Isomorphism problem and homological properties of DG-free algebras

Abstract

A differential graded (DG) free algebra $\mathcal{A}$ is a connected cochain DG algebra such that its underlying graded algebra is $${\mathcal{A}}^{\#}=\mathbb{k}〈x_1,x_2,\dots ,x_n〉, \text{with} \left|x_i\right|=1, \forall i\in \{1,2,\dots ,n\}.$$

We prove that the differential structures on DG free algebras are in one to one correspondence with the set of crisscross ordered n-tuples of n × n matrices. We also give a criterion to judge whether two DG free algebras are isomorphic. As an application, we consider the case of n = 2. Based on the isomorphism classification, we compute the cohomology graded algebras of non-trivial DG free algebras with two generators, and show that all those non-trivial DG free algebras are Koszul and Calabi-Yau.

Keywords:

• Calabi-Yau
• DG algebra
• free algebra
• homologically smooth
• isomorphism problem
• Koszul

2010 Mathematics Subject Classification:

• Primary 16E45
• 16E65
• 16W20
• 16W50

Additional information

Funding

X. M. is supported by NSFC (Grant no. 11871326), the China Postdoctoral Science Foundation (Grant nos. 20090450066 and 201003244), the Key Disciplines of Shanghai Municipality (Grant no. S30104) and the Innovation Program of Shanghai Municipal Education Commission (Grant no. 12YZ031).