Divisors on coherent schemes and homogeneous spaces

We investigate the positivity and extension of invertible sheaves on group homogeneous spaces over coherent bases. Bypassing the failure of standard limit arguments and the classical Weil–Cartier correspondence, we develop a valuative divisor theory on locally coherent schemes. This establishes an exact correspondence between effective valuative divisors and rank-one reflexive sheaves, yielding a non-Noetherian Ramanujam–Samuel theorem. To homologically control special fibre degenerations, we study morphisms of (N)-type; these govern the descent of generically trivial invertible sheaves and establish the theorems of the cube and the square without smoothness hypotheses. Utilizing the Picard-admissibility of group actions, we construct ample invertible sheaves explicitly from one-codimensional orbit boundaries. This achieves the rigid extension of generic polarizations to integral models over Prüfer bases, structurally generalizing Raynaud’s classical proof of his quasi-projectivity theorems.