terms. It can be expressed as a more general twisted homogeneous coordinate ring, where the key object is a sheaf of orders over ${\ensuremath{\mathcal{O}}}_{E^{G}}$. In fact, the fat point modules over $A^{G,\mu}$ — which are also $B^{G,\mu}$-modules — can be explained by an equivalence of categories consequent to this work.

Other examples of twists are also studied, one of which relates to Rogalski and Zhang’s classification of AS-regular algebras of dimension 4 with three degree 1 generators and an additional ${\ensuremath{\mathbb{Z}}}^{2}$-grading [@rogalski2012regular]. Their work classified such algebras into 8 families up to isomorphism; we demonstrate that several of these families are related by cocycle twists.

I would first like to thank EPSRC for their generous grant enabling me to study for a PhD, since undoubtedly I could not have afforded to do so otherwise.

My supervisor, Professor Toby Stafford, has been a superb guide and mentor throughout the last few years; I have certainly come to understand why his former students hold him in such high esteem. Whenever I may have been feeling frustrated with maths, a meeting with Toby would invariably result in me leaving full of his contagious enthusiasm.

I would also like to acknowledge the fellow PhD 