Algebraic $K$-theory of planar cuspidal curves

Abstract: In this paper, we evaluate the algebraic $K$-groups of a planar cuspidal curve over a perfect $\mathbb{F}_p$-algebra relative to the cusp point. A conditional calculation of these groups was given earlier by Hesselholt, assuming a conjecture on the structure of certain polytopes. Our calculation here, however, is unconditional and illustrates the advantage of the new setup for topological cyclic homology by Nikolaus-Scholze, which is used throughout. The only input necessary for our calculation is the evaluation by the Buenos Aires Cyclic Homology group and by Larsen of the structure of Hochschild complex of the coordinate ring as a mixed complex, that is, as an object of the infinity category of chain complexes with circle action.