Connected simplicial algebras are Andr\'e-Quillen complete

TL;DR

This paper proves that all connected simplicial commutative algebras over a field are complete with respect to Andre9-Quillen homology, establishing convergence of the unstable Adams spectral sequence for these algebras.

Contribution

It introduces a modified Adams tower construction for simplicial algebras and proves their Andre9-Quillen completeness, linking it to cosimplicial resolutions.

Findings

Connected simplicial algebras are Andre9-Quillen complete.

The modified Adams tower converges for these algebras.

Establishes convergence of the unstable Adams spectral sequence.

Abstract

We modify a classical construction of Bousfield and Kan to define the Adams tower of a simplicial nonunital commutative algebra over a field k. We relate this construction to Radulescu-Banu's cosimplicial resolution, and prove that all connected simplicial algebras are complete with respect to Andr\'e-Quillen homology. This is a convergence result for the unstable Adams spectral sequence for commutative algebras over k.