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 $A$.



> 
> 1. 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?
> 2. 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.**



> 
> 3. 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 $\overset{!}{\otimes}$.



Closed symmetric monoidal structure on the derived category of modules whose unit is a dualizing complex?
Let me answer the question in the title and explain the notation $- \otimes^{!}-$, forgive me for ignoring some finiteness conditions.
Let's start with something somewhat simpler - take any field $k$, and for any finitely generated $k$-algebra $A$ with structure map $f:k \to A$, let $R\_A = f^{!}(k)$, the canonical (or rigid) dualizing complex of $A$. Let $D\_A(-) = RHom\_A(-,R\_A)$ be the dualizing functor.


Given any functor $F:D\_f(A) \to D\_f(B)$ between such $k$-algebras, we can define its twist as follows: $F^{!}(-) = D\_B(F(D\_A(-))$.
For example, if $g:A \to B$ is a $k$-algebra map and you take $G(-) = B\otimes^L\_A -$ then (with suitable boundedness conditions), you will get that $G^{!}$ is just $g^{!}$, so this construction can be thought of a generalization of the construction of the twisted inverse image from the usual inverse image functor.


Now, we can co the same construction for functors in several variables. For instance, twisting $-\otimes^L\_A -$, we would get the following functor:


$D\_A(D\_A(-)\otimes^L\_A D\_A(-))$, and so it makes sense to denote this by $-\otimes^{!}-$. Observe now that since $D\_A(D\_A(-)) \cong 1$, at least when restricted to things with f.g cohomologies, the operation $-\otimes^{!}-$ will remain associative and commutative up to natural isomorphisms. What is the unit of this operation? it is $D\_A(A) = R\_A$, the dualizing complex.


You have asked about another operation, which I will write as
$(M,N) \mapsto RHom\_{A\otimes^L\_k A} (A, M\otimes^L\_k N)$.


How are they related? turns out, they are the same!
This was essentially shown for $k$-algebras in Theorem 4.1 of


Avramov, L. L., Iyengar, S. B., Lipman, J., & Nayak, S. (2010). Reduction of derived Hochschild functors over commutative algebras and schemes. Advances in Mathematics, 223(2), 735-772.


(you don't need $k$ to be a field).


It is also true for commutative DG-algebras (again, under some assumptions), see Theorem 5.5 of my preprint <https://arxiv.org/abs/1510.05583>


