Artin-Schelter regularity of twisted tensor products

Nakayama automorphisms of twisted tensor products

“…The usual tensor product is a special kind of twisted tensor products when the graded twisting map τ is chosen to be the filliping map, and we still use the notation ⊗ without superscript for it. The name "R-smash product" is used in [3,12,18], which is the same to twisted tensor product, we prefer the later one here.…”

“…Let A = k x 1 , · · · , x n /(G A ) and B = k y 1 , · · · , y m /(G B ) be two connected graded algebras. Since τ is just a graded linear map, we require τ being determined by its action on the generating set of B ⊗ A as in [18]. That is, we ask the linear map τ has form of…”

“…Sketch of proof. We describe the construction of complex F φ briefly, which may be useful in Example 2.15 and Example 5.3, and see [18] for the completed proof in a more general situation.…”

“…Roughly speaking, twisted tensor product A ⊗ τ B of algebras A and B is an associative algebra, which is A ⊗ B as vector spaces with the multiplication defined by a twisting map τ : B ⊗ A → A ⊗ B. The notion also has been called R-smash product in several recent papers ( [3,12,18]).…”

“…Let A and B be two connected graded algebras with B having a pure resolution, and let A ⊗ τ B be a twisted tensor product for some graded twisting map τ = (σ, 0). We construct a minimal free resolution of trivial module of A ⊗ τ B (see Proposition 2.10), by the complex obtained in [18,Theorem 0.2]. This free resolution helps us bridge the relationships among the Ext-algebras E(A), E(B) and E(A ⊗ τ B).…”