Journal of the Ramanujan Mathematical Society
Volume 38, Issue 4, December 2023 pp. 309–324.
An affineness criterion for algebraic groups and applications
Authors:
C. Sancho de Salas, F. Sancho de Salas and J. B. Sancho de Salas
Author institution:
Departamento de
Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain.
Summary:
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 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 system {P}(H{0} (X,L)). As an
application of the affineness criterion, we give a new and simple proof of
the Chevalley-Barsotti Theorem on the structure of algebraic groups.
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