A comparison of dg algebra resolutions with prime residual characteristic

TL;DR

This paper compares dg algebra resolutions over local rings with residue field characteristic p, revealing structural relationships and properties related to minimal models, homotopy Lie algebras, and rigidity of deviations.

Contribution

It provides a detailed description of minimal models in terms of acyclic closures and establishes new properties of homotopy Lie algebras and deviations in this setting.

Findings

The homotopy Lie algebra is abelian for maps with residual characteristic p.

Acyclic closures are quotients of minimal models in this context.

Deviations exhibit rigidity properties indicating (quasi-)complete intersection structures.

Abstract

In this article we fix a prime integer $p$ and compare certain dg algebra resolutions over a local ring whose residue field has characteristic $p$. Namely, we show that given a closed surjective map between such algebras there is a precise description for the minimal model in terms of the acyclic closure, and that the latter is a quotient of the former. A first application is that the homotopy Lie algebra of a closed surjective map with residual characteristic $p$ is abelian. We also use these calculations to show deviations enjoy rigidity properties which detect the (quasi-)complete intersection property.