Manuel Ojanguren

Variétés unirationelles non rationelles: Au-delà de l"exemple d"Artin et Mumford. (Non rational unirational varieties: Beyond the Artin-Mumford example)

Keywords: non-rational but unirational variety ; Grothendieck invariants ; rationality ; Brauer invariants Reference CMA-ARTICLE-1989-001doi:10.1007/BF01850658 Record created on 2008-12-16, modified on 2016-08-08

A purity theorem for the witt group

Abstract Let A be a regular local ring and K its field of fractions. We denote by W the Witt group functor that classifies quadratic spaces. We say that purity holds for A if W(A) is the intersection of all W(A p ) ⊂ W(K), as p runs over the height-one prime ideals of A. We prove purity for every regular local ring containing a field of characteristic ≠ 2. The question of purity and of the injectivity of W(A) into W(K) for arbitrary regular local rings is still open.

Cancellation of Azumaya algebras

1. Let R be a commutative ring with 1 and &4(R) the category of Azumaya algebras over R. It was proved in [4] that if R is semilocal then, for X, Y, A E Obj &z(R) the “cancellation law” X @ -4 s X (3 Y T A s Y holds. In this note we prove a theorem which gives a set of equivalent conditions for cancellation over any ring. WC derive, as corollaries, the above result, a theorem of Knus and the “cancellation law” for Az(k[x]) where k is a perfect field. We show (Proposition 2) that cancellation does not always hold over polynomial rings in two variables. To show this, we first prove that if D is any noncommutative division ring, there exist nonfree projective ideals in D[x,y]. This result seems to be of some independent interest. For unexplained terms we refer to Bass [2].

Espaces principaux homogènes localement triviaux

Keywords: local triviality of principal homogeneous space ; reductive group scheme Reference CMA-ARTICLE-1992-001doi:10.1007/BF02699492 Record created on 2008-12-16, modified on 2016-08-08

A note on the fundamental theorem of projective geometry

Keywords: foundations of geometry Reference CMA-ARTICLE-1969-001doi:10.1007/BF02564531 Record created on 2008-12-16, modified on 2016-08-08

Rationally trivial Hermitian spaces are locally trivial

Abstract. Let R be a regular local ring, K its field of fractions and A an Azumaya algebra with involution over R. Let h be an

A splitting theorem for quadratic forms

\epsilon

Modules and quadratic forms over polynomial algebras

-hermitian space over A. We show that if

WITT GROUPS OF THE PUNCTURED SPECTRUM OF A 3-DIMENSIONAL REGULAR LOCAL RING AND A PURITY THEOREM

\bold{h}\otimes_R K

Ketu and the second invariant of a quadratic space

is hyperbolic over