Linkage of Pfister forms over semi-global fields

Abstract We study linkage of $$(d+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -fold quadratic Pfister forms over function fields in one variable over a henselian valued field of 2-cohomological dimension d . Specifically, we characterise this property in terms of linkage of quadratic Pfister forms over function fields over the residue field of the henselian valued field; in full generality in characteristic different from 2, and for most complete discretely valued fields in characteristic 2. As an application, we obtain a proof that $$(d+2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -fold quadratic Pfister forms over function fields in one variable over a d -dimensional higher local field are linked.