphisms. The pointed case.
  6. Projective actions on sections of invariant invertible sheaves
  7. Representability of the Pic funtor on rational varieties
  8. The affineness criterion for algebraic groups.
  9. Invariance of invertible sheaves by 1-dimensional deformations
  10. Proof of the Chevalley-Barsotti Theorem on algebraic groups

Key Result

Proposition 2.1

If $X$ and $S$ are varieties, then the natural morphism is an epimorphism with kernel $\{(\lambda,\lambda^{-1}), \lambda\in k^{\rm x}\}\simeq k^{\rm x}$.

Theorems & Definitions (48)

  • Proposition 2.1: Rosenlicht2, Thm. 2
  • Corollary 2.2
  • Remark 2.3
  • Proposition 2.4
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Definition 3.4: Raynaud V.4.1
  • Definition 3.5: Mumford, Definition 1.6
  • Remark 3.6
  • ...and 38 more