# 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?