# Can one always find a bigger global resolution

Let $$X$$ be a scheme. Let $$E$$ be a perfect complex of coherent sheaves on $$X$$ and suppose it admits two global resolutions $$F$$ and $$F'$$. By global resolution I mean that both $$F$$ and $$F'$$ are quasi-isomorphic to $$E$$ and both are complexes of vector bundles.

$$\bf{Question:}$$ Is it possible to find a global resolution $$H$$ of $$E$$ such that $$F, F'$$ map mono-morphically into $$H$$?

I know how to do this when $$E$$ has perfect amplitude $$[0, 1]$$ but I don't see how to generalize to say $$[0, n]$$ for arbitrary $$n$$.

• Why do you want this? – Sasha Jan 12 at 7:21