Witt groups of hermitian forms over ∗-fields

Orderings, valuations, and Hermitian forms over *-fields

“…The next two definitions were introduced by Craven [Cr4]; they give a version of semiordering which works well with * -orderings, in that virtually all results concerning semiorderings in the commutative theory can be extended to these. Again, their existence implies that D is infinite-dimensional over its center.…”

“…As in the commutative theory, it is also possible to make the definitions relative to a preordering (cf. [Cr4,§3]…”

“…As for orderings of commutative fields, one can topologize the sets of Baer orderings, * -orderings, c-orderings, etc. The actual work which has been done in this regard is only for Baer orderings [Cr2], [Le2], * -orderings [Cr2], [Cr4], [Cr5], * -semiorderings [Cr2] and weak * -orderings [Le2]. In all of these cases, the topology is generated by the subbasis consisting of all sets H(a), defined to be the set of all orderings in which a ∈ S (D) × is positive.…”

“…As in the commutative case, the spaces are compact, Hausdorff and totally disconnected. In the case of * -orderings, this topology can also be obtained as the Zariski topology on W R (D, * ), a ring of equivalence classes of hermitian forms defined in [Cr4] (see §4 for more information). In the sequel, we shall write X D for the topological space of all * -orderings on (D, * ).…”

“…We shall follow [L] and [Cr4,§4] in defining compatibility here. We say a * -valuation v and a Baer ordering P are compatible if the valuation ring A v is convex with respect to P , and are fully compatible if 0 < a ≤ b with respect to P implies…”