DG polynomial algebras and their homological properties

Abstract

In this paper, we introduce and study differential graded (DG for short) polynomial algebras. In brief, a DG polynomial algebra A is a connected cochain DG algebra such that its underlying graded algebra A^# is a polynomial algebra $\mathbb{K}[x_1, x_2, \ldots, x_{n}]$ with |x_i| = 1, for any i ∈ {1, 2, ..., 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 Gorenstein DG algebras. Furthermore, it is proved that the DG polynomial algebra A is a Calabi-Yau DG algebra when its differential ∂_A ≠ 0 and the trivial DG polynomial algebra (A, 0) is Calabi-Yau if and only if n is an odd integer.