Quasi-coherent cohomology in non-commutative algebraic geometry

Question

In non-commutative algebraic geometry, the motto so to speak is to replace the study of a scheme $X$ with the study of the category $D_{qcoh}(X)$ of quasi-coherent sheaves and study the properties thereof. My question is what is the correct analogue for the cohomology of quasi-coherent sheaves in that setting?

For a quasi-coherent sheaf $\mathcal{F}$ on $X$, we can write $$H^n(X,\mathcal{F})\cong \text{Ext}^n(\mathcal{O}_X,\mathcal{F})$$ which would suggest that quasi-coherent sheaf cohomology of a category $\mathcal{C}$ of an object $\mathcal{F}\in \text{Ob}(\mathcal{C})$, we have $$H^\bullet(\mathcal{C},\mathcal{F}):= \text{Hom}_{\mathcal{C}}(G,\mathcal{F})$$ for the "correct" analogue of $\mathcal{O}_X$. The problem is, I don't know what the correct analogue is....

Any pointers?

Answer 1

The structure sheaf is not intrinsic to the derived category (for instance, the standard derived equivalence of the abelian variety and its dual takes the structure sheaf of the former to a skyscraper sheaf on the latter). Therefore, there is no preferred choice for $G$, and thus, if you want to talk about sheaf cohomology, you should specify such $G$ as an additional piece of data.