  In this paper, we will investigate further properties of A-schemes. The
category of A-schemes possesses many properties of the category of coherent
schemes, and in addition, it is co-complete and complete. There is the
universal compactification, namely, the Zariski-Riemann space in the category
of A-schemes. We compare it with the conventional Zariski-Riemann space, and
characterize the latter by a left adjoint.
