# 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
• Do you want the mono-morphism to be also a quasi-isomorphism? – Sándor Kovács Jan 28 at 20:41
• @SándorKovács no I don't think I need the mono to be a quasi-isomorphism :). – Anette Jan 28 at 21:07

Lemma Let $$A$$ and $$B$$ be complexes of sheaves. Then there exists a complex of sheaves $$C$$ such that both $$A$$ and $$B$$ admit a monomorphism into $$C$$ and the monomorphism $$A\hookrightarrow C$$ is a quasi-isomorphism. Furthermore, if both $$A$$ and $$B$$ are complexes of locally free sheaves, then $$C$$ can be chosen to be a complex of locally free sheaves.
Proof: Let $$D$$ denote the mapping cone of the identity morphism $$B\to B$$. Note that then by the definition of the mapping cone $$B$$ admits a monomorphism into $$D$$. Further note that $$D$$ is quasi-isomorphic to the zero complex. Now let $$C=A\oplus D$$. This has the required properties. $$\square$$
OK, so now take $$A=F$$ and $$B=F'$$. Note that this did not require $$F'$$ to be also quasi-isomorphic to $$E$$, only that it consists of locally free sheaves.