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Suppose $(A, (\mu_k))$ is a (curved) $A_\infty$ algebra, and let $(\tilde A, (\tilde\mu_k))$ be its minimal model. Now, we have two Hochschild cohomology rings $HH^*(A)$ and $HH^*(\tilde A)$. (It may be also related to $HH^*(A,\tilde A)$ or $HH^*(\tilde A,A)$ for the diagonal $A_\infty$ bimodules.)

Are there any natural relations between $HH^*(A)$ and $HH^*(\tilde A)$ ? What I'm interested most is the following question:

Question: Does there exist a natural ring homomorphism between them? (Any reference would be greatly appreciated.)

The ring structure is defined by $f\cdot g= \pm\mu(\cdots f(\cdots)\cdots g(\cdots) \cdots)$

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    $\begingroup$ I am not so great with references, but don't we have a spectral sequence $HH^*(H_*(A)) \implies HH^*(A)$ (coming from filtering the bar construction), so by its naturality we have a map of the spectral sequences for $A$ and for $\tilde{A}$ which is an iso on these pages, hence an isomorphism on the $E_\infty$ pages? So the answer to your question is, the complexes are quasiisomorphic. $\endgroup$
    – Connor Malin
    Commented Jul 26, 2021 at 19:20
  • $\begingroup$ @ConnorMalin Thank you. Does this spectral sequence relation imply other additional relations than quasi-isomorphisms, such like ring homomorphisms? $\endgroup$
    – Hang
    Commented Jul 27, 2021 at 16:29
  • $\begingroup$ The spectral sequence is used to prove the map is a quasi-isomorphism. Whatever structure Hochschild cohomology has will be preserved since we are applying Hochschild cohomology to a map of A infinity objects. $\endgroup$
    – Connor Malin
    Commented Jul 27, 2021 at 20:31
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    $\begingroup$ If I'm not mistaken, there is an infinity quasi isomorphism from the hochschild complex of the minimal model to that of the original algebra, constructed as in the homotopy transfer theorem $\endgroup$
    – Fernando Muro
    Commented Jul 27, 2021 at 22:13

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