Nagata embedding and A-schemes

We define the notion of normal A-schemes, and approximable A-schemes. Approximable A-schemes inherit many good properties of ordinary schemes. As a consequence, we see that the Zariski-Riemann space can be regarded in two ways – either as the limit space of admissible blow ups, or as the universal compactification of the given non-proper scheme. We can prove Nagata embedding using Zariski-Riemann spaces.