Excision and algebraic K-theory

I will report on joint work with Tamme. I will recall that algebraic K-theory does not satisfy excision and describe an effective way of studying the failure for excision in any localizing invariant. As applications we find a short proof of Suslin's excision theorem, that what we call truncating invariants satisfy excision, and new ways of calculating algebraic K-groups of certain rings. Examples of truncating invariants include periodic cyclic homology over the rationals and the fibre of the cyclotomic trace. If time permits I will also indicate joint work with Tamme and Meier, where we study chromatic localizations of algebraic K-theory. As applications of our approach we show that K(1)-localized algebraic K-theory satisfies excision (first proven by Bhatt--Clausen--Mathew) and that the K(n)-localized K-theory of any bounded above connective ring spectrum vanishes for n at least 2.

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