Nakayama automorphism is used to study group actions and Hopf algebra actions on Artin-Schelter regular algebras of global dimension three.

As the definition of free class of differential modules over a commutative ring in [ 1 Avramov , L. L. , Buchweitz , R.-O. , Iyengar , S. ( 2007 ). Class and rank of differential modules . Invent. Math. 169 : 1 – 35 . [Crossref], [Web of Science ®] [Google Scholar] ], we define DG free class for semifree DG modules over an Adams connected DG algebra A. For any DG A-modules M, we define its cone length as the least DG free classes of all semifree resolutions of M. The cone length of a DG A-module plays a similar role as projective dimension of a module over a ring does in homological ring theory. The left (resp., right) global dimension of an Adams connected DG algebra A is defined as the supremum of the set of cone lengths of all DG A-modules (resp., A ^op -modules). It is proved that the definition is a generalization of that of graded algebras. Some relations between the global dimension of H(A) and the left (resp. right) global dimension of A are discovered. When A is homologically smooth, we prove that the left (right) global dimension of A is finite and the dimension of D(A) and D ^c (A) are not bigger than the DG free class of a minimal semifree resolution X of the DG A ^e -module A.

Let $A$ be a connected cochain differential graded (DG, for short) algebra. This note shows that $A$ is a $0$ -Calabi-Yau DG algebra if and only if $A$ is a Koszul DG algebra and $\mathrm {Tor}_A^0(\Bbbk _A,{}_A\Bbbk )$ is a symmetric coalgebra. Let $V$ be a finite dimensional vector space and $w$ a potential in $T(V)$ . Then the minimal subcoalgebra of $T(V)$ containing $w$ is a symmetric coalgebra, which implies that a locally finite connected cochain DG algebra is $0$ -CY if and only if it is defined by a potential $w$ .

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.