# zbMATH — the first resource for mathematics

Let $$A=k\oplus_{n>0}A_n$$ be a connected graded $$k$$-algebra and let $$\text{Gr-}A$$ denote the category of graded right $$A$$-modules with morphisms being graded homomorphisms of degree 0. If $$\{\tau_n\mid n\in\mathbb{Z}\}$$ is a set of graded $$k$$-linear bijections of degree 0 from $$A$$ to itself satisfying $$\tau_n(y\tau_m(z))=\tau_n(y)\tau_{n+m}(z)$$ for all $$l,m,n\in\mathbb{Z}$$ and all $$y\in A_m$$, $$z\in A_l$$, we define a new graded associative multiplication $$*$$ on the underlying graded $$k$$-vector space $$\oplus_{n\geq 0}A_n$$ by $$y*z=y\tau_m(z)$$ for all $$y\in A_m$$, $$z\in A_l$$. The graded algebra with the new multiplication $$*$$ is called a twisted algebra of $$A$$.
Theorem. Let $$A$$ and $$B$$ be two connected graded algebras generated in degree 1. The categories $$\text{Gr-}A$$ and $$\text{Gr-}B$$ are equivalent if and only if $$A$$ is isomorphic to a twisted algebra of $$B$$. If algebras are noetherian, then Gelfand-Kirillov dimension, global dimension, injective dimension, Krull dimension, and uniform dimension are preserved under twisting. Moreover, we prove the following Theorem: The following properties are preserved under twisting for connected graded noetherian algebras: (a) Artin-Schelter Gorenstein (or Artin-Schelter regular). (b) Auslander Gorenstein (or Auslander regular) and Cohen-Macaulay. Some of these results are also generalized to certain semigroup-graded algebras.