DG polynomial algebras and their homological properties

TL;DR

This paper introduces and analyzes differential graded polynomial algebras, exploring their structures, automorphisms, and homological properties, including conditions for being Calabi-Yau, thus advancing understanding of their algebraic and geometric features.

Contribution

It classifies differential structures on DG polynomial algebras, computes automorphism groups, and establishes their homological and Calabi-Yau properties, providing new insights into their algebraic nature.

Findings

All possible differential structures are described.

DG automorphism groups are computed.

DG polynomial algebras are shown to be homologically smooth and Gorenstein.

Abstract

In this paper, we introduce and study differential graded (DG for short) polynomial algebras. In brief, a DG polynomial algebra $\mathcal{A}$ is a connected cochain DG algebra such that its underlying graded algebra $\mathcal{A}^{\#}$ is a polynomial algebra $\mathbb{k}[x_1,x_2,\cdots, x_n]$ with $|x_i|=1$, for any $i\in \{1,2,\cdots, n\}$. We describe all possible differential structures on DG polynomial algebras; compute their DG automorphism groups; study their isomorphism problems; and show that they are all homologically smooth and Gorestein DG algebras. Furthermore, it is proved that the DG polynomial algebra $\mathcal{A}$ is a Calabi-Yau DG algebra when its differential $\partial_{\mathcal{A}}\neq 0$ and the trivial DG polynomial algebra $(\mathcal{A}, 0)$ is Calabi-Yau if and only if $n$ is an odd integer.