Let $A$ be non-positively graded commutative DG-algebra almost of finite type over a field $k$ of characteristic $0$. Most of these assumptions (affine, commutative, characteristic, bound) are only to ensure that the question isn't answered negatively by exhibiting a not so convincing counter-example.

Let $Coh(A)$ be the stable infinity category of DG-modules over $A$ with coherent cohomologies in finitely many degrees. Let $Perf(A)$ be the stable infinity category of perfect $DG$ modules over $A$.

- Does there exist a
closed symmetric monoidal structure(in the $\infty-$sense)on $Perf(A)$which induces the familiar symmetric monoidal structure on the homotopy category $Ho(Perf(A))$ corresponding to $\{\mathcal{Hom}_A(-,-),-\otimes-\}$ (both understood as derived bi-functors between triangulated categories)? If so, is this easy? If not where can I find a proof?- Same question for $Coh(A)$ the stable $\infty$-category of perfect complexes.

**Should we expect that the stable category of all modules** $QCoh(A)$ to **admit a closed symmetric monoidal structure?** In the classical non-derived setting $QCoh(A)$ is a symmetric monoidal abelian category however **it doesn't have an internal hom** as the sheaf hom is not quasi-coherent in general. Despite this I've seen in several places claims that the derived $\infty$-category $QCoh(A)$ has a ** closed symmetric monoidal structure** which is odd for me even from the classical perspective. What am I missing?

Assume $A$ is perfect as a bi-module over itself and suppose there's monoidal structure on $Perf(A)$ as above. Let $A\otimes_k A \to A$ be the diagonal morphism and $\pi_1, \pi_2$ the 2 projections $A \to A \otimes_k A$. and consider the following bi-functor:

$$Perf(A)\times Perf(A) \to Perf(A)$$

$$(M,N) \mapsto M \overset{!}{\otimes}_A N := \mathcal{Hom}_{A \otimes_k A}(A, \pi_1^* M \otimes_{A \otimes_k A} \pi_2^* N)$$

I've seen in several places **cryptic remarks suggesting that this functor gives a symmetric monoidal structure on the derived category** for which the **unit is a dualizing complex.**

- Does the
shriek tensor product$\overset{!}{\otimes}$ defined above extend toa symmetric monoidal structureon $Perf(A)$? If sois it closed?What is itsunit?What will be the correct generalization for non-smooth $A$? Would one must enlarge to the category $IndCoh$ or $QCoh$?

The main point of this question is trying to understand if there's something behind said cryptic remarks about $\overset{!}{\otimes}$.