Dualization in algebraic K-theory and the invariant e¹ of quadratic forms over schemes

Marek Szyjewski. "Dualization in algebraic K-theory and the invariant e¹ of quadratic forms over schemes." Fundamenta Mathematicae 215.3 (2011): 233-299. <http://eudml.org/doc/283342>.

@article{MarekSzyjewski2011,
abstract = { 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. },
author = {Marek Szyjewski},
journal = {Fundamenta Mathematicae},
keywords = {Witt group; scheme; -theory of an exact category; dualization; exact category with duality; invariant of quadratic forms; discriminant; bisimplicial space; projective variety; conic},
language = {eng},
number = {3},
pages = {233-299},
title = {Dualization in algebraic K-theory and the invariant e¹ of quadratic forms over schemes},
url = {http://eudml.org/doc/283342},
volume = {215},
year = {2011},
}

TY - JOUR
AU - Marek Szyjewski
TI - Dualization in algebraic K-theory and the invariant e¹ of quadratic forms over schemes
JO - Fundamenta Mathematicae
PY - 2011
VL - 215
IS - 3
SP - 233
EP - 299
AB - 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.
LA - eng
KW - Witt group; scheme; -theory of an exact category; dualization; exact category with duality; invariant of quadratic forms; discriminant; bisimplicial space; projective variety; conic
UR - http://eudml.org/doc/283342
ER -