Milnor K-theory of local rings [Elektronische Ressource] / vorgelegt von Moritz Kerz

MilnorK-theory of local ringsDissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) dermathematischen Fakultät der Universität Regensburgvorgelegt vonMoritz Kerz aus Frankfurt am Main20082Promotionsgesuch eingereicht am: 7. April 2008Die Arbeit wurde angeleitet von: Prof. Dr. Uwe JannsenPrüfungsausschuss:Prof. Dr. Felix Finster (Vorsitzender)Prof. Dr. Uwe Jannsen (Erstgutachter)Prof. Dr. Stefan Müller-Stach (Zweitgutachter)Prof. Dr. Alexander Schmidt3Wann, wenn nicht jetzt, sollen wir den Stein schleuderngegen Goliaths Stirn?Primo LeviSummaryThis thesis examines MilnorK-theory of local rings. We will prove the Beilinson-Lichtenbaumconjecture relating MilnorK-groups of equicharacteristic regular local rings with infinite residuefields to motivic cohomology groups, the Gersten conjecture for Milnor K-theory and in thefinite residue field case we will show that (n,n)-motivic cohomology of an equicharacteristicregular local ring is generated by elements of degree 1.MilnorK-theory of fields originated in Milnor’s seminal Inventiones article from 1970 [28].There he defined Milnor K-groups and proposed his famous conjectures, now known as theMilnor conjectures, which on the one hand relate Milnor K-theory to quadratic forms and onthe other hand to Galois cohomology. Following Milnor’s ideas the theory of MilnorK-groupsof fields developed swiftly.