Witt kernels a survey Detlev Hoffmann - - PowerPoint PPT Presentation

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”

• f 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 “normal form”

• f forms φ

φ diagonalizable φ ∼ = a1, . . . , an if φ ∼ = hyperbolic (see below) always 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 “normal form”

• f forms φ

φ diagonalizable φ ∼ = a1, . . . , an φ ∼ = [a1, b1] ⊥ . . . ⊥ [an, bn] if φ ∼ = hyperbolic (see below) always where [a, b] = ax2 + xy + by2 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 “normal form”

• f forms φ

φ diagonalizable φ ∼ = a1, . . . , an φ ∼ = [a1, b1] ⊥ . . . ⊥ [an, bn] if φ ∼ = hyperbolic (see below) always where [a, b] = ax2 + xy + by2 2-dim isotropic forms hyperbolic plane H = xy 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 “normal form”

• f forms φ

φ diagonalizable φ ∼ = a1, . . . , an φ ∼ = [a1, b1] ⊥ . . . ⊥ [an, bn] if φ ∼ = hyperbolic (see below) always where [a, b] = ax2 + xy + by2 2-dim isotropic forms metabolic plane hyperbolic plane H = xy Ma = ( 0 1

1 a )

Ma ∼ = Mb ⇔ aF 2 = bF 2 M0 = H 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 “normal form”

• f forms φ

φ diagonalizable φ ∼ = a1, . . . , an φ ∼ = [a1, b1] ⊥ . . . ⊥ [an, bn] if φ ∼ = hyperbolic (see below) always where [a, b] = ax2 + xy + by2 2-dim isotropic forms metabolic plane hyperbolic plane H = xy Ma = ( 0 1

1 a )

Ma ∼ = Mb ⇔ aF 2 = bF 2 M0 = H 1, −1 [0, 0] hyperb./metab. forms

• rthogonal sum of hyperb./metabolic planes

Witt decomposition

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 4 / 25

Notations and facts

B2 Q1 Q2 “normal form”

• f forms φ

φ diagonalizable φ ∼ = a1, . . . , an φ ∼ = [a1, b1] ⊥ . . . ⊥ [an, bn] if φ ∼ = hyperbolic (see below) always where [a, b] = ax2 + xy + by2 2-dim isotropic forms metabolic plane hyperbolic plane H = xy Ma = ( 0 1

1 a )

Ma ∼ = Mb ⇔ aF 2 = bF 2 M0 = H 1, −1 [0, 0] hyperb./metab. forms

• rthogonal sum of hyperb./metabolic planes

Witt decomposition φ ∼ = φan ⊥ φ0 with φan anisotropic, unique up to ∼ =, and φ0 metab. (only dim unique) φ0 hyperbolic, 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 φan ∼ = ψan. For Q1, B2: WF = Witt ring of F, i.e. the Witt equivalence classes

• f forms over F together with addition induced by ⊥ and

multiplication induced by ⊗. For Q2: WqF = Witt group of F, i.e. the Witt equivalence classes of forms over F together with addition induced by ⊥; this is not a ring, but can be made into a WF-module via c ⊗ [a, b] =

• ca, b

c

• .

For a field extension K/F: the Witt kernel W (K/F) (resp. Wq(K/F)) is the kernel of the natural restriction homomorphism WF → WK (resp. WqF → WqK).

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): a1, . . . , an = 1, −a1 ⊗ . . . ⊗ 1, −an. PnF = isometry classes of n-fold Pfister forms. (Q2): a1, . . . , an]] = a1, . . . , an−1 ⊗ [1, an]. P(q)

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): a1, . . . , an = 1, −a1 ⊗ . . . ⊗ 1, −an. PnF = isometry classes of n-fold Pfister forms. (Q2): a1, . . . , an]] = a1, . . . , an−1 ⊗ [1, an]. P(q)

n F = isometry

classes of n-fold quadratic Pfister forms. Pfister forms are central to the theory and have many nice and important properties: PnF additively generates I nF, 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−1FWqF.

Pfister forms are either anisotropic or hyperbolic (Q1, Q2) resp. metabolic (B2). Pfister forms π are round: π ∼ = aπ (a ∈ F ∗) ⇐ ⇒ a ∈ DF(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

• ver F stay anisotropic over K. In particular, W (K/F) = 0 (Q1, B2) resp.

Wq(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

• ver F stay anisotropic over K. In particular, W (K/F) = 0 (Q1, B2) resp.

