# Descent in algebraic K-theory

Wintersemester 2017/18

Time and place: Mondays 14-16, SFB seminar room

Wintersemester 2017/18

Time and place: Mondays 14-16, SFB seminar room

Course description

We will study the algebraic K-theory of derived schemes. Our focus will be on questions of descent, for which purpose we will revisit the celebrated paper of Thomason-Trobaugh [TT] and see how it can be extended to the world of derived algebraic geometry. More specifically, we will discuss two types of descent problems: first, descent for the Zariski and Nisnevich topologies; and secondly, descent by derived blow-ups (which has recently been established by Kerz-Strunk-Tamme [KST]). As far as time permits, we will also see some applications to the K-theory of classical schemes, including a pro-cdh-descent result and the resolution of Weibel's conjecture on negative K-theory (both also obtained in [KST]).

We will assume familiarity with the language of ∞-categories or homotopical algebra, but not necessarily with derived algebraic geometry or algebraic K-theory.

Lecture notes

Lecture 0: Overview (pdf)

Literature

[KST] Moritz Kerz, Florian Strunk, Georg Tamme, *Algebraic K-theory and descent for blow-ups*, arXiv:1611.08466

[L] Jacob Lurie, *Spectral Algebraic Geometry*, pdf

[TT] R.W. Thomason and T. Trobaugh, *Higher algebraic K-theory of schemes and of derived categories*

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