Characterizing Quaternion Rings

We consider the problem of classifying noncommutative R-algebras of low rank over an arbitrary base ring R. We unify and generalize the many definitions of quaternion ring, and give several necessary and sufficient conditions which characterize them. Let R be a commutative, connected Noetherian ring (with 1). Let B be an algebra over R, an associative ring with 1 equipped with an embedding R →֒ B of rings whose image lies in the center of B; we identify R with its image R · 1 ⊂ B. Assume further that B is a finitely generated, projective R-module. The problem of classifying algebras of small rank has an extensive history. The identification of quadratic rings over Z by their discriminants is classical. Commutative rings of rank at most 5 over R = Z have been classified by Bhargava [3], building work of many others; his 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 [23]. 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 4. 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. Any quadratic algebra B, i.e. an algebra of rank rk(B) = 2, is necessarily commutative (see Lemma 2.7) 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 1.13). We define the geometric degree of B to be the maximum of degS(B ⊗R S) with R → S a homomorphism of (commutative) rings. We prove the following result (Corollary 2.15). Theorem A. Let B be an R-algebra and suppose there exists a ∈ R such that a(a− 1) is a nonzerodivisor. Then the following are equivalent. (i) B has degree 2; (ii) B has geometric degree 2; (iii) B 6= R has a standard involution. Note that if 2 6= 0 ∈ R, then one can take a = −1 in the above theorem. In view of the above result, it is natural then to consider the class of R-algebras with a standard involution. Classically, when R = F is a field and B is a noncommutative division ring, we know that B is a quaternion algebra over F , a central simple algebra of rank 4. Extensions of this fundamental result to other base rings have been considered Date: July 13, 2009.