chapter  7
32 Pages

Superidentities and Kemer’s Solution for Non-Affine Algebras

Exercises for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

As indicated in §1.3.3, “super” means “Z2-graded.” In this chapter we develop a theory of superidentities of the superalgebra A = A0 ⊕ A1, and present Kemer’s “Grassmann trick,” (Theorem 7.2.1) that any PI-algebra in characteristic 0 corresponds naturally to a PI-superalgebra that is PI2-equivalent to a suitable affine superalgebra. This assertion is false in nonzero characteristic. (Indeed, if true, it would verify Specht’s conjecture, but we shall present counterexamples in Chapter 9.)

Following Kemer, we then backtrack and modify the results of Chapter 6 for superalgebras. As in Chapter 6, the main step is what we call Kemer’s Super-PI Representability Theorem:

Theorem 7.0.1. Every affine ungraded-PI superalgebra is PI2-equivalent to a finite dimensional superalgebra.