Wq(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

• ver F stay anisotropic over K. In particular, W (K/F) = 0 (Q1, B2) resp.

Wq(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 = ⇒ φK anisotropic. In particular: W (K/F) = 0.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 8 / 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 = ⇒ φK anisotropic. In particular: W (K/F) = 0.

Remark

In B2, the behaviour of W (K/F) w.r.t. separable/purely inseparable is “opposite” to that in Q1: Q1: W (K/F) = 0 if K/F purely inseparable; B2: W (K/F) = 0 if K/F separable.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 8 / 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 = ⇒ φK anisotropic. In particular: W (K/F) = 0.

Remark

In B2, the behaviour of W (K/F) w.r.t. separable/purely inseparable is “opposite” to that in Q1: Q1: W (K/F) = 0 if K/F purely inseparable; B2: W (K/F) = 0 if K/F separable. Unfortunately, generally algebraic extensions K/F cannot be written as F ⊆ F insep ⊆ K with F insep/F purely inseparable and K/F insep separa-

• ble. Extensions were this is possible are called balanced.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 8 / 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 = ⇒ φK anisotropic. In particular: W (K/F) = 0.

Remark

In B2, the behaviour of W (K/F) w.r.t. separable/purely inseparable is “opposite” to that in Q1: Q1: W (K/F) = 0 if K/F purely inseparable; B2: W (K/F) = 0 if K/F separable. So in B2 we can restrict ourselves to consider inseparable extensions but must include those that aren’t necessarily purely inseparable. In Q2, every type of algebraic extension must be considered, making this the trickiest case.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 8 / 25

Witt kernels for algebraic extensions: Quadratic extensions

Proposition

Q1 (Pfister 1966?) K = F( √ d): φ anisotropic over F, isotropic over K = ⇒ φ ∼ = a1, −d ⊥ . . . (a ∈ F ∗). In particular, W (K/F) = 1, −dWF.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 9 / 25

Witt kernels for algebraic extensions: Quadratic extensions

Proposition

Q1 (Pfister 1966?) K = F( √ d): φ anisotropic over F, isotropic over K = ⇒ φ ∼ = a1, −d ⊥ . . . (a ∈ F ∗). In particular, W (K/F) = 1, −dWF. Q2 (Baeza 1974, 1978) K = F(℘−1(d)) (℘−1(d) =root of X 2 + X + d): φ anisotropic over F, isotropic over K = ⇒ φ ∼ = a[1, d] ⊥ . . .. In particular, Wq(K/F) = WF · [1, d]. K = F( √ d): φ anisotropic over F, isotropic over K = ⇒ φ ∼ = [a, b] ⊥ [ad, c] ⊥ . . . In particular, Wq(K/F) = 1, dWqF.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 9 / 25

Witt kernels for algebraic extensions: Quadratic extensions

Proposition

Q1 (Pfister 1966?) K = F( √ d): φ anisotropic over F, isotropic over K = ⇒ φ ∼ = a1, −d ⊥ . . . (a ∈ F ∗). In particular, W (K/F) = 1, −dWF. Q2 (Baeza 1974, 1978) K = F(℘−1(d)) (℘−1(d) =root of X 2 + X + d): φ anisotropic over F, isotropic over K = ⇒ φ ∼ = a[1, d] ⊥ . . .. In particular, Wq(K/F) = WF · [1, d]. K = F( √ d): φ anisotropic over F, isotropic over K = ⇒ φ ∼ = [a, b] ⊥ [ad, c] ⊥ . . . In particular, Wq(K/F) = 1, dWqF. B2 (Laghribi 2004) K = F( √ d): φ anisotropic over F, isotropic over K = ⇒ φ ∼ = a1, d + c2 ⊥ . . . (a, c ∈ F, a = 0). In particular, W (K/F) is generated by forms of type 1, t with t ∈ K ∗2.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 9 / 25

Witt kernels for algebraic extensions: Quadratic extensions

Remark

The above implies (in all cases) that quadratic extensions K/F are excellent extensions: φ form over F, then ∃ form ψ over F with (φK)an ∼ = ψK (the anisotropic part over K is defined over F). This is useful when determining certain Witt kernels of extensions K/F that contain quadratic subextensions.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 10 / 25

Witt kernels of biquadratic extensions

Using the fact that quadratic extensions are excellent and that their kernels are generated by the (obvious) 1-fold Pfister forms, we then readily get:

