Normality and $K_1$-stability of Roy's elementary orthogonal group

ABSTRACT:

A. Roy introduced the elementary orthogonal group $\EO_A(Q\perp H(A)^m)$
of a quadratic space with a hyperbolic summand over a commutative ring 
$A$. This construction of Roy generalized the earlier work's of
Dickson-Siegel-Eichler-Dieudonn\'{e} over fields.

In this talk, we shall discuss the normality of the elementary orthogonal
group (Dickson--Siegel--Eichler--Roy or DSER group) $\EO_A(Q\perp H(A)^m)$
under some conditions  on the hyperbolic rank. We also establish stability
results for $K_1$ of  Roy's elementary orthogonal group under different
stable range conditions. The stability problem for $K_1$ of quadratic
forms was studied in 1960's and in early 1970's by H. Bass, A. Bak, A.
Roy, M. Kolster and L.N. Vaserstein. We obtain a Dennis-Vaserstein type
decomposition theorem for the elementary orthogonal group (DSER group)
which is used to deduce the stability theorem.