Jan Van Geel

Quadratic forms isotropic over the function field of a conic

The behaviour of quadratic forms under the extension to the function field of a conic is studied. A recent result of Rost is utilized to explain some known results in an elementary way and also to obtain some new results.

Orders in quaternion algebras over global function fields having the cancellation property

Let A be a central simple algebra, with centre a global function field K defined over F,, the field with q elements. Let 44, be the set of all valuations on K. For every finite subset SC M, we can consider the ring R = R, = n,., s R,, where R, is the valuation ring associated to o; R is a Dedekind domain. The valuations u in M, S can be identified with the primes p of R, called the finite primes; the valuations u in S, the infinite primes, are identified with the prime ideals pL’ in R,. If 0 is an R-order in A, we denote with LF,(O) the set of isomorphism classes of locally free left O-ideals and with CL(O) the locally free class group of 0, i.e., the set of stable isomorphism classes of locally free O-ideals. There is a map

Diagonalisation of idempotent matrices

Let R be a ring, associative with unit element and with an involution * on it. An m x n matrix A is said to have a Moore-Penrose (MP) inverse with respect to the involution * iff there exists an n x m matrix X such that AXA = A; X,4X=X, (AX)* = AX; (XA)* = XA. The solution, if it exists, is unique and denoted by A +. Several authors considered the problem of characterising matrices over certain domains for which an MP-inverse exists; cf. [ 1, 3, 61. There results were generalised by Puystjens and Robinson; cf. [4]. The latter noted that if an m x IZ matrix A over a ring is of the form