Witt's Extension Theorem for Quadratic Spaces over Semiperfect Rings

2014/08/03 by Uriya A. First, First, Uriya A.
Mathematics · #11E08 #11E39 #11E81 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:11E08 #msc:11E39 #msc:11E81

paper · pdf · doi:10.48550/arxiv.1408.0522

25 pages; some proofs were shortened, replaced with appropriate references

arxiv created 2015/08/13 · arxiv updated 2015/08/14

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.

Related

• Subordinate quadratic forms and isometric maps over semirings
• Springer's odd degree extension theorem for quadratic forms over semilocal rings
• Patching and Weak Approximation in Isometry Groups
• Sesquilinear forms over rings with involution
• Quadratic D-forms with applications to hermitian forms