Deformations of $\mathbb{E}_{\infty}$-groups of units and logarithmic derivatives of $\mathbb{E}_{\infty}$-rings

TL;DR

This paper extends classical deformation results of units in commutative rings to $ ext{E}_ ext{infty}$-ring spectra and introduces a spectral map generalizing the logarithmic derivative via module derivations.

Contribution

It generalizes deformation theory of units to $ ext{E}_ ext{infty}$-rings and constructs a spectral map that broadens the concept of the logarithmic derivative.

Findings

Deformation results are extended to $ ext{E}_ ext{infty}$-ring spectra.

A new spectral map generalizing the logarithmic derivative is constructed.

The work bridges classical algebraic concepts with modern homotopy-theoretic frameworks.

Abstract

We extend a classical fact about deformations of groups of units of commutative rings to $\mathbb{E}_{\infty}$-ring spectra, and we use this result to provide a map of spectra generalizing the ordinary logarithmic derivative induced by an $R$-module derivation.