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Given a field k of characteristic ≠ 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;
Definition source: English Wiktionary via Wiktextract
Topics: algebra, mathematics, sciences
Examples
- 2011, Marco Schlichting, Higher Algebraic K-theory, Guillermo Cortiñas (editor), Topics in Algebraic and Topological K-Theory, Springer, Lecture Notes in Mathematics 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].
Meaning relationships
Synonyms: none provided
Antonyms: none provided
Given a field k of characteristic ≠ 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; (algebraic geometry, by extension) 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 (category theory, by extension) given a category with duality, the quotient of isometry classes of symmetric spaces, modulo metabolic spaces.
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
Definition source: English Wiktionary via Wiktextract
Usage: broadly
Topics: algebra, algebraic-geometry, geometry, mathematics, sciences
No example sentence was provided for this meaning.
Meaning relationships
Synonyms: none provided
Antonyms: none provided
Given a field k of characteristic ≠ 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; (algebraic geometry, by extension) 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 (category theory, by extension) given a category with duality, the quotient of isometry classes of symmetric spaces, modulo metabolic spaces.
given a category with duality, the quotient of isometry classes of symmetric spaces, modulo metabolic spaces.
Definition source: English Wiktionary via Wiktextract
Usage: broadly
Topics: algebra, category-theory, computing, engineering, mathematics, natural-sciences, physical-sciences, sciences
No example sentence was provided for this meaning.
Meaning relationships
Synonyms: none provided
Antonyms: none provided
Named after German mathematician Ernst Witt (1911–1991), who introduced the concept in 1937.
Across languages
Translations
2 source translations are retained for this English entry.
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French:
groupe de Witt
— group of equivalence classes found within a field, variety or category
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German:
Witt-Gruppe
— group of equivalence classes found within a field, variety or category
