Faithfully Quadratic Rings

The aim of this paper is to lay the groundwork for a theory of quadratic forms over several significant, and quite extensive, classes of preordered rings. By “quadratic forms” we understand, here, diagonal quadratic forms with unit coefficients; and “ring” stands for commutative unitary ring where 2 in invertible. We achieve this by the use, in the ring context, of our abstract theory of quadratic forms, the theory of special groups, expounded in [11] . This is done as follows: fix a preordered ring (p-ring) 〈A, T 〉; then (1) The “intrinsic” theory of quadratic forms in 〈A, T 〉 is based on the following notion of T -isometry: Two n-dimensional quadratic forms φ = ∑n i=1 aiX 2 i ψ = ∑n i=1 biX 2 i , with ai, bi ∈ A× are T -isometric, φ ≈ T ψ, if there is a sequence φ0, φ1, . . . , φk of n-dimensional diagonal forms over A×, so that φ = φ0, ψ = φk and for every 1 ≤ i ≤ k, φi is either isometric to φi−1 in the sense that there is a matrix M ∈ GLn(A) such that φi = Mφi−1M , or there are t1, . . . , tn ∈ T× such that φi = 〈 t1x1, . . . , tnxn 〉 and φi−1 = 〈x1, . . . , xn 〉. In the classical case of the reduced theory of quadratic forms over fields, ≈ T is an algebraic equivalent of isometry defined in terms of total signature (Pfister’s local-global principle). (2) To 〈A, T 〉 we associate a structure GT (A) = A×/T× endowed with the product operation induced by A× and the binary isometry, ≡T , induced by ≈ T restricted to binary forms (plus −1 = −1/T× as a distinguished element) 〈GT (A),≡T ,−1 〉 is not quite a special group in the sense of [11], Definition 1.2, p.2, but it satisfies some of its axioms, constituting a proto-special group as in Definition 6.3 of [17]. Though related, the internal and the formal approach (via GT (A)) are far from identical. (3) In section 2 (item 2.1) we introduce three axioms in terms of the value representation relation D v T on 〈A, T 〉 defined by: for a, b1, . . . , bn ∈ A× a ∈ D v T (b1, . . . , bn) ⇔ ∃ t1, . . . , tn ∈ T such that a = ∑n i=1 tibi. When satisfied in 〈A, T 〉, these axioms are sufficient − and under reasonable assumptions, also necessary − to guarantee identity between the intrinsic and formal approaches and, in fact, more: (3.i) The structure GT (A) is a special group. (3.ii) T -isometry and value representation in 〈A, T 〉 are faithfully coded by the corresponding formal notions in GT (A). Example 9.7 shows that (3.ii) is not an automatic consequence of (3.i). We call T -faithfully quadratic any p-ring 〈A, T 〉 verifying these axioms. In fact, this setting as well as the consequences (3.i) and (3.ii) apply, more generally, to forms with entries in certain subgroups of A×, called T subgroups, and also to the case where T = A; in this latter case, T -isometry is just matrix isometry. Briefly (and approximately) stated, the axioms for T -quadratically faithful rings express the properties of value representation known as transversality (for 2-forms), inductive characterization and Witt-cancellation (for 1-forms). It is worth noticing that T -isometry as defined above, coincides with isometry in the sense of equality of signs at all natural T -signatures existing in 〈A, T 〉 (cf. Definition 1.19). For rings A verifying the 2-transversality axiom for T = A, the mod 2 K-theory of A obtained from that in [23] coincides with the K-theory of the special group G(A) as defined in [12] and [16].