Quadratic forms of projective spaces over rings

V. M. Levchuk ; O. A. Starikova

Sbornik. Mathematics, Tome 197 (2006) no. 6, pp. 887-899

Résumé

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 $2\in R^*$. 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 $|R^*:R^{*2}|$ greater than 1. In the case of index 2 this problem has a solution given in Theorem 2.1 for $1+R^{*2}\subseteq R^{*2}$ (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 $F[[x,y]]/\langle x^{2},xy,y^{2}\rangle$ with non-principal maximal ideal, where $F=2F$ is a field with an invertible non-square in $1+F^{2}$ and $|F^{*}:F^{*2}|=2$. 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. Bibliography: 28 titles.