Classical groups and their generalizations are central objects in Algebraic $K$-theory. Orthogonal groups are one type of classical groups. We shall discuss a generalized version of elementary orthogonal groups.

Let $R$ be a commutative ring in which $2$ is invertible. Let $Q$ be a non-degenerate quadratic space over $R$ of rank $n$ and let $\mathbb{H}(R)^m$ denote the hyperbolic space of rank $m$. We consider the elementary orthogonal transformations of the quadratic space $Q \perp \mathbb{H}(R)^m$. These transformations were introduced by Amit Roy in $1968$. Earlier forms of these transformations over fields were considered by Dickson, Siegel, Eichler and Dieudonné. We call the elementary orthogonal transformations as Dickson–Siegel–Eichler–Roy elementary orthogonal transformations or Roy’s elementary orthogonal transformations. The group generated by these transformations is called DSER elementary orthogonal group. We shall discuss the structure of this group.

As part of the solution to the famous Serre’s problem on projective modules, D. Quillen had proved the remarkable Local-Global criterion for a module $M$ to be extended. This result is known as Quillen’s Patching Theorem or Quillen’s Local-Global Principle. The Bass–Quillen conjecture is a natural generalization of Serre’s problem. In this talk, we shall see the solution of the quadratic version of the Bass–Quillen conjecture over an equicharacteristic regular local ring.

The DSER elementary orthogonal group is a normal subgroup of the orthogonal group. We shall also discuss some generalizations of classical groups over form rings and their comparison with the DSER elementary orthogonal group.

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Last updated: 22 Sep 2023