Dualization in algebraic $K$-theory and the invariant $e^1$ of quadratic forms over schemes

Marek Szyjewski¹

¹ Mieszka I 15/97 40-877 Katowice, Poland

Fundamenta Mathematicae, Tome 215 (2011) no. 3, pp. 233-299

Résumé

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 $e^0 : W(F)\rightarrow\mathbb{Z}/2$, and the discriminant $e^1$ with values in $k_1 (F)= F^{\ast }/F^{\ast 2}$, which behaves well on the fundamental ideal $I(F)={\rm ker}(e^0)$. 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: $\bullet$ to establish a theory of the invariant $e^1$ in this generality;$\bullet$ to provide computations involving this invariant and show its usefulness. We define a relative version of $e^1$ for pairs of quadratic forms with the same value of~$e^0$. This is first done in terms of loops in some bisimplicial set whose fundamental group is the $K_1$ 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 $e^1$ 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 $e^1$ on the pairs of metabolic forms and define the codomain for $e^1$ by factoring out this subgroup from some obvious subquotient of $K_1$. This proves to be a correct definition of the small $k_1$ for categories; it agrees with Milnor's usual $k_1$ 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.

Détail

DOI : 10.4064/fm215-3-3

Keywords: classical witt theory field study quadratic forms begins simple invariants dimension form modulo called dimension index denoted rightarrow mathbb discriminant values ast ast which behaves fundamental ideal ker here sophisticated situation considered quadratic forms scheme generally exact category duality purposes bullet establish theory invariant generality bullet provide computations involving invariant its usefulness define relative version pairs quadratic forms value first done terms loops bisimplicial set whose fundamental group underlying exact category translated language term double exact sequences which allows carry out actual computations unexpected difficulty value relative vanish even forms metabolic make invariant defined witt classes study subgroup generated values pairs metabolic forms define codomain factoring out subgroup obvious subquotient proves correct definition small categories agrees milnors usual fields provide applications invariant computing pairs forms projective line forms conics