G-algebras, Twistings, and Equivalences of Graded Categories

Given Z-graded rings A and B, we ask when the graded module categories gr-A and gr-B are equivalent. Using Z-algebras, we relate the Morita-type results of Ahn-Marki and del Rio to the twisting systems introduced by Zhang, and prove, for example: Theorem If A and B are Z-graded rings, then: (1) A is isomorphic to a Zhang twist of B if and only if the Z-algebras A = circle plus(i, j is an element of Z) A(j-i) and B = circle plus(i, j is an element of Z) Bj-i are isomorphic. (2) If A and B are connected graded with A1 not equal 0, then gr-A similar or equal to gr-B if and only if A and B are isomorphic. This simplifies and extends Zhang's results.