it enables the use of faithful flatness arguments. Our main results are stated later in this introduction, namely in Proposition \[prop: twoconstrsaresame\] and Theorems \[thm: thebigone\], \[thm: sklyanin\] and \[thm: geometricthcrdesc\].

The motivation for our work was an example of Odesskii, appearing in the survey paper [@odesskii2002elliptic] on *elliptic algebras*. Although the example as we state it does not use a specific type of algebra, the original example in [@odesskii2002elliptic] used a *4-dimensional Sklyanin algebra*. In Chapter \[chap: sklyanin\] we will study cocycle twists of such Sklyanin algebras, although they will also be discussed briefly in §\[subsec: examples\] (see Definition \[def: sklyanin relations\] for a presentation by generators and relations).

In order to state Odesskii’s example, we need to make the following assumptions. Let $k= {\ensuremath{\mathbb{C}}}$ and consider an associative $k$-algebra $A$ that is generated as a $k$-algebra by $x_0,x_1,x_2$ and $x_3$. We will assume that the Klein four-group $G=(C_2)^2= \langle g_1, g_2 \rangle$ acts on $A$ by algebra automorphisms, i.e. $G \rightarrow
\text{Aut}_{\text{alg}}(A)$. Denoting the action of $g \in G$ by the superscript $-^g$, the action is defined on generators by $$\label{eq: odesskiiaction}
x_i^{g_{1}} = x_{i+2},\;\;\; x_i^{g_{2}} = (-1)^i x_{i},$$ where $i \in \{0,1,2,3\}$ and indices are taken modulo 