Annihilators and kernels in Kato's cohomology in positive characteristic and in Witt groups in characteristic 2

• Applicant: Professor Dr. Detlev Hoffmann
• Subject Area: Mathematics
• Term: from 2013 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 248466702

Groups, rings and fields are fundamental objects in algebra. Groups are objects consisting of elements that can be combined using a group operation to form new elements according to strict rules. The rotations of a cube in space or the integers together with the usual addition form a group. Rings consist of elements that form a group under an addition, but that can also be multiplied with each other, with the interaction of addition and multiplication being governed by certain rules. E.g., the integers with usual addition and multiplication form a ring. Fields are rings in which multiplication has particularly nice properties, e.g. there always exists the reciprocal of a nonzero element. Examples of fields are the rational and the real numbers. Algebraists are often interested in objects defined over rings or fields, such as equations given by a polynomial in one or more variables with coefficients in a ring or field. Such equations may be transformed according to certain rules, and a natural question is to decide when an equation can be transformed into another, i.e. when two equations are "equivalent". To do so, one tries to find certain invariants that equivalent equations share and these invariants can in turn be elements in a group or ring. Quadratic forms can be considered as polynomial equations of degree 2 in several variables and they have been studied extensively for centuries. An important result in their classification is the proof of the co-called Milnor conjectures by Voevodsky (Fields medal in 2002 for this and related work), where quadratic forms are related to so-called Galois cohomology groups and Milnor K-groups in the case of fields where 2 is different from 0. Analogous results for fields with 2 equal to 0 have been obtained by Kazuya Kato in the 1980s where he used what we call the Kato cohomology groups. These are important in the study of fields in which a prime number p equals 0 and they have many applications, e.g. in number theory (class field theory). To gain a better understanding of these important groups, we plan to study two questions: 1. Given an element in Kato cohomology, which elements in Kato cohomology become zero when multiplied by that given element, i.e. we want to determine the "annihilator" of that given element. 2. Which elements in Kato cohomology become zero when passing from a field to a bigger field, i.e. we want to determine the "kernel of the restriction map" for that field extension. We also want to study analogous questions for the Witt groups of fields in which 2 equals 0. These Witt groups essentially classify quadratic (resp. bilinear) forms over such fields and by Kato's results, the questions for Witt groups are intimately linked to those for Kato's cohomology.

• DFG Programme: Research Grants
• International Connection: Chile
• Partner Organisation: Comisión Nacional de Investigación Cientifica; y Tecnológica - CONICYT
• Participating Persons: Professor Dr. Roberto Aravire; Professor Dr. Ricardo Baeza