$K$-Theory of Truncated Polynomials

Noah Riggenbach

$K$-Theory of Truncated Polynomials

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arxiv 2211.11110 v2 pith:CZZTZPNO submitted 2022-11-20 math.KT math.ATmath.NT

Noah Riggenbach This is my paper · ORCID

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keywords termscohomologyperfectoidprismaticringringstheoryalgebraic

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abstract

We study the algebraic $K$-theory of rings of the form $R[x]/x^e$. We do this via trace methods and filtrations on topological Hochschild homology and related theories by quasisyntomic sheaves. We produce computations for $R$ a perfectoid ring in terms of the big Witt vectors of $R$, for $R$ a smooth curve over a perfectoid ring in terms of the prismatic cohomology of $R$, and for $R$ a complete mixed characteristic discrete valuation rings with perfect residue field in terms of the prismatic cohomology and Hodge-Tate divisor of $R$.

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Cyclotomic synthetic spectra
math.KT 2024-11 conditional novelty 8.0 of 10

The motivic filtration on THH(R;Z_p) is shown to be a p-typical cyclotomic synthetic spectrum, with applications to TC and syntomic cohomology bounds.

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