Corollary

(i) (Q1) K = F(√a, √ b) = ⇒ W (K/F) = 1, −aWF + 1, −bWF (Elman-Lam-Wadsworth 1976); (ii) (Q2) K = F(√a, √ b) = ⇒ Wq(K/F) = 1, aWqF + 1, bWqF (Mammone-Moresi 1995); (iii) (Q2) K = F(√a, ℘−1(c)) = ⇒ Wq(K/F) = 1, aWqF + WF · [1, c] (Ahmad 2004); (iv) (Q2) K = F(℘−1(c), ℘−1(d)) = ⇒ Wq(K/F) = WF · [1, c] + WF · [1, d] (Ahmad 2004).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 11 / 25

Witt kernels of multiquadratic extensions

(Q1) If K = F(√c, √ d, √e) then generally, 1, −cWF + 1, −dWF + 1, −eWF W (K/F) . Examples of strict inclusion: Elman-Lam-Tignol-Wadsworth 1983, based on examples of indecomposable division algebras of index 8 and exponent 2 (Amitsur-Rowen-Tignol 1979).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 12 / 25

Witt kernels of multiquadratic extensions

(Q1) If K = F(√c, √ d, √e) then generally, 1, −cWF + 1, −dWF + 1, −eWF W (K/F) . Examples of strict inclusion: Elman-Lam-Tignol-Wadsworth 1983, based on examples of indecomposable division algebras of index 8 and exponent 2 (Amitsur-Rowen-Tignol 1979). (Q1) For “nice” fields (e.g. F global or F a function field of transcendence degree 1 over a real (or algebraically) closed field): W (F(

• di, i ∈ I)/F) =
• i∈I

1, −diWF (Elman-Lam-Wadsworth 1979).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 12 / 25

Witt kernels of multiquadratic extensions

(Q1) If K = F(√c, √ d, √e) then generally, 1, −cWF + 1, −dWF + 1, −eWF W (K/F) . Examples of strict inclusion: Elman-Lam-Tignol-Wadsworth 1983, based on examples of indecomposable division algebras of index 8 and exponent 2 (Amitsur-Rowen-Tignol 1979). (Q1) For “nice” fields (e.g. F global or F a function field of transcendence degree 1 over a real (or algebraically) closed field): W (F(

• di, i ∈ I)/F) =
• i∈I

1, −diWF (Elman-Lam-Wadsworth 1979). (Q2) K = F(√a1, . . . , √an), E = F(℘−1(c)). Then Wq(K/F) = n

i=1

ai WqF, Wq(KE/F) = W (K/F) + WF · [1, c] (Laghribi 2006 for K/F but with a gap in the proof; Aravire-Laghribi 2011).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 12 / 25

Witt kernels of multiquadratic extensions

Aravire and Laghribi compute the kernel HnF → HnK (resp. HnKE) for Kato’s cohomology and then use Kato’s isomorphism I n

q F/I n+1 q

∼ = HnF.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 13 / 25

Witt kernels of multiquadratic extensions

Aravire and Laghribi compute the kernel HnF → HnK (resp. HnKE) for Kato’s cohomology and then use Kato’s isomorphism I n

q F/I n+1 q

∼ = HnF.

Question

In Q2, are there examples of separable triquadratic extensions whose Witt kernel is not the “expected” one as in the Q1 example by Elman-Lam-Tignol-Wadsworth?

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 13 / 25

Witt kernels of degree 4 extensions

(Q1) [K : F] = 4, K = F(α), p(t) = t4 + at2 + bt + c ∈ F[t] the minimal polynomial of α, f (t) the cubic resolvent of p(t). Then W (K/F) is generated by d with F( √ d) ⊂ K, and by forms

• f type

−r, f (r) (Sivatski 2010).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 14 / 25

Witt kernels of degree 4 extensions

(Q1) [K : F] = 4, K = F(α), p(t) = t4 + at2 + bt + c ∈ F[t] the minimal polynomial of α, f (t) the cubic resolvent of p(t). Then W (K/F) is generated by d with F( √ d) ⊂ K, and by forms

• f type

−r, f (r) (Sivatski 2010). (Q2) K = F(

4

√ d): Wq(K/F) is generated by forms of type d, a]],

• c, dc2(x4 + d2y4)]] (Ahmad 2005).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 14 / 25

Witt kernels of degree 4 extensions

(Q1) [K : F] = 4, K = F(α), p(t) = t4 + at2 + bt + c ∈ F[t] the minimal polynomial of α, f (t) the cubic resolvent of p(t). Then W (K/F) is generated by d with F( √ d) ⊂ K, and by forms

• f type

−r, f (r) (Sivatski 2010). (Q2) K = F(

4

√ d): Wq(K/F) is generated by forms of type d, a]],

• c, dc2(x4 + d2y4)]] (Ahmad 2005).

