  Among all affine, flat, finitely presented group schemes, we focus on those
that are pure, this includes all groups which are extensions of a finite
locally free group by a group with connected fibres. We prove that over an
arbitrary base ring, pure group schemes have a classifying space satisfying the
resolution property, an embedding into some GLn, a tensor generator for their
category of finite type representations, and can be reconstructed from their
category of projective finite type representations. In the case of an Artinian
base ring, the same is true for all affine, flat, finitely presented group
schemes, this answers a question of Conrad. We also prove that quotients of
pure groups by closed pure subgroups over an arbitrary base scheme are
Zariski-locally quasi-projective. This answers a question of Raynaud, in the
case of affine groups. We give various applications.
