2010/01/26 by Mathieu Florence, Florence, Mathieu
Mathematics · #14E05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #math.AG #msc:14E05
paper · pdf · doi:10.48550/arxiv.1001.4602
15 pages
arxiv created 2010/01/26 · openalex publication_date 2010/01/26 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a field, and A a finite-dimensional k-algebra. Let d be an integer. Denote by Gr(d,A) the Grassmannian of d-subspaces of A (viewed as a k-vector space), and by GL1(A) the algebraic k-group whose points are invertible elements of A. The group GL1(A) acts naturally on Gr(d,A) (by the formula g.E=gE). The aim of this paper is to study some birational properties of this action. More precisely, let r be the gcd of d and dim(A). Under some hypothesis on A (satisfied if A/k is étale), I show that the variety Gr(d,A) is birationally and GL1(A)-equivariantly isomorphic to the product of Gr(r,A) by a projective space (on which GL1(A) acts trivially). By twisting, this result has some corollaries in the theory of central simple algebras. For instance, let B and C be two central simple algebras over k, of coprime degrees. Then the Severi-Brauer variety SB(B ⊗ C) is birational to the product of SB(B) × SB(C) by an affine space of the correct dimension. These corollaries are in the spirit of Krashen's generalized version of Amitsur's conjecture.