(Q2) It should(?) be possible to treat separable degree 4 extensions in a way similar to the case Q1. The only remaining case would then be that of extensions K = F(α) with α root of an irreducible polynomial X 4 + aX 2 + b with (aF 2 + b) ∩ F 2 = ∅ — these are exactly the degree 4 extensions that do not contain inseparable quadratic subextensions (non balanced!).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 14 / 25

Witt kernels of degree 4 extensions

(Q1) [K : F] = 4, K = F(α), p(t) = t4 + at2 + bt + c ∈ F[t] the minimal polynomial of α, f (t) the cubic resolvent of p(t). Then W (K/F) is generated by d with F( √ d) ⊂ K, and by forms

• f type

−r, f (r) (Sivatski 2010). (Q2) K = F(

4

√ d): Wq(K/F) is generated by forms of type d, a]],

• c, dc2(x4 + d2y4)]] (Ahmad 2005).

(Q2) It should(?) be possible to treat separable degree 4 extensions in a way similar to the case Q1. The only remaining case would then be that of extensions K = F(α) with α root of an irreducible polynomial X 4 + aX 2 + b with (aF 2 + b) ∩ F 2 = ∅ — these are exactly the degree 4 extensions that do not contain inseparable quadratic subextensions (non balanced!). There are (to my knowledge) no “general” results in Q1, Q2 for higher even degree extensions without further assumptions on the field and/or the type of extension.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 14 / 25

Witt kernels of purely inseparable extensions (B2)

Proposition (H 2005)

Let K/F be purely inseparable of exponent 1 (i.e. K 2 ⊂ F). Then: W (K/F) is the ideal in WF generated by {1, t | t ∈ (K ∗)2}.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 15 / 25

Witt kernels of purely inseparable extensions (B2)

Proposition (H 2005)

Let K/F be purely inseparable of exponent 1 (i.e. K 2 ⊂ F). Then: W (K/F) is the ideal in WF generated by {1, t | t ∈ (K ∗)2}.

Remarks

One can show an even more precise statement about the isotropy behaviour of bilinear forms under such extensions (such extensions are excellent). K/F purely inseparable of exponent 1, [K : F] = 2n = ⇒ K = F(√a1, . . . , √an) (∃ai ∈ F ∗). (H 2005) More generally, W (K/F) has been determined for all finite purely inseparable extensions that possess a higher 2-basis {α1, . . . , αn}, i.e. one has K = F(α1, . . . , αn) ∼ = F(α1) ⊗F · · · ⊗F F(αn) .

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 15 / 25

Witt kernels for function field extensions

Here, by function field we will always mean the function field of a single irreducible polynomial (function field of a “hypersurface”).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 16 / 25

Witt kernels for function field extensions

Here, by function field we will always mean the function field of a single irreducible polynomial (function field of a “hypersurface”). X = (X1, . . . , Xn) an n-tuple of variables, f (X) ∈ F[X] irreducible. The function field F(f ) of f over F is defined as F(f ) = Quot(F[X]/(f ))

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 16 / 25

Witt kernels for function field extensions

Here, by function field we will always mean the function field of a single irreducible polynomial (function field of a “hypersurface”). X = (X1, . . . , Xn) an n-tuple of variables, f (X) ∈ F[X] irreducible. The function field F(f ) of f over F is defined as F(f ) = Quot(F[X]/(f ))

Question

Can one determine W (F(f )/F) resp. Wq(F(f )/F)?

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 16 / 25

Witt kernels for function field extensions

Here, by function field we will always mean the function field of a single irreducible polynomial (function field of a “hypersurface”). X = (X1, . . . , Xn) an n-tuple of variables, f (X) ∈ F[X] irreducible. The function field F(f ) of f over F is defined as F(f ) = Quot(F[X]/(f ))

Question

Can one determine W (F(f )/F) resp. Wq(F(f )/F)?

