Smooth Affine PI Algebras

Abstract

We wish to discuss a certain class of algebras, finitely generated as algebras over an algebraically closed field k, and satisfying the identities of n by n matrices. The free algebra on m generators satisfying these identities we denote by k it is usually called the algebra of generic matrices. It has many good properties which distinguish it from one of its random homomorphic images. Of course it has the freeness property. Another property is that all points of its Spec are accessible from Spec n by curves. In the commutative case, for n equal to 1, a homomorphic image is called smooth if locally it spectrum looks like k 8 , that is to say like the Spec of the polynomial ring in 5 variables. More precisely, upon completion at a point it should become power series in 8 variables. In the noncommutative case for n = 2, the dimension of the 2X2 algebra of generic matrices is 5. Thus it is impossible to say that a noncommutative homomorphic image of say dimension 1 or 2 should look like the Spec of the ‘polynomials’ in 1 or 2 variables.