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Complete filtered $A_\infty$-algebras model certain deformation problems inthe noncommutative setting. The formal deformation theory of a grouprepresentation is a classical example. With such applications in mind, weprovide the $A_\infty$ analogs of several key theorems from the Maurer-Cartantheory for $L_\infty$-algebras. In contrast with the $L_\infty$ case, ourresults hold over a field of arbitrary characteristic. We first leverage someabstract homotopical algebra to give a concise proof of the$A_\infty$-Goldman-Millson theorem: The nerve functor, which assigns asimplicial set $\mathcal{N}_{\bullet}(A)$ to an $A_\infty$-algebra $A$, sendsfiltered quasi-isomorphisms to homotopy equivalences. We then characterize thehomotopy groups of $\mathcal{N}_\bullet(A)$ in terms of the cohomology algebra$H(A)$, and its group of quasi-invertible elements. Finally, we return to thecharacteristic zero case and show that the nerve of $A$ is homotopy equivalentto the simplicial Maurer-Cartan set of its commutator $L_\infty$-algebra. Thisanswers a question posed by N. de Kleijn and F. Wierstra in arXiv:1809.07743.

On the Goldman-Millson theorem for $A_\infty$-algebras in arbitrary characteristic