Remark

The case n = 1 corresponds to the case of simple algebraic extensions.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 16 / 25

Witt kernels for function field extensions

Here, by function field we will always mean the function field of a single irreducible polynomial (function field of a “hypersurface”). X = (X1, . . . , Xn) an n-tuple of variables, f (X) ∈ F[X] irreducible. The function field F(f ) of f over F is defined as F(f ) = Quot(F[X]/(f ))

Question

Can one determine W (F(f )/F) resp. Wq(F(f )/F)?

Remark

The case n = 1 corresponds to the case of simple algebraic extensions.

Remark

An easy argument shows: K/F purely transcendental = ⇒ anisotropic forms/F stay anisotropic/K. In particular, W (K/F) = 0 resp. Wq(K/F) = 0.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 16 / 25

Knebusch’s Norm Principle

Theorem (Knebusch 1973 in case Q1, B2, Baeza 1990 in case Q2)

f ∈ F[X] irreducible with leading coefficient 1 (w.r.t. lexicographical

• rdering of monomials). Let φ be an anisotropic form over F. Then φK

hyperbolic (Q1, Q2) resp. metabolic (B2) ⇐ ⇒ f φ ∼ = φ.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 17 / 25

Knebusch’s Norm Principle

Theorem (Knebusch 1973 in case Q1, B2, Baeza 1990 in case Q2)

f ∈ F[X] irreducible with leading coefficient 1 (w.r.t. lexicographical

• rdering of monomials). Let φ be an anisotropic form over F. Then φK

hyperbolic (Q1, Q2) resp. metabolic (B2) ⇐ ⇒ f φ ∼ = φ. While this gives us an intrinsic description of forms in the Witt kernel, it doesn’t really tell us anything about how generators of the Witt kernel look like.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 17 / 25

Witt kernels for function fields of quadratic forms

An important case is the function field of a quadratic form (quadric): q(X) = n

i=1 aijXiXj, where we allow degenerate quadratic forms in char

2. q(X) irreducible if dim(q) ≥ 2 and q anisotropic, the only relevant case. We also talk about the function field of a bilinear form b(X, Y ), where we mean the function field of the quadratic form qb(X) = b(X, X).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 18 / 25

Witt kernels for function fields of quadratic forms

An important case is the function field of a quadratic form (quadric): q(X) = n

i=1 aijXiXj, where we allow degenerate quadratic forms in char

2. q(X) irreducible if dim(q) ≥ 2 and q anisotropic, the only relevant case. We also talk about the function field of a bilinear form b(X, Y ), where we mean the function field of the quadratic form qb(X) = b(X, X). (Q1) q an anisotropic n-Pfister, n ≥ 1. Then W (F(q)/F) = qWF (Arason-Pfister 1971, Elman-Lam 1972).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 18 / 25

Witt kernels for function fields of quadratic forms

An important case is the function field of a quadratic form (quadric): q(X) = n

i=1 aijXiXj, where we allow degenerate quadratic forms in char

2. q(X) irreducible if dim(q) ≥ 2 and q anisotropic, the only relevant case. We also talk about the function field of a bilinear form b(X, Y ), where we mean the function field of the quadratic form qb(X) = b(X, X). (Q1) q an anisotropic n-Pfister, n ≥ 1. Then W (F(q)/F) = qWF (Arason-Pfister 1971, Elman-Lam 1972). (Q1) Fitzgerald (1981) gave explicit generators for W (F(q)/F) for

• ther special types of anisotropic q, in particular for all q with

dim(q) ≤ 5. In all cases considered, the generators are certain Pfister forms. E.g. φ anisotropic over F and hyperbolic over F(d, −a, −b, ab) ⇐ ⇒ φ ∼ =⊥n

i=1 ri

a, b, si with si ∈ DF( d ).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 18 / 25

Witt kernels for function fields of quadratic forms

(Q2) q ∈ PnF (resp. P(q)

n F) anisotropic =

⇒ Wq(F(q)/F) = qWqF (resp. Wq(F(q)/F) = WF · q) (Laghribi 2002, H-Laghribi 2004).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 19 / 25

Witt kernels for function fields of quadratic forms

(Q2) q ∈ PnF (resp. P(q)

n F) anisotropic =

⇒ Wq(F(q)/F) = qWqF (resp. Wq(F(q)/F) = WF · q) (Laghribi 2002, H-Laghribi 2004). (Q2) Laghribi (2005) derives results analogous to those of Fitzgerald for Witt kernels Wq(F(q)/F) for function fields of forms of small dimension.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 19 / 25

