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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".

Thanks in advance.

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    $\begingroup$ If you want an approach to noncommutative projective geometry, this survey arxiv.org/abs/math/9910082 of Stafford-Van den Bergh describes work following the philosophy due to Artin-Tate-van den Bergh of using a category of graded modules modulo torsion to stand as a noncommutative projective space. This ultimately relies on a noncommutative version of Serre's theorem, which in the commutative setting says that this category is equivalent to that of (quasi-) coherent sheaves. I know this doesn't quite answer your question as posed but hope it's helpful. $\endgroup$ – Jan Grabowski Aug 20 '20 at 10:29
  • $\begingroup$ Have you tried making sense of "very ample line objects" in a dg-category? $\endgroup$ – user147129 Aug 20 '20 at 14:28
  • $\begingroup$ @Riza That is what I was trying to make sense of actually, but I have not had luck so far. $\endgroup$ – Federico Barbacovi Aug 20 '20 at 14:30
  • $\begingroup$ @Jan thanks, I'll take a look at the reference! $\endgroup$ – Federico Barbacovi Aug 20 '20 at 14:30
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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.

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