The question is basically the one outlined in the title. Let $\mathcal{T}$ be a triangulated category containing infinite direct sums (e.g. $D\_{qc}(X)$ for some separated, finite type over a field $k$, scheme $X$) and consider the subcategory $\mathcal{E}$ generated by an object $E$ of $\mathcal{T}$. Here by subcategory generated I mean the smallest thick full triangulated subcategory containing all direct sums (and thus homotopy colimits). Is it true that $\mathcal{E}$ contains all the cohomology sheaves of $E$? Does it contain only some of them? It is clear that if I were to consider the subcategory generated in $\mathcal{T}^c$ (hence admitting only finite direct sums) this would necessarily be true. Indeed, considering the geometric example $\mathcal{T} = D\_{qc}(X)$, then the cohomology sheaves of a perfect complex are not necessarily perfect.
Thanks.



Can homotopy colimits recover cohomology sheaves?
No. 


Let $j:\mathbb{A}^2\_k\smallsetminus\{0\}\to \mathbb{A}^2\_k$ be the canonical open embedding. Then the derived pushforward $Rj\_\*$ is fully faithful and colimit-preserving. In particular, the subcategory of $D\_{qc}(\mathbb{A}^2)$ generated under colimits by $\mathscr{A}:=Rj\_\*\mathcal{O}$ is contained in this subcategory (in fact it coincides with the category of $\mathscr{A}$-modules, which is a subcategory because $\mathscr{A}$ is an idempotent algebra). However $H\_0\mathscr{A}=k[x,y]$ is not.


