Intuition for points of the moduli of objects for a dg-category

Question

Problem summary: I'm trying to get some intuition for what the moduli space of objects for a dg-category (as in this paper by Brav and Dyckerhoff) actually looks like/how to give an alternative identification of the moduli of objects in certain cases.

Edited to add: Adding as per David's suggestion below: the original paper on the topic by Toën and Vaquié, and this paper on a coherent version of the construction by Lowrey that I was previously unaware of.

---

Let $\mathcal{C}$ be a compactly generated dg-category. Then the moduli of objects is defined such that on an affine $U$, $\mathcal{M}_{\mathcal{C}}(U)$ is the space of exact functors from compact objects $\mathcal{C}^c$ in $\mathcal{C}$ to perfect complexes on $U$. When $\mathcal{C}$ is smooth, such functors are corepresented by left proper objects $c\in \mathcal{C}^c\otimes\text{Perf}(U)$ -- this means that inner hom out of $c$ is a continuous functor $\mathcal{C}^c\otimes\text{Perf}(U) \to \text{Perf}(U)$ with a continuous right adjoint.

I'd like to get a better grasp on what these left proper objects actually look like. For instance, if $\mathcal{C} = \text{QCoh}(T^\ast C)$ for $C$ a smooth projective curve, then $\mathcal{M}_\mathcal{C}$ should be (I think) the moduli space of Higgs sheaves on $C$. If $\mathcal{C} = k[x]\text{-mod}$, then I think that being left proper should be related to some sort of ''finite support'' condition. But I'm not sure how to actually prove either of these.

To be concrete, let's suppose that $A$ is a regular Noetherian $k$-algebra for $k$ a field of characteristic zero, and consider th dg-category $A\text{-mod}$. If I have everything straight in my head (already a bold assumption), then I think $\text{Hom}_A(M,-): A\text{-mod}\to k\text{-mod}$ is continuous with a right adjoint precisely if $M$ is a perfect complex. I also think that in this case the right adjoint is the functor $\text{Hom}_k(M^\vee, -)$ where $M^\vee = \text{Hom}_A(M,A)$. If this is all the case, then what additional constraint is imposed on $M$ by continuity of its right adjoint?

Answer 1

[It would be good to mention the original paper of Toën-Vaquié https://arxiv.org/abs/math/0503269 where these moduli spaces of objects are defined, and maybe that of Lowrey for the ``coherent" version https://arxiv.org/abs/1110.5117 ]

Having a continuous right adjoint (in the compactly generated presentable / "large" dg category setting) is equivalent to preserving compact objects. So you're asking for perfect A-modules to be taken to bounded complexes of vector spaces, which geometrically means taking Hom from a sheaf with proper support (so cohomology is finite dimensional) -- or as you say, in the affine setting, this implies having finite support. So for example in the Higgs sheaves context you're only seeing Higgs sheaves which are proper over $C$ -- i.e. supported on spectral varieties that are finite over the curve, as opposed to "going off to infinity" in the cotangent bundle (like Higgs sheaves with poles do).

There are many references for sheaves with proper support, not sure what the best one is, but in any case https://arxiv.org/abs/1312.7164 talks about representability results for these kinds of proper functors (taking proper functors to Vect exchanges Coh and Perf).