The Quadratic Analogue of Serre’s Conjecture

Abstract

This chapter will be a short exposition on some results in the literature (up to about 1977) concerning the quadratic analogue of Serre’s Conjecture. In this investigation, one considers f.g. projective modules P (say over R =k[x₁ ..., x _d ], k a field), equipped with, respectively, the following three types of structures, S: (1) quadratic forms, (2) symmetric bilinear forms, or (3) symplectic forms. In the spirit of Serre’s Problem on projective modules over the polynomial ring R, the main question to be asked in this chapter is the following:

Is the object (P, S) over k[x₁, ..., x _d ] necessarily extended, in a suitable sense, from an object of a similar kind over k? For convenience of the exposition, we shall restrict our attention to cases (2) and (3), and skip the more difficult case of quadratic forms. All rings considered in this chapter will be assumed to be commutative (with an identity).