Algebraic $K$-Theory and Trace Methods

Algebraic $K$-theory captures deep arithmetic and geometric information about rings, but direct computation of $K_n(A)$ for $n \geq 2$ is notoriously difficult. The cyclotomic trace $\text{trc}: K(A) \to \text{TC}(A)$ to topological cyclic homology provides a powerful computational tool. Following the Nikolaus-Scholze framework, we define $\text{TC}(A)$ via the equalizer of Frobenius and canonical maps on $\text{THH}(A)^{tC_p}$, and demonstrate how this leads to explicit calculations of $K$-groups for $p$-adic integers and truncated polynomial algebras.