Submission: 2009, Aug 18
Let K be a field and A a finite dimensional algebra, of dimension n. Let 0 <: r < n be an integer. The Grassmannian Gr(r,A) is naturally endowed with an action of the algebraic group PGL_1(A). Under some assumptions on A (satisfied if A/K is étale) , we prove that there exists a birational, PGL_1(A)-equivariant isomorphism between Gr(r,A) and the product of Gr(gcd(r,n),A) by an affine space, on which PGL_1(A) acts trivially. We then derive some corollaries of this theorem. Among these, let us mention the following. i) Let A and B be two central simple algebras over K, of coprime degree. Then SB(A \otimes B) is birational to the product of SB(A)xSB(B) by an affine space of the correct dimension. ii) Let A be a central simple algebra over K, of degree n. Let 0 < r < n be an integer. Then the generalized Severi-Brauer variety SB(r,A) is birational to the product of SB(gcd(r,n),A) by an affine space of the correct dimension.
2000 Mathematics Subject Classification:
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