It is well-known that if $ X $ is a proper scheme over a ring $ A $, the global sections $ \mathcal{O}_{X} ( X ) $ of $ X $ are integral over $ A $. If $ X $ is moreover projective over $ A $, the global sections form a finitely generated module over $ A $.
We also know that the property of being finitely generated as a module is equivalent to being integral and being finite generated as an algebra. The natural question is then the following:
Is there some weaker property $ P $ of schemes such that being projective implies being $ P $ and $ X $ having $ P $ guarantees that the global sections of $ X $ are finitely generated over $ A $ as an algebra?
