\mathcal{G}(1,\gamma)^{G,\mu} \cong
\mathcal{G}(1,\overline{\gamma}).$$

### Contents {#subsec: contents}

We now give a brief description of the contents of this thesis. The remainder of the current chapter consists of §\[sec: notation\], where our basic conventions and notation are given.

In Chapter \[chap: background\] we describe some of the background material that is needed. We begin with §\[sec: noncommgeom\], which consists of an introduction to noncommutative algebraic geometry emphasising the aspects that are most pertinent to cocycle twists. This is followed by §\[sec: twistsofalgebras\], in which we study several related concepts involving the twisting of algebras; cocycle twists, Zhang twists and crossed products are all defined. The chapter ends with the short section §\[sec: goldietheory\], which describes the Goldie theory that we will use.

The next chapter is the first containing original work. In §\[sec: construction\] two twisting constructions are defined and then subsequently shown to be the same in Proposition \[prop: twoconstrequal\]. The preservation of properties under such twists is then treated in §\[sec: preservation\]. A result of particular note is Corollary \[cor: asreg\], where the property of being AS-regular is shown to be preserved under cocycle twists. In §\[sec: modules\] we study the relationship between point modules over an algebra and 