# ag.algebraic geometry – Epi-mono factorisations in schemes via scheme-theoretic image

Suppose that $$f : X rightarrow Y$$ is a morphism of schemes.
Let $$Z hookrightarrow Y$$ the scheme-thereotic image of $$f$$.
Under what conditions is the morphism $$X rightarrow Z$$ an epimorphism?

If both $$X = mathrm{Spec} , B$$ and $$Y = mathrm{Spec} , A$$ are affine, then $$Z = mathrm{Spec}, A/I$$, where $$I$$ is the kernel of $$A rightarrow B$$. Then $$A/I rightarrow B$$ is injective, hence a monomorphism. Therefore $$X rightarrow Z$$ is an epimorphism.

I would think that if $$f$$ is quasi-compact or $$X$$ is integral, the scheme theoretic image has nice properties ($$f(X)$$ is dense in $$Z$$) and hopefully the result is true.