Twisted Graded Algebras and Equivalences of Graded Categories

Abstract

Let A = ⊕ n ⩾0 An be a connected graded k-algebra and let Gr-A denote the category of graded right A-modules with morphisms being graded homomorphisms of degree 0. If {τn ∣ n ∈ Z} is a set of graded k-linear bijections of degree 0 from A to itself satisfying τ n ( y τ m ( z ) ) = τ n ( y ) τ n + m ( z ) for all l, m, n ∈ Z and all y ∈ Am, z ∈ Al, we define a new graded associative multiplication * on the underlying graded k-vector space ⊕n⩾0 An by y * z = yτm(z) for all y ∈ Am, z ∈ Al. 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. Then the categories Gr-A and 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: Artin-Schelter Gorenstein (or Artin-Schelter regular); Auslander Gorenstein (or Auslander regular) and Cohen-Macaulay. Some of these results are also generalized to certain semigroup-graded algebras.