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

Tom 215 / 2011

Marek Szyjewski Fundamenta Mathematicae 215 (2011), 233-299 MSC: Primary 11E081. DOI: 10.4064/fm215-3-3


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.


  • Marek SzyjewskiMieszka I 15/97
    40-877 Katowice, Poland

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