a collection of our main results regarding Constructions 1 and 2, the first of which shows that they are in fact the same.

\[prop: twoconstrsaresame\] Assume that $A$ and $G$ satisfy the base assumptions of §\[subsec: theoryoftwist\]. Then for any normalised 2-cocycle $\mu$ one has an isomorphism of $k$-algebras $A^{G,\mu} \cong (AG_{\mu})^G$.

The equivalence of the two twisting constructions allows us to adopt the notation $A^{G,\mu}$ for a cocycle twist formed using *either* construction. Our main tool for proving the preservation of properties under twisting is provided by the following lemma; the twisted bimodule structures involved are defined prior to Proposition \[prop: fflat\].

\[lem: bimoddecomp\] Under the base assumptions of §\[subsec: theoryoftwist\], the twisted group algebra $AG_{\mu}$ has the following decomposition as an $(A^{G,\mu},A^{G,\mu})$-bimodule: $$AG_{\mu} \cong \bigoplus_{g \in G} {^{\text{id}}(A^{G,\mu})^{\phi_{g}}}.$$ Here $\phi_{g}$ is some $k$-algebra automorphism of $A^{G,\mu}$, with $\phi_e=\text{id}$. Each summand is free as a left and right $A^{G,\mu}$-module. Consequently, $AG_{\mu}$ is a faithfully flat extension of $A^{G,\mu}$ on both the left and the right. Furthermore, $AG_{\mu}$ is a faithfully flat extension of $A$ on both the left and the right too.

Figure \[fig: twists\] illustrates how we can use Proposition \[lem: bimoddecomp\] to adopt a new strategy when 