G-algebras, Twistings, and Equivalences of Graded Categories

Abstract

Given ${\mathbb Z}$-graded rings A and B, we ask when the graded module categories gr-A and gr-B are equivalent. Using ${\mathbb Z}$-algebras, we relate the Morita-type results of Áhn-Márki and del Río to the twisting systems introduced by Zhang, and prove, for example: Theorem If A and B are ${\mathbb Z}$ -graded rings, then: (1) A is isomorphic to a Zhang twist of B if and only if the ${\mathbb Z}$ -algebras $\overline{A} = \bigoplus_{i,j \in {\mathbb Z}} A_{j-i}$ and $\overline{B} = \bigoplus_{i,j \in {\mathbb Z}} B_{j-i}$ are isomorphic. (2) If A and B are connected graded with A ₁ ≠ 0, then gr-A ≃ gr- B if and only if $\overline{A}$ and $\overline{B}$ are isomorphic. This simplifies and extends Zhang’s results.