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Twisted graded algebras and equivalences of graded categories. (English) Zbl 0852.16005
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.

MSC:
16D90 Module categories in associative algebras
16W50 Graded rings and modules (associative rings and algebras)
16S80 Deformations of associative rings
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