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

TL;DR

This paper extends the Goldman-Millson theorem to complete filtered $A_ olinebreak_ ext{infinity}$-algebras over fields of arbitrary characteristic, establishing homotopy equivalences and characterizing homotopy groups in deformation theory.

Contribution

It provides the first $A_ olinebreak_ ext{infinity}$-analog of the Goldman-Millson theorem valid in arbitrary characteristic, with new homotopical and cohomological characterizations.

Findings

Nerve functor preserves homotopy equivalences for filtered quasi-isomorphisms.

Homotopy groups of the nerve are described via cohomology algebra and quasi-invertible elements.

In characteristic zero, the nerve is homotopy equivalent to the Maurer-Cartan set of the commutator $L_ olinebreak_ ext{infinity}$-algebra.

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.