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Let $k$ be an algebraically closed field of characteristic $0$.

Let $A,B$ be commutative differential graded algebras (cdga) over $k$ such that $H^{i}(A)=H^{i}(B) =0 \ (i>0)$.

Here, differentials are defined cohomologically, i.e, $$ \cdots \rightarrow A^{i-1} \overset{d_A}{\rightarrow} A^i \overset{d_A}{\rightarrow} A^{i+1} \overset{}{\rightarrow} \cdots $$ where $d_A$ is the differential of $A$. In other word, $A,B$ are connective.

We also denote the dg categories of dg $A$-modules and dg $B$-modules by $D_{dg}(A),D_{dg}(B)$, respectively.

Question If $D_{dg}(A),D_{dg}(B)$ are quasi-equivalent as dg categories, then $A, B$ are quasi-isomorphic as cdgas ?

Edit(10/4,5) Also, are there any candidate of conditions for cdgas for the reconstruction theorem(including the version below) to hold? Or is it solved under some assumptions?

Any comments and references are welcome. Thank you !

Please take a look at the useful comments.

The same question is in MSE.

Edit (a variant of the question, 9/20):

Let $\tilde{D}_{dg}(A),\tilde{D}_{dg}(B)$ be dg-derived categories, i.e, $\tilde{D}_{dg}(A) = D_{dg}(A)/ Ac(A), \tilde{D}_{dg}(B) = D_{dg}(B)/ Ac(B)$, where $Ac(A) \subset D_{dg}(A), Ac(B) \subset D_{dg}(B)$ are the full sub dg categories of acyclic complexes of $D_{dg}(A), D_{dg}(B)$, respectively.

If $\exists F: \tilde{D}_{dg}(A) \rightarrow \tilde{D}_{dg}(B)$ are quasi-equivalent of dg categories and $H^0(F)$ is a monoidal functor between monoidal categories $H^0(\tilde{D}_{dg}(A)), H^0(\tilde{D}_{dg}(B))$, then $A,B$ are quasi-isomorphic ?

Edit (classical case, 9/21):

When we consider two commutative rings $A,B$, where we regard them as cdgas in degree $0$. Even in this case, is the question true ?

Edit (10/4) Can my series of questions be solved from Toen's dg Morita theory or Lurie's theory of higher algebra? Or is this exactly what should be studied in the future?

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    $\begingroup$ I believe the key phrase here is "(derived) Morita theory." There is a wide literature on this, for example, see here, here, here, or here, to start. It may also help to reference Cohn or Cisinski and Tabuada for a quick overview of model structures on DG categories, as well as other work of Tabuada on those model structures. $\endgroup$
    – Stahl
    Commented Sep 19, 2023 at 3:05
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    $\begingroup$ Putting together results from these references, I think that if $A$ and $B$ are Morita equivalent $k$-algebras, viewed as DG-$k$ algebras in degree $0,$ they should have quasi-equivalent DG-derived categories, but not necessarily be quasi-isomorphic as DG-$k$-algebras. $\endgroup$
    – Stahl
    Commented Sep 19, 2023 at 3:16
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    $\begingroup$ There is also a difference between dgas and cdgas. The dg-category $D_{dg}(A)$ depends only on $A$ as a dga/$E_1$-algebra, and there can be several non-equivalent $E_\infty$-algebra structures on such $A$. So the answer to the question is "no" for this reason. $\endgroup$
    – Marc Hoyois
    Commented Sep 19, 2023 at 9:40
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    $\begingroup$ @MarcHoyois "and there can be several non-equivalent $E_\infty$-algebra structures on such $A$" --- this is actually false, when one works over a field of characteristic zero. This is Theorem A of arxiv.org/abs/1904.03585 . $\endgroup$
    – Dan Petersen
    Commented Sep 19, 2023 at 16:35
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    $\begingroup$ @Stahl: the OP assumes commutativity, and ordinary commutative $k$-algebras are Morita equivalent iff they're isomorphic (because the module category knows the center of a ring). So this approach won't work for producing a counterexample. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 20, 2023 at 3:14

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