Abstract
Let M be a nonsingular quadratic module over a commutative ring R. The primary goal of this chapter is the proof of the following fact: If M is free of finite rank, then C(M) has a special element, or, equivalently, the Arf algebra A(M) of M is isomorphic to R[X](X2 - aX - b) for some a and b in R with a2 + 4b ∈ R*. This has the important consequence that the results developed in Chapters 7 and 8 for C(M) hold for any free M. In addition, we will establish that the discriminant a2 + 4b of X2 - aX - b is the classical discriminant of the quadratic module M (except for a factor 2 if rank M is odd). It was established in Chapter 9D that the Arf algebra A(M) is a separable quadratic algebra. The structural features of this algebra will play a decisive role in the proofs. Throughout, R is an arbitrary commutative ring.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
