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

Abstract

Complete filtered $A_{\infty }$-algebras model certain deformation problems in the noncommutative setting. The formal deformation theory of a group representation is a classical example. With such applications in mind, we provide the $A_{\infty }$ analogs of several key theorems from the Maurer-Cartan theory for $L_{\infty }$-algebras. In contrast with the $L_{\infty }$ case, our results hold over a field of arbitrary characteristic. We first leverage some abstract homotopical algebra to give a concise proof of the $A_{\infty }$-Goldman-Millson theorem: The nerve functor, which assigns a simplicial set ${\mathcal{N}}_{\bullet }(A)$ to an $A_{\infty }$-algebra $A$, sends filtered quasi-isomorphisms to homotopy equivalences. We then characterize the homotopy 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 the characteristic zero case and show that the nerve of $A$ is homotopy equivalent to the simplicial Maurer-Cartan set of its commutator $L_{\infty }$-algebra. This answers a question posed by N. de Kleijn and F. Wierstra in arXiv:1809.07743.

Minor revision. Remark 6.4 added concerning curved algebras. 44 pages. To appear in the Journal of Algebra

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