Quadratic forms of projective spaces over rings

Тип публикации: статья из журнала

Год издания: 2006

Идентификатор DOI: 10.1070/SM2006v197n06ABEH003782

Аннотация: In the passage from fields to rings of coefficients quadratic forms with invertible matrices lose their decisive role. It turns out that if all quadratic forms over a ring are diagonalizable, then in effect this is always a local principal ideal ring R with 2 epsilon R*. The problem of the construction of a 'normal' diagonal form of a quadratic form over a ring R faces obstacles in the case of indices vertical bar R* : R *2 vertical bar greater than 1. In the case of index 2 this problem has a solution given in Theorem 2.1 for 1 + R (*2) subset of R-*2 (an extension of the law of inertia for real quadratic forms) and in Theorem 2.2 for 1 + R-2 containing an invertible non-square. Under the same conditions on a ring R with nilpotent maximal ideal the number of classes of projectively congruent quadratic forms of the projective space associated with a free R-module of rank n is explicitly calculated (Proposition 3.2). Up to projectivities, the list of forms is presented for the projective plane over R and also (Theorem 3.3) over the local ring F [[x, y]]/(x(2), xy, y(2)) with non-principal maximal ideal, where F = 2F is a field with an invertible non-square in I + F-2 and vertical bar F-* : F-*2 vertical bar = 2. In the latter case the number of classes of non-diagonalizable quadratic forms of rank 0 depends on one's choice of the field F and is not even always finite; all the other forms make up 21 classes.

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Издание

Журнал: SBORNIK MATHEMATICS

Выпуск журнала: Vol. 197, Is. 05.06.2017

Номера страниц: 887-899

ISSN журнала: 10645616

Место издания: LETCHWORTH

Издатель: LONDON MATHEMATICAL SOCIETY RUSSIAN ACAD SCIENCES

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