  We prove a relative version of a theorem on torsors on the projective line
due to Philippe Gille. As a consequence we obtain a ``weak homotopy
invariance'' result for torsors under reductive group schemes defined over
arbitrary semi-local regular domains. Specifically, only regular semi-local
domains with infinite residue fields are regarded in this preprint. However,
all results of the present preprint are true (after minor modifications) for
arbitrary semi-local regular domains. This will be the topic of our next
preprint.

Algebraic Geometry;K-Theory and Homology