Witt kernels — a survey Detlev Hoffmann detlev.hoffmann@math.tu-dortmund.de TU Dortmund Emory, 19 May 2011 Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 1 / 25
The general question We often study algebraic objects defined over fields. A natural and important question then becomes: how do these objects behave under field extensions. Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 2 / 25
The general question We often study algebraic objects defined over fields. A natural and important question then becomes: how do these objects behave under field extensions. Example Consider central simple algebras over a field F and a field extension K / F . Which division algebras over F will have zero divisors over K ? How much will the index go down? Keyword: index reduction formulas. Which algebras over F become split over K ? Keyword: relative Brauer group Br( K / F ). Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 2 / 25
Our question We are interested in quadratic resp. symmetric bilinear forms over F (always assumed to be nondegenerate). The analogous problems are: Question Which anisotropic forms over F become isotropic over K? How much will the Witt index go up? Which anisotropic forms becomes hyperbolic/metabolic over K? I.e. Determine the Witt kernel. Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 3 / 25
Our question We are interested in quadratic resp. symmetric bilinear forms over F (always assumed to be nondegenerate). The analogous problems are: Question Which anisotropic forms over F become isotropic over K? How much will the Witt index go up? Which anisotropic forms becomes hyperbolic/metabolic over K? I.e. Determine the Witt kernel. Fo this talk, we mainly look a the second question. We need to distinguish three cases: (Q1) Quadratic forms in characteristic not 2. (Q2) Quadratic forms in characteristic 2. (B2) Symmetric bilinear forms in characteristic 2. Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 3 / 25
Notations and facts B2 Q1 Q2 “normal form” of forms φ 2-dim isotropic forms hyperb./metab. forms Witt decomposition Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 4 / 25
Notations and facts B2 Q1 Q2 φ diagonalizable “normal form” φ ∼ = � a 1 , . . . , a n � of forms φ if φ �∼ = hyperbolic always (see below) 2-dim isotropic forms hyperb./metab. forms Witt decomposition Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 4 / 25
Notations and facts B2 Q1 Q2 φ ∼ φ diagonalizable = “normal form” φ ∼ = � a 1 , . . . , a n � [ a 1 , b 1 ] ⊥ . . . ⊥ [ a n , b n ] of forms φ if φ �∼ = hyperbolic where always [ a , b ] = ax 2 + xy + by 2 (see below) 2-dim isotropic forms hyperb./metab. forms Witt decomposition Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 4 / 25
Notations and facts B2 Q1 Q2 φ ∼ φ diagonalizable = “normal form” φ ∼ = � a 1 , . . . , a n � [ a 1 , b 1 ] ⊥ . . . ⊥ [ a n , b n ] of forms φ if φ �∼ = hyperbolic where always [ a , b ] = ax 2 + xy + by 2 (see below) 2-dim hyperbolic plane H = xy isotropic forms � 1 , − 1 � [0 , 0] hyperb./metab. forms Witt decomposition Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 4 / 25
Notations and facts B2 Q1 Q2 φ ∼ φ diagonalizable = “normal form” φ ∼ = � a 1 , . . . , a n � [ a 1 , b 1 ] ⊥ . . . ⊥ [ a n , b n ] of forms φ if φ �∼ = hyperbolic where always [ a , b ] = ax 2 + xy + by 2 (see below) 2-dim metabolic plane hyperbolic plane H = xy M a = ( 0 1 isotropic 1 a ) M a ∼ forms = M b � 1 , − 1 � [0 , 0] ⇔ aF 2 = bF 2 M 0 = H hyperb./metab. forms Witt decomposition Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 4 / 25
Notations and facts B2 Q1 Q2 φ ∼ φ diagonalizable = “normal form” φ ∼ = � a 1 , . . . , a n � [ a 1 , b 1 ] ⊥ . . . ⊥ [ a n , b n ] of forms φ if φ �∼ = hyperbolic where always [ a , b ] = ax 2 + xy + by 2 (see below) 2-dim metabolic plane hyperbolic plane H = xy M a = ( 0 1 isotropic 1 a ) M a ∼ forms = M b � 1 , − 1 � [0 , 0] ⇔ aF 2 = bF 2 M 0 = H hyperb./metab. orthogonal sum of hyperb./metabolic planes forms Witt decomposition Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 4 / 25
