Witt's Extension Theorem for Quadratic Spaces over Semiperfect Rings

TL;DR

This paper extends Witt's theorem to quadratic spaces over semiperfect rings, showing isometries of summands extend to the whole space and deriving new cancellation results for various forms.

Contribution

It proves Witt's extension theorem for quadratic spaces over semiperfect rings, generalizing previous results and establishing new cancellation theorems for forms over different rings.

Findings

Isometries of summands extend to entire quadratic spaces over semiperfect rings

Unimodular quadratic spaces over semiperfect rings cancel from orthogonal sums

Cancellation theorems for sesquilinear and hermitian forms over specific rings

Abstract

We prove that every isometry of between (not-necessarily orthogonal) summands of a unimodular quadratic space over a semiperfect ring can be extended an isometry of the whole quadratic space. The same result was proved by Reiter for the broader class of semilocal rings, but with certain restrictions on the base modules, which cannot be removed in general. Our result implies that unimodular quadratic spaces over semiperfect rings cancel from orthogonal sums. This improves a cancellation result of Quebbemann, Scharlau and Schulte, which applies to quadratic spaces over hermitian categories. Combining this with other known results yields further cancellation theorems. For instance, we prove cancellation of (1) systems of sesquilinear forms over henselian local rings, and (2) non-unimodular hermitian forms over (arbitrary) valuation rings. Finally, we determine the group generated by the reflections of a unimodular quadratic space over a semiperfect ring.