# Is there a notion of projective dg category?

Since the paper Smooth and proper noncommutative schemes and gluing of DG categories by Orlov, dg categories are considered the non-commutative counterpart of algebraic geometry. More specifically, we call a dg category a non-commutative scheme if it is an admissible dg subcategory of the dg category $$\mathfrak{Perf}(X)$$ for a smooth projective scheme $$X$$. Now, many properties of a scheme $$X$$ defined over a field $$k$$ can be translated into properties of the category $$\mathfrak{Perf}(X)$$ (a dg enhancement of $$Perf(X)$$), e.g.

(1) the scheme $$X$$ is proper over $$k$$ if and only if for any $$E,F \in \mathfrak{Perf}(X)$$ we have (here I am identifying $$E$$ and $$F$$ with their image in the homotopy category of $$\mathfrak{Perf}(X)$$, which by definition is $$Perf(X)$$) $$\sum_{n \in \mathbb{Z}} \text{dim} \, \text{Hom}_{Perf(X)}(E,F[n]) < +\infty$$

(2) the scheme $$X$$ is smooth over $$k$$ if and only if the diagonal bimodule associated to $$\mathfrak{Perf}(X)$$ is perfect in the derived category of $$\mathfrak{Perf}(X)-\mathfrak{Perf}(X)$$ bimodules.

From the above, one can then generalise these notions to that and smooth and proper dg categories. My question is whether there exists a similar analogy for the notion of projective scheme, and therefore the notion of a "projective dg category".

If $$X$$ is a smooth projective threefold with a flopping curve $$C$$ then typically the variety $$Y$$ obtained from $$X$$ by a flop in $$C$$ is not projective, but smooth, proper, and derived equivalent to $$X$$. This shows that projectivity is not invariant under derived equivalence, hence does not correspond to a property of the derived category.