Witt group

Etymology
Named after German mathematician (1911–1991), who introduced the concept in 1937.

Noun

 * 1)  Given a field k of characteristic &ne; 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;  given a variety X, the quotient of the Grothendieck group of isometry classes of quadratic spaces on X, with respect to orthogonal sum, modulo the subgroup generated by metabolic spaces  given a category with duality, the quotient of isometry classes of symmetric spaces, modulo metabolic spaces.
 * 2) * 2011, Marco Schlichting, Higher Algebraic K-theory, Guillermo Cortiñas (editor), Topics in Algebraic and Topological K-Theory, Springer, 2008, page 167,
 * The second reason for this emphasis is that an analog of the Thomason-Waldhausen Localization Theorem also holds for many other (co-) homology theories besides K-theory, among which Hochschild homology, (negative, periodic, ordinary) cyclic homology [49], topological Hochschild (and cyclic) homology [2], triangular Witt groups [6] and higher Grothendieck–Witt groups [77].
 * The second reason for this emphasis is that an analog of the Thomason-Waldhausen Localization Theorem also holds for many other (co-) homology theories besides K-theory, among which Hochschild homology, (negative, periodic, ordinary) cyclic homology [49], topological Hochschild (and cyclic) homology [2], triangular Witt groups [6] and higher Grothendieck–Witt groups [77].

Usage notes

 * A Witt group of a field can be made into a commutative ring by equipping it with a ring product defined as the tensor product of quadratic forms.
 * This construction is sometimes called a ; the term is ambiguous, however, being also often used for a completely unrelated ring of Witt vectors.

Translations

 * French: groupe de Witt
 * German: Witt-Gruppe