Rings of Low Rank with a Standard Involution

We consider the problem of classifying (possibly noncommutative) R-algebras of low rank over an arbitrary base ring R. We first classify algebras by their degree, and we relate the class of algebras of degree 2 to algebras with a standard involution. We then investigate a class of exceptional rings of degree 2 which occur in every rank n ≥ 1 and show that they essentially characterize all algebras of degree 2 and rank 3. Let R be a commutative ring (with 1). Let B be an algebra over R, an associative ring with 1 equipped with an embedding R ↪→ B of rings (mapping 1 ∈ R to 1 ∈ B) whose image lies in the center of B; we identify R with its image in B. Assume further that B is a finitely generated, faithfully projective R-module of constant rank. The problem of classifying algebras B of low rank has an extensive history. The identification of quadratic rings over Z by their discriminants is classical and goes back as far as Gauss. Commutative rings of rank at most 5 over R = Z have been classified by Bhargava [1], building on work of others; this beautiful work has rekindled interest in the subject and has already seen many applications. Progress on generalizing these results to arbitrary commutative base rings R (or even arbitrary base schemes) has been made by Wood [11]. A natural question in this vein is to consider noncommutative algebras of low rank, and in this article we treat algebras of rank at most 3. The category of R-algebras (with morphisms given by isomorphisms) has a natural decomposition by degree. The degree of an R-algebra B, denoted degR(B), is the smallest positive integer n such that every x ∈ B satisfies a monic polynomial of degree n with coefficients in R. Any quadratic algebra B, i.e. an algebra of rank 2, is necessarily commutative (see Lemma 2.9) and has degree 2. Moreover, a quadratic algebra has a unique R-linear (anti)involution : B → B such that xx ∈ R for all x ∈ B, which we call a standard involution. The situation is much more complicated in higher rank. In particular, the degree of B does not behave well with respect to base extension (Example 1.20). We define the geometric degree of B to be the maximum of degS(B⊗R Date: March 15, 2013. 1991 Mathematics Subject Classification. 16G30, 11E20, 16W10.