  The present paper contains new geometric theorems in mixed characteristic
case. We derive a bunch of cohomological consequences using these geometric
theorems. Among them an isotropy result for quadratic spaces, a purity result
for quadratic spaces, a result on the Grothendieck--Serre conjecture for the
special linear group of an Azumaya algebra. The Gersten conjecture for the
functor K2 is proved. Bloch-Ogus type result is obtained as well. Suslin's
exact sequence is derived and its application to a finiteness result is given.
A version of the Roitman theorem is proved.
