Deformations of algebraic schemes via Reedy-Palamodov cofibrant resolutions

TL;DR

This paper develops a new approach to study deformations of algebraic schemes over differential graded Artin algebras using Reedy-Palamodov resolutions, linking deformation theory with DG-Lie algebras.

Contribution

It introduces a novel method employing Reedy-Palamodov cofibrant resolutions to analyze scheme deformations via DG-Lie algebras, extending existing deformation frameworks.

Findings

Describes the DG-Lie algebra controlling deformations of schemes.

Establishes a link between local Tate-Quillen resolutions and deformation theory.

Provides a new algebraic analog of Palamodov's resolvent for complex spaces.

Abstract

Let $X$ be a Noetherian separated and finite dimensional scheme over a field $\mathbb{K}$ of characteristic zero. The goal of this paper is to study deformations of $X$ over a differential graded local Artin $\mathbb{K}$-algebra by using local Tate-Quillen resolutions, i.e., the algebraic analog of the Palamodov's resolvent of a complex space. The above goal is achieved by describing the DG-Lie algebra controlling deformation theory of a diagram of differential graded commutative algebras, indexed by a direct Reedy category.