Witt kernels for function fields of quadratic forms

(Q2) q ∈ PnF (resp. P(q)

n F) anisotropic =

⇒ Wq(F(q)/F) = qWqF (resp. Wq(F(q)/F) = WF · q) (Laghribi 2002, H-Laghribi 2004). (Q2) Laghribi (2005) derives results analogous to those of Fitzgerald for Witt kernels Wq(F(q)/F) for function fields of forms of small dimension. (B2) Laghribi (2005) determined explicit generators for W (F(b)/F) for arbitrary anisotropic bilinear b. We will generalize these results below.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 19 / 25

Witt kernels for function fields of quadratic forms

(Q2) q ∈ PnF (resp. P(q)

n F) anisotropic =

⇒ Wq(F(q)/F) = qWqF (resp. Wq(F(q)/F) = WF · q) (Laghribi 2002, H-Laghribi 2004). (Q2) Laghribi (2005) derives results analogous to those of Fitzgerald for Witt kernels Wq(F(q)/F) for function fields of forms of small dimension. (B2) Laghribi (2005) determined explicit generators for W (F(b)/F) for arbitrary anisotropic bilinear b. We will generalize these results below. (Q1) Jonathan Shick (1994) proved results on Witt kernels for function fields of hyperelliptic curves.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 19 / 25

Witt kernels for function fields of quadratic forms

(Q2) q ∈ PnF (resp. P(q)

n F) anisotropic =

⇒ Wq(F(q)/F) = qWqF (resp. Wq(F(q)/F) = WF · q) (Laghribi 2002, H-Laghribi 2004). (Q2) Laghribi (2005) derives results analogous to those of Fitzgerald for Witt kernels Wq(F(q)/F) for function fields of forms of small dimension. (B2) Laghribi (2005) determined explicit generators for W (F(b)/F) for arbitrary anisotropic bilinear b. We will generalize these results below. (Q1) Jonathan Shick (1994) proved results on Witt kernels for function fields of hyperelliptic curves. There have been almost no other results concerning generators of W (F(f )/F) or Wq(F(f )/F) for other types of irreducible f ∈ F[X].

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 19 / 25

Witt kernels for function fields of quadratic forms

(Q2) q ∈ PnF (resp. P(q)

n F) anisotropic =

⇒ Wq(F(q)/F) = qWqF (resp. Wq(F(q)/F) = WF · q) (Laghribi 2002, H-Laghribi 2004). (Q2) Laghribi (2005) derives results analogous to those of Fitzgerald for Witt kernels Wq(F(q)/F) for function fields of forms of small dimension. (B2) Laghribi (2005) determined explicit generators for W (F(b)/F) for arbitrary anisotropic bilinear b. We will generalize these results below. (Q1) Jonathan Shick (1994) proved results on Witt kernels for function fields of hyperelliptic curves. There have been almost no other results concerning generators of W (F(f )/F) or Wq(F(f )/F) for other types of irreducible f ∈ F[X]. We will change that in case B2!

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 19 / 25

The Witt kernel of function fields in case B2: The easy case

Here: X = (X1, · · · , Xn), X 2 = (X 2

1 , · · · , X 2 n ).

Proposition

(char(F) = 2) Let f (X) ∈ F[X] be irreducible. If f (X) / ∈ F[X 2], then W (F(f )/F) = 0.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 20 / 25

The Witt kernel of function fields in case B2: The easy case

Here: X = (X1, · · · , Xn), X 2 = (X 2

1 , · · · , X 2 n ).

Proposition

(char(F) = 2) Let f (X) ∈ F[X] be irreducible. If f (X) / ∈ F[X 2], then W (F(f )/F) = 0.

Proof.

