Riemannian Geometry on Associative Varieties

Abstract

We prove that the classical algebraic varieties over algebraically closed fields can be defined over arbitrary fields $k.$ Then we prove that for associative algebras $A$, there exist local representing objects $A_M$ for simple modules $M.$ Replacing the localization in maximal ideals in the commutative situation with the local representations in simple modules in the associative, we define an associative generalization of varieties. Now we realize that replacing $\mathbb R[x_1,\dots,x_n]=\mathbb R[n]$ with $C^\infty(\mathbb R^n),$ we can do differential geometry for associative $\mathbb R[n]$-algebras. This says that we can define a Riemannian geometry on associative varieties. This gives us the definition of connections and algebraic geodesic curves, introducing real geometry into associative algebras.