Inventiones mathematicae

Inventiones math. 9, 318-344 (1970)

Algebraic X-Theory and Quadratic Forms

John Milnor (Cambridge, Massachusetts)

The first section of this paper defines and studies a graded ring K^F associated to any field F. By definition, K„F is the target group of the universal n-linear function from F* x • • • x F* to an additive group, ing the condition that a^ x • • x a„ should map to zero whenever ^i + ^i+i = l- Here F* denotes the multiplicative group F—0^

Section2 constructs a homomorphism d: K„F-^K„^^F associated with a discrete valuation on F with residue class field F. These morphisms д are used to compute the ring K^ F{t) of a rational function field, using a technique due to John Tate.

Section 3 relates K^F to the theory of quadratic modules by defining certain "Stiefel-Whitney invariants" ofa quadratic module over a field F of characteristic Ф2. The definition is closely related to Delzant [5].

Let W be the Witt ring of anisotropic quadratic modules over F, and let I a Wbe the maximal ideal, consisting of modules of even rank. Section 4 studies the conjecture that the associated graded ring

{ W / I , I / P , P / I\ . . . )

is canonically isomorphic to K^F/IK^F. Section 5 computes the Witt ring of a field F{t) of rational functions.

Section 6 describes the conjecture that K^F/IK^F is canonically isomorphic to the cohomology ring H*{Gf;Z/2Z); where Gp denotes the Galois group of the separable closure of F. An appendix, due to Tate, computes K^F/IK^F for a global field.

Throughout the exposition I have made free use of unpublished theorems and ideas due to Bass and Tate. I want particularly to thank Tate for his generous help,

§ LThe Ring K^ F

To any field F we associate a graded ring

K^FHK , F , K , F , K2F , . . . )

as follows. By definition, X^F is just the multiplicative group F* written additively. To keep notation straight, we introduce the canonical morphism , _. ^ ^ _