Thursday, May 18, 2023 at 10:00 a.m. (sharp)
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PhD Candidate: Hyungseop Kim
Supervisor: Michael Groechenig
Thesis title: Descent techniques in algebraic K-theory
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We investigate two different approaches to describing algebraic K-theory of schemes through descent techniques, one of global nature and the other of local nature.
The first half of the thesis is devoted to the study of an adelic descent statement for algebraic K-theory of Noetherian schemes, or more generally for any localizing invariants in place of algebraic K-theory. Given a Noetherian scheme $X$ of finite Krull dimension, Beilinson’s cosemisimplicial ring $A^{\bullet}_{\mathrm{red}}(X)$ of reduced adeles on $X$ provides a resolution of the structure sheaf of $X$. We prove that for any localizing invariant $E$ of small stable $\infty$-categories, e.g., nonconnective algebraic K-theory of Bass-Thomason, there is a natural equivalence $E(X)\simeq\lim_{\Delta_{s}}E(A^{\bullet}_{\mathrm{red}}(X))$. This can be viewed as a variant of the formal glueing problem for algebraic K-theory which concerns all irreducible closed subsets at once. We prove the descent statement by first converting the question to a cubical descent statement, and then constructing exact sequences of perfect module categories over adele rings.
In the second half of the thesis, we turn our attention to the study of $p$-adic K-theory of characteristic $p$ rings. Specifically, we provide an alternative proof of Kelly-Morrow’s generalization of Geisser-Levine theorem to the Cartier smooth case. Our approach puts emphasis on utilizing motivic filtration and descent spectral sequence. Using the homological smoothness of Cartier smooth rings, we first compute their prismatic cohomology and syntomic cohomology complexes. Through motivic filtration, this computation gives a description of topological cyclic homology for Cartier smooth rings. Then, we use the pro-\’etale descent spectral sequence for topological cyclic homology and rigidity properties of the cyclotomic trace and syntomic cohomology complexes to deduce the result, computing algebraic K-theory of local Cartier smooth rings in terms of their logarithmic de Rham-Witt groups. We also collect some direct consequences of our arguments to prismatic cohomology complexes of Cartier smooth rings and their $p$-torsion free liftings to mixed characteristic.
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The draft of the thesis can be found here: main
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