Quadratic forms of projective spaces over rings

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.