Characterizing Quaternion Rings over an Arbitrary Base

We consider the class of algebras of rank 4 equipped with a standard involution over an arbitrary base ring. In particular, we characterize quaternion rings, those algebras defined by the construction of the even Clifford algebra. A quaternion algebra is a central simple algebra of dimension 4 over a field F . Generalizations of the notion of quaternion algebra to other commutative base rings R have been considered by Kanzaki [10], Hahn [7], Knus [13], and most recently Gross and Lucianovic [6]. In this article, we pursue these generalizations further. Let R be a commutative Noetherian 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 this image in B. Assume further that B is a finitely generated, projective R-module of constant rank. A standard involution on B is an R-linear (anti-)involution : B → B such that xx ∈ R for all x ∈ B. Given an algebra B with a standard involution, we define the reduced trace trd : B → R by x 7→ x + x and the reduced norm nrd : B → R by x 7→ xx. Then every element x ∈ B satisfies the polynomial μ(x;T ) = T 2 − trd(x)T + nrd(x). A free quaternion ring B over a PID or local ring R is an R-algebra of rank 4 with a standard involution such that the characteristic polynomial χ(x;T ) of left multiplication by x on B is equal to μ(x;T ) = (T 2 − trd(x)T + nrd(x)). The result of Gross and Lucianovic is as follows. Proposition ([6, Proposition 4.1]). Let R be a PID or local ring. Then there is a bijection between the space of ternary quadratic forms over R under a twisted action of GL3(R) and isomorphism classes of free quaternion rings over R. In this correspondence, one associates to a ternary quadratic form q the even Clifford algebra C(q). (Here, the usual action of GL3(R) on quadratic forms is twisted by the determinant; see Proposition 3.1 for more details.) In this article, we generalize this result and treat an arbitrary commutative base ring R. A ternary quadratic module is a triple (M, I, q) whereM is a projectiveR-module of rank 3, I is an invertible R-module (projective of rank 1), and q : M → I is a quadratic map. To a ternary quadratic module (M, I, q), one can associate the even Clifford algebra C(M, I, q), an R-algebra of rank 4 with standard involution. A quaternion ring is an R-algebra B such that there exists a ternary quadratic module (M, I, q) with B ∼= C(M, I, q). Date: June 12, 2010. Subject classification: 11R52, 11E20, 11E76.