On Grassmannians over * -rings PII: S 0 0 2 4 -3 7 9 5( 0 2) 0 0 3 4 4 -0

Abstract By [M. Golasiński, F. Gómez Ruiz, Polynomial and regular maps into Grassmannians, submitted] the tangent bundle T G n,r (K) of the Grassmannian G n,r (K) is diffeomorphic to the manifold Idem n,r (K) of idempotent n × n matrices with rank r for K the reals, complex numbers or quaternions. Furthermore, a regular deformation retraction of affine varieties n,r (K) : Idem n,r (K) → G n,r (K) can be described explicitly. Given a ring R with involution a notion of a Grassmannian and its normal and tangent bundle over R is studied. Then the above results are raised in that context provided appropriate properties for the ring R. In particular, this holds for any C * -algebra and a localization of some coordinate rings