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Orders, in non-Eichler (R)-algebras over global function fields, having the cancellation property. (English) Zbl 0666.16003

We study R-orders \(\theta\) in central simple algebras A over global function fields K. We say that \(\theta\) has the cancellation property iff all stably free \(\theta\)-ideals are free. If Eichler’s condition (with respect to R) is satisfied for the algebra A then all R-orders in A have the cancellation property. However, very few non Eichler (R) algebras contain R-orders with the cancellation property. A first necessary condition is that the centre K of A must be a rational function field.
In this paper we obtain a complete characterization of the non-Eichler (R) algebras which contain R-orders with the cancellation property. This characterization, although obtained by analytic-numbertheoretic methods, is of geometrical nature. We also determine all hereditary R-orders in these algebras which have the cancellation property.
Reviewer: M.Denert

MSC:

16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16P10 Finite rings and finite-dimensional associative algebras
11R58 Arithmetic theory of algebraic function fields
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