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# K-theory of the integers and the Kummer-Vandiver conjecture

## January 31 at 11:15 – 12:45 CET

Christian Dahlhausen: The spectrum level rank filration

This talk covers the results [Rog00, §1–§4] whose proofs rely on [Rog92, §1–9]. First recall some relevant notions from topology which are needed. Introduce the rank filtration F• K(R) on the algebraic K-theory spectrum K(R) of a ring R and relate its subquotients ̄F• K(R) with the relative smash product of the universal bundle EGL•(R) with the stable building D(R•) over GL•(R) R [Prop. 2.2]. In order to understand better the second component D(R•) introduce the poset filtration on it [Def. 3.1] and then relate the subquotients of this filtration with the relative smash product of GL•(R)/Pω with the subquotients on the poset filtration on stable apartments [Prop. 4.2] which will be analysed in the subsequent talk.