In the classical Witt theory over a field F, the study of quadratic forms begins with two simple invariants: the dimension of a form modulo 2, called the dimension index and denoted e⁰: W(F) → ℤ/2, and the discriminant e¹ with values in k₁(F) = F*/F*², which behaves well on the fundamental ideal I(F)= ker(e⁰). Here a more sophisticated situation is considered, of quadratic forms over a scheme and, more generally, over an exact category with duality. Our purposes are: ∙ to establish a theory of the invariant e¹ in this generality; ∙ to provide computations involving this invariant and show its usefulness. We define a relative version of e¹ for pairs of quadratic forms with the same value of e⁰. This is first done in terms of loops in some bisimplicial set whose fundamental group is the K₁ of the underlying exact category, and next translated into the language of 4-term double exact sequences, which allows us to carry out actual computations. An unexpected difficulty is that the value of the relative e¹ need not vanish even if both forms are metabolic. To make the invariant well defined on the Witt classes, we study the subgroup H generated by the values of e¹ on the pairs of metabolic forms and define the codomain for e¹ by factoring out this subgroup from some obvious subquotient of K₁. This proves to be a correct definition of the small k₁ for categories; it agrees with Milnor's usual k₁ in the case of fields. Next we provide applications of this new invariant by computing it for some pairs of forms over the projective line and for some forms over conics.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-3-3,
author = {Marek Szyjewski},
title = {Dualization in algebraic K-theory and the invariant e$^1$ of quadratic forms over schemes},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {233-299},
zbl = {1270.11040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-3-3}
}
Marek Szyjewski. Dualization in algebraic K-theory and the invariant e¹ of quadratic forms over schemes. Fundamenta Mathematicae, Tome 215 (2011) pp. 233-299. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-3-3/