Witt ring

The concept of the Witt ring comes from algebra . He is supposed to cover the square spaces over a ring , i.e. H. the modules with a symmetrical bilinear shape. It was introduced by Ernst Witt in 1937 . ${\displaystyle W(R)}$ ${\displaystyle R}$ ${\displaystyle R}$

Definition for any ring

Be a commutative ring . ${\displaystyle R}$

The set of square spaces , i.e. H. der -modules with symmetrical bilinear form, has a ring structure with the orthogonal direct sum as addition and the tensor product as multiplication. Two square spaces are said to be stable equivalent if there are such that is isomorphic to . ${\displaystyle R}$ ${\displaystyle \oplus }$ ${\displaystyle \otimes }$ ${\displaystyle S_{1},S_{2}}$ ${\displaystyle T_{1},T_{2}}$ ${\displaystyle S_{1}\oplus T_{1}}$ ${\displaystyle S_{2}\oplus T_{2}}$

Stable equivalence is an equivalence relation . The set of equivalence classes forms a ring with the links induced by and , which is referred to as the Witt ring . ${\displaystyle \oplus }$ ${\displaystyle \otimes }$ ${\displaystyle W(R)}$

Equivalent definition for body

Be a body of characteristic . The hyperbolic plane is the one with the symmetric bilinear form , the metabolic quadratic form is an orthogonal direct sum of hyperbolic planes. ${\displaystyle K}$ ${\displaystyle \operatorname {char} (K)\neq 2}$ ${\displaystyle H}$ ${\displaystyle K^{2}}$ ${\displaystyle b(x,y)=2xy}$

For such bodies, the Witt ring can be defined equivalently as the set of equivalence classes for the equivalence relation: and are equivalent if there is a metabolic quadratic form with or . ${\displaystyle W(K)}$ ${\displaystyle S_{1}}$ ${\displaystyle S_{2}}$ ${\displaystyle M}$ ${\displaystyle S_{2}=S_{1}\oplus M}$ ${\displaystyle S_{1}=S_{2}\oplus M}$

Examples

• For every algebraically closed field is . ${\displaystyle K}$ ${\displaystyle W(K)\simeq \mathbb {Z} /2\mathbb {Z} }$
• For the field of real numbers is . ${\displaystyle W(\mathbb {R} )\simeq \mathbb {Z} }$
• For the ring of whole numbers is (Hasse-Minkowski). ${\displaystyle W(\mathbb {Z} )=\mathbb {Z} }$
• For the field of rational numbers is (weak form of the Hasse-Minkowski theorem ). ${\displaystyle W(\mathbb {Q} )=W(\mathbb {R} )\bigoplus \oplus _{p}W(F_{p})}$
• For a finite field with is . ${\displaystyle F_{q}}$ ${\displaystyle q\equiv 1\mod 4}$ ${\displaystyle W(F_{q})\simeq \mathbb {Z} /4\mathbb {Z} }$
• For a finite field with is . ${\displaystyle F_{q}}$ ${\displaystyle q\equiv 3\mod 4}$ ${\displaystyle W(F_{q})\simeq \mathbb {Z} /2\mathbb {Z} \left[F^{*}/F^{*2}\right]}$
• For a local body with maximum ideal the norm is . ${\displaystyle K}$ ${\displaystyle {\mathfrak {m}}}$ ${\displaystyle N({\mathfrak {m}})\equiv 1\mod 4}$ ${\displaystyle W(K)=\mathbb {Z} /2\mathbb {Z} \left[\mathbb {Z} /2\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z} \right]}$
• For a local body with maximum ideal the norm is . ${\displaystyle K}$ ${\displaystyle {\mathfrak {m}}}$ ${\displaystyle N({\mathfrak {m}})\equiv 3\mod 4}$ ${\displaystyle W(K)=\mathbb {Z} /4\mathbb {Z} \left[\mathbb {Z} /2\mathbb {Z} \right]}$
• For each body the torsion component is generated by von Pfister forms . The order of each torsion element is a power of two. ${\displaystyle K}$ ${\displaystyle W(K)}$

literature

• John Milnor , Dale Husemöller : Symmetric bilinear forms . Springer, 1973.

Individual evidence

1. ↑ Witt, Theory of Square Shapes in Any Bodies, J. Reine Angew. Math., Volume 176, 1937, pp. 31-44