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Person# Jean Fasel

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a d

Projective module

In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties o

Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebr

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Let A be a d-dimensional smooth algebra over a perfect field of characteristic not 2. Let Um(n+1)(A)/En+1 (A) be the set of unimodular rows of length n + 1 up to elementary transformations. If n >= (d + 2)/2, it carries a natural structure of group as discovered by van der Kallen. If n = d >= 3, we show that this group is isomorphic to a cohomology group H-d (A, G(d+1)). This extends a theorem of Morel, who showed that the set Um(d+1)(A)/SLd+1(A) is in bijection with H-d (A, G(d+1))/SLd+1(A). We also extend this theorem to the case d = 2. Using this, we compute the groups Um(d+1)(A)/Ed+1(A) when A is a real algebra with trivial canonical bundle and such that Spec(A) is rational. We then compute the groups Um(d+1)(A)/SLd+1 (A) when d is even, thus obtaining a complete description of stably free modules of rank d on these algebras. We also deduce from our computations that there are no stably free non free modules of top rank over the algebraic real spheres of dimension 3 and 7.

2011Let A be a commutative noetherian ring of Krull dimension 3. We give a necessary and sufficient condition for A-projective modules of rank 2 to be free. Using this, we show that all the finitely generated projective modules over the algebraic real 3-sphere are free. (C) 2010 Elsevier Inc. All rights reserved.

2010Let A be a noetherian commutative Z[1/2]-algebra of Krull dimension d and let P be a projective A-module of rank d. We use derived Grothendieck-Witt groups and Euler classes to detect some obstructions for P to split off a free factor of rank one. If d

2009