Linkage of quadratic Pfister forms

Wed, 18/01/2017 - 10:30
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Quadratic Pfister forms are a special class of quadratic forms that arise naturally as norm forms of composition algebras.  The Witt group I_q F of quadratic forms (modulo hyperbolic forms) over a field F is a module over the Witt ring of bilinear forms.  This gives a most important filtration { I_q^n F }.  The n-fold Pfister forms, which are tensor products of n Pfister forms, generate I_q^n F.

 

We call a set of quadratic n-fold Pfister forms linked if they all share a common (n-1)-fold Pfister factor.  Since we wish to develop a characteristic-free theory, we need to consider the characteristic 2 case, where one has to distinguish between right linkage and left linkage.

 

To a certain type of set of s n-fold Pfister forms, we associate an invariant in I_q^{n+1} F which lives in I_q^{n+s-1} F when the set is linked.  We study the properties of this invariant and compute necessary conditions for a set to be linked.

 

We also consider the related notion of linkage for quaternion algebras via linkage of the associated norm forms.