For a prime $p\ge 5$, we compute the algebraic $K$-theory modulo $p$ and ${v}_{1}$ of the mod $p$ Adams summand, using topological cyclic homology. On the way, we evaluate its modulo $p$ and ${v}_{1}$ topological Hochschild homology. Using a localization sequence, we also compute the $K$-theory modulo $p$ and ${v}_{1}$ of the first Morava $K$-theory.

This paper gives an exposition of algebraic K-theory, which studies functors ${K}_{n}:\text{Rings}\to \text{Abelian}\phantom{\rule{4.0pt}{0ex}}\text{Groups}$, $n$ an integer. Classically $n=0,1$ introduced by Bass in the mid 60’s (based on ideas of Grothendieck and others) and $n=2$ introduced by Milnor [Introduction to algebraic K-theory, Annals of Math. Studies, 72, Princeton University Press, 1971: Zbl 0237.18005]. These functors are defined and applications to topological K-theory (Swan), number theory, topology and geometry (the Wall finiteness obstruction to a CW-complex being finite,...

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