Noncommutative algebras of dimension three over integral schemes

Abstract:

In this article we describe the algebraic data which is equivalent to giving an associative, noncommutative algebra ${\mathcal {O}_X}$ over an integral $k$-scheme $Y$ (where $k$ is an algebraically closed field of characteristic $\ne 3$), which is locally free of rank $3$. The description allows us to conclude that, essentially, all such are locally upper triangular $2 \times 2$ matrices, with degenerations of a restricted form allowed.