Witt group

英語（米国） (en-US) 英語（イギリス） (en-GB)

名詞

広義

(algebra) 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;

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The classification of quadratic forms over a field often relies on computations in the Witt group.

詳細

意味（1）

(algebra) 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;

A general method of studying the Witt group of a smooth variety is through the graded group associated to the filtration induced by the filtration of the Witt group of the function field by powers of the fundamental ideal of even rank forms.

詳細

意味（2）

(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

意味（3）

(category theory, by extension) given a category with duality, the quotient of isometry classes of symmetric spaces, modulo metabolic spaces.

Witt groups （ plural ）