Let $A$ be a finite dimensional central simple algebra(CSA) over $k$. I want to view the definition of CSA geometrically.
So If I treat $A$ just as a algebra it means that $A$ is a $k-$ vector space with with distinguished element $1 \in A$ and there is multiplication map $A \otimes_{k} A \rightarrow k$. Suppose if we fix basis say $\lbrace u_{i} \rbrace$, then $u_{i}u_{j} = \sum c_{ijk} u_{k}$, where $c_{ijk} \in k$. If we assume $A$ is associative then it just says that there are relations $\mathcal{R}$ between $c_{ijk}$, so geometrically it gives a closed subscheme $Spec(k[c_{ijk}]/\mathcal{R})$ inside $Spec(k[c_{ijk}])$. What is the geometric interpretation of "simple algebra" condition and "central algebra" condition on that subscheme. Can someone help me in understanding this definition geometrically?