F(f )/F can be realized as a purely transcendental extension followed by a separable extension. Earlier result: Witt kernels for such extensions are trivial.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 20 / 25

Witt kernels of function fields in case B2: Notations

Some notations (where we assume more generally char(F) = p > 0 and wlog F non perfect): X = (X1, . . . , Xn), X p = (X p

1 , . . . , X p n ), X ′ = (X1, . . . , Xn−1);

For h(X) ∈ F[X]: Coef(h) = {all F-coefficients of h};

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 21 / 25

Witt kernels of function fields in case B2: Notations

Some notations (where we assume more generally char(F) = p > 0 and wlog F non perfect): X = (X1, . . . , Xn), X p = (X p

1 , . . . , X p n ), X ′ = (X1, . . . , Xn−1);

For h(X) ∈ F[X]: Coef(h) = {all F-coefficients of h}; For f (X) ∈ F[X p]: Apply invertible linear change of variables and scale: f f (X) = X pm

n

+ gm−1(X ′)X p(m−1)

n

+ . . . + g0(X ′) with gi(X ′) ∈ F[X ′p]. One can show: F p(Coef( f )) is independent of the change of variables and subsequent scaling;

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 21 / 25

Witt kernels of function fields in case B2: Notations

Some notations (where we assume more generally char(F) = p > 0 and wlog F non perfect): X = (X1, . . . , Xn), X p = (X p

1 , . . . , X p n ), X ′ = (X1, . . . , Xn−1);

For h(X) ∈ F[X]: Coef(h) = {all F-coefficients of h}; For f (X) ∈ F[X p]: Apply invertible linear change of variables and scale: f f (X) = X pm

n

+ gm−1(X ′)X p(m−1)

n

+ . . . + g0(X ′) with gi(X ′) ∈ F[X ′p]. One can show: F p(Coef( f )) is independent of the change of variables and subsequent scaling; Assume p = char(F) = 2: Suppose [F 2(Coef( f )) : F 2] = 2r. Put PF(f ) = { a1, . . . , ar | F 2(Coef( f )) = F 2(a1, . . . , ar)} .

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 21 / 25

Witt kernels of function fields in case B2: Main result

Theorem (Dolphin-H 2009)

Let f ∈ F[X 2] be irreducible. Then W (F(f )/F) is generated by PF(f ). More precisely: If b is anisotropic and b ∈ W (F(f )/F), then ∃πi ∈ PF(f ), ci ∈ F ∗, 1 ≤ i ≤ t, s.t. b ∼ = c1π1 ⊥ . . . ⊥ ctπt.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 22 / 25

Witt kernels of function fields in case B2: Main result

Theorem (Dolphin-H 2009)

Let f ∈ F[X 2] be irreducible. Then W (F(f )/F) is generated by PF(f ). More precisely: If b is anisotropic and b ∈ W (F(f )/F), then ∃πi ∈ PF(f ), ci ∈ F ∗, 1 ≤ i ≤ t, s.t. b ∼ = c1π1 ⊥ . . . ⊥ ctπt. For the proof, one needs differential forms.

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 22 / 25

Strategy of the proof

Here, K = F(f ) denotes the function field, Ωn(F) the space of absolute K¨ ahler n-differential forms over F (where we also consider more generally fields of characteristic p > 0), νn(F) the kernel of the Artin-Schreier map ℘ : Ωn(F) → Ωn(F)/dΩn−1(F).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 23 / 25

Strategy of the proof

Here, K = F(f ) denotes the function field, Ωn(F) the space of absolute K¨ ahler n-differential forms over F (where we also consider more generally fields of characteristic p > 0), νn(F) the kernel of the Artin-Schreier map ℘ : Ωn(F) → Ωn(F)/dΩn−1(F). The strategy: First, determine the kernel Ωn(K/F) of the map Ωn(F) → Ωn(K) (proof by induction on the number of variables, the 1-variable case uses some sort of descent argument). For p = 2, using generalizations of result by Aravire-Baeza, compute the kernel νn(K/F) = νn(F) ∩ Ωn(K/F) and use Kato’s isomorphism νn(F) ∼ = I nF/I n+1F to compute the kernel of the map I nF/I n+1F → I nK/I n+1K. Now use some standard procedure to get to W (K/F).

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 23 / 25

An application to Milnor K-theory

Let F be a field of characteristic p > 0 and let K = F(f ) be a function field in the previous sense, f ∈ F[X p]. Using our determination of the kernel Ωn(K/F) and the Bloch-Kato-Gabber isomorphism νn(F) ∼ = Kn(F)/pKn(F), Stephen Scully (2010) explicitly determined the generators of the kernel of the map Kn(F)/pKn(F) → Kn(K)/pKn(K) : The kernel is generated by symbols {a1, . . . , an} such that F p(Coef( f )) ⊆ F p(a1, . . . , an) .

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 24 / 25

Thank you!

Detlev Hoffmann (TU Dortmund) Witt kernels — a survey Emory, 19 May 2011 25 / 25