diagrams – The obstacles to the abelian sheaf being almost coherent

Let $ X $ to be a Noetherian spectral topological space. A necessary condition for an abelian sheaf on $ X $ be almost coherent with respect to an affine system structure on $ X $ does his superior cohomology disappear?

I do not think that's a sufficient condition. For example, let's take a single-point space $ X = {pt } $ and a sheaf $ F $ with $ F (X) = mathbb {Z} $. The superior cohomology of $ F $ disappears for size reasons. Yes $ F $ were almost consistent with an affine system structure on $ X $it would be a module on a commutative unital ring $ R $ with exactly one prime ideal. Since there is no non-zero nilpotents in $ End ( mathbb {Z}) $, elements of the ideal first would annihilate $ mathbb {Z} $. So, $ mathbb {Z} $ would also be a module on the quotient of $ R $ by his prime ideal (which is a field). This is a contradiction because $ mathbb {Z} $ is free abelian.

The question is: what are the other reasonably easy elements to formulate so that an abelian package is almost coherent? Is it possible to give a necessary and sufficient condition that is not entirely tautological?