# On deformations of diagrams of commutative algebras

## TL;DR

This paper investigates classical deformation problems of diagrams of commutative algebras over a field of characteristic zero, identifying homotopy classes of controlling DG-Lie algebras using model structures.

## Contribution

It characterizes the homotopy types of DG-Lie algebras controlling these deformations via projective and Reedy model structures, providing new insights into their classification.

## Key findings

- Identifies homotopy classes of DG-Lie algebras controlling deformations.
- Uses projective and Reedy model structures for classification.
- Provides an elementary introduction to relevant model structures.

## Abstract

In this paper we study classical deformations of diagrams of commutative algebras over a field of characteristic 0. In particular we determine several homotopy classes of DG-Lie algebras, each one of them controlling this above deformation problem: the first homotopy type is described in terms of the projective model structure on the category of diagrams of differential graded algebras, the others in terms of the Reedy model structure on truncated Bousfield-Kan approximations. The first half of the paper contains an elementary introduction to the projective model structure on the category of commutative differential graded algebras, while the second half is devoted to the main results.