# Is there any numerical obstruction for all perfect complexes on a scheme being strictly perfect?

Let $X$ be a scheme and $E^{\cdot}$ be a cochain complex of sheaves of $\mathcal{O}_X$-modules.

We call $E^{\cdot}$ a strictly perfect complex if $E^{\cdot}$ is a bounded (in both direction) complex of finite dimensional algebraic vector bundles on $X$.

Moreover we call $E^{\cdot}$ a perfect complex if for all $x\in X$, there is an open neighborhood $U$ of $x$ and a strictly perfect complex $F^{\cdot}$ on $U$ together with a quasi-isomorphism $F^{\cdot}\overset{\sim}{\rightarrow} E^{\cdot}|_{U}$.

It is obvious that a strictly perfect complex must be perfect and it also has been proved in "SGA6" or "Higher Algebraic K-theory of Schemes and of Derived categories" Section 2 that any perfect complex on $X$ is strictly perfect if the scheme $X$ is quasi-compact and has an ample line bundle.

Nevertheless, for general $X$, perfect complex are not necessarily strictly perfect. The reason is that we cannot glue together the algebraic vector bundles and the quasi-isomorphisms on each open subset of $X$.

$\textbf{My question}$ is: is there any numerical invariant of the scheme $X$ (or the perfect complex $E^{\cdot}$) which is the obstruction of $E^{\cdot}$ being strictly perfect on $X$?