Notations and facts B2 Q1 Q2 φ ∼ φ diagonalizable = “normal form” φ ∼ = � a 1 , . . . , a n � [ a 1 , b 1 ] ⊥ . . . ⊥ [ a n , b n ] of forms φ if φ �∼ = hyperbolic where always [ a , b ] = ax 2 + xy + by 2 (see below) 2-dim metabolic plane hyperbolic plane H = xy M a = ( 0 1 isotropic 1 a ) M a ∼ forms = M b � 1 , − 1 � [0 , 0] ⇔ aF 2 = bF 2 M 0 = H hyperb./metab. orthogonal sum of hyperb./metabolic planes forms φ ∼ = φ an ⊥ φ 0 with Witt φ an anisotropic, unique up to ∼ =, and decomposition φ 0 metab. φ 0 hyperbolic, unique (only dim unique) Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 4 / 25
Witt ring and Witt group Two forms φ and ψ are Witt equivalent if φ an ∼ = ψ an . For Q1, B2: WF = Witt ring of F , i.e. the Witt equivalence classes of forms over F together with addition induced by ⊥ and multiplication induced by ⊗ . For Q2: W q F = Witt group of F , i.e. the Witt equivalence classes of forms over F together with addition induced by ⊥ ; this is not a ring, � � ca , b but can be made into a WF -module via � c � ⊗ [ a , b ] = . c For a field extension K / F : the Witt kernel W ( K / F ) (resp. W q ( K / F )) is the kernel of the natural restriction homomorphism WF → WK (resp. W q F → W q K ). Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 5 / 25
Pfister forms n -fold Pfister forms are defined as follows: (Q1, B2): � � a 1 , . . . , a n � � = � 1 , − a 1 � ⊗ . . . ⊗ � 1 , − a n � . P n F = isometry classes of n -fold Pfister forms. � ⊗ [1 , a n ]. P ( q ) (Q2): � � a 1 , . . . , a n ]] = � � a 1 , . . . , a n − 1 � n F = isometry classes of n -fold quadratic Pfister forms. Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 6 / 25
Pfister forms n -fold Pfister forms are defined as follows: (Q1, B2): � � a 1 , . . . , a n � � = � 1 , − a 1 � ⊗ . . . ⊗ � 1 , − a n � . P n F = isometry classes of n -fold Pfister forms. � ⊗ [1 , a n ]. P ( q ) (Q2): � � a 1 , . . . , a n ]] = � � a 1 , . . . , a n − 1 � n F = isometry classes of n -fold quadratic Pfister forms. Pfister forms are central to the theory and have many nice and important properties: P n F additively generates I n F , the n -th power of the fundamental ideal of even-dimensional forms IF in WF . P ( q ) n F generates I n q F = I n − 1 FW q F . Pfister forms are either anisotropic or hyperbolic (Q1, Q2) resp. metabolic (B2). Pfister forms π are round : π ∼ = a π ( a ∈ F ∗ ) ⇐ ⇒ a ∈ D F ( π ) ( a is represented by π ). Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 6 / 25
Witt kernels for algebraic extensions: Artin-Springer Theorem Artin-Springer Theorem If K / F is an algebraic extension with [ K : F ] odd, then anisotropic forms over F stay anisotropic over K. In particular, W ( K / F ) = 0 (Q1, B2) resp. W q ( K / F ) = 0 (Q2). Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 7 / 25
Witt kernels for algebraic extensions: Artin-Springer Theorem Artin-Springer Theorem If K / F is an algebraic extension with [ K : F ] odd, then anisotropic forms over F stay anisotropic over K. In particular, W ( K / F ) = 0 (Q1, B2) resp. W q ( K / F ) = 0 (Q2). Remark This was published in an article by T.A. Springer in 1952, but already shown in 1937 by E. Artin in a communication to Witt. Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 7 / 25
Witt kernels for algebraic extensions: Artin-Springer Theorem Artin-Springer Theorem If K / F is an algebraic extension with [ K : F ] odd, then anisotropic forms over F stay anisotropic over K. In particular, W ( K / F ) = 0 (Q1, B2) resp. W q ( K / F ) = 0 (Q2). Remark This was published in an article by T.A. Springer in 1952, but already shown in 1937 by E. Artin in a communication to Witt. In Q1: K / F purely inseparable = ⇒ K / F odd, so in this situation, to determine W ( K / F ), it suffices to consider separable extensions as each algebraic extension K / F can be written F ⊆ F sep ⊆ K with F sep / F separable, K / F sep purely inseparable. Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 7 / 25
Witt kernels for algebraic extensions: Separable extensions in case B2 Proposition (Knebusch 1973) ⇒ In the situation of B2: φ anisotropic form over F and K / F separable = φ K anisotropic. In particular: W ( K / F ) = 0 . Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 8 / 25
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