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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.

  1. 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 $\overset{!}{\otimes}$.

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  • $\begingroup$ What exactly is $\pi_1^*M$? $\endgroup$ – მამუკა ჯიბლაძე May 13 '17 at 20:07
  • $\begingroup$ @მამუკაჯიბლაძე The usual pullback of modules $M \otimes_A ( A\otimes_kA)$. $\endgroup$ – Saal Hardali May 13 '17 at 20:08
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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

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