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 to a symmetric monoidal structure on $Perf(A)$? If so is it closed? What is its unit? 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 $\overset{!}{\otimes}$.