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I apologize in advance if the question is too basic.
Let $A$ be a differential graded algebra and $H^{0}A$ the 0-cohomology of $A$ (which is an ordinary ring) and $A^0$ is the 0-level of $ A$ . An element $a\in A^0$ is said to be homotopy invertible if there exists $x\in A^0$ such that $[xa-1]=[ax-1]=0\in H^0 A$.

I was wondering if there exists a weak equivalence of differential graded algebras $A\rightarrow B$ such that an element $ b\in B^0$ is strictly invertible if and only if it is homotopy invertible. Thank you in advance.

PS: The differential graded algebra A is not supposed to be bounded.

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    $\begingroup$ In order to take cohomology classes you must first assume that $xa-1$ and $ax-1$ are cocycles, which amounts to $xa$ and $ax$ being cocycles. $\endgroup$ Oct 3, 2013 at 17:43
  • $\begingroup$ Fernando, it is by definition since I wrote [ax-1]... Sorry if it was not clear. $\endgroup$
    – Ilias A.
    Oct 3, 2013 at 18:44
  • $\begingroup$ I have undeleted this question which you have deleted six years ago. -- Please do not self-delete your useful questions! $\endgroup$
    – Stefan Kohl
    Apr 30, 2021 at 21:10

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