trying to show that properties are preserved under cocycle twists. Instead of the direct route from $A$ to $A^{G,\mu}$, we can attempt to push and pull properties along the faithfully flat extensions via the twisted group algebra $AG_{\mu}$.

(1.7,1.7) – (4,2.9); (7.2,1.8) – (5,2.9); (1.7,1.3) .. controls (3.45,0.8) and (5.45,0.8) .. (7.2,1.3);

at (4.5,3) [$AG_{\mu}$]{}; at (1.5,1.5) [$A$]{}; at (7.5,1.5) [$A^{G,\mu}\cong(AG_{\mu})^G$]{};

at (2.5,2.5) ; at (6.5,2.5) ; at (4.5,0.7) ;

Using the new technique highlighted by Figure \[fig: twists\] allows us to prove the following main theorem concerning the preservation of properties under cocycle twists. It brings together Lemma \[lem: hilbseries\], Corollaries \[cor: uninoeth\] and \[cor: asreg\], and Propositions \[prop: koszul\] and \[prop: cohenmac\]. Connected graded algebras and Hilbert series are defined in Definition \[def: conngraded\] and Definition \[def: hilbseries\] respectively. Definitions of the remaining properties involved are deferred until their appearance in §\[sec: preservation\].

\[thm: thebigone\] Further to the base assumptions of §\[subsec: theoryoftwist\], assume that $A$ is connected graded and the action of $G$ on $A$ by $k$-algebra automorphisms preserves this grading. Then for any normalised 2-cocycle $\mu$, $A$ has one of the following properties if and only if $A^{G,\mu}$ does:

-   it is strongly noetherian;

-   it 