$K$-Theory of Truncated Polynomials

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 $R$.

arXiv comment39 pages, fixed several typos, moved some of the more technical discussion in the introduction into a new section, rewrote and added several parts of [v1] to improve readability following a referee report, and added the ramified case of Theorem 1.13(now Theorem 1.6)