2
$\begingroup$

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 